| /* |
| * Copyright (c) 2008-2012 Stefan Krah. All rights reserved. |
| * |
| * Redistribution and use in source and binary forms, with or without |
| * modification, are permitted provided that the following conditions |
| * are met: |
| * |
| * 1. Redistributions of source code must retain the above copyright |
| * notice, this list of conditions and the following disclaimer. |
| * |
| * 2. Redistributions in binary form must reproduce the above copyright |
| * notice, this list of conditions and the following disclaimer in the |
| * documentation and/or other materials provided with the distribution. |
| * |
| * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS "AS IS" AND |
| * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE |
| * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE |
| * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE |
| * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL |
| * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS |
| * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) |
| * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT |
| * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY |
| * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF |
| * SUCH DAMAGE. |
| */ |
| |
| |
| #include "mpdecimal.h" |
| #include <stdio.h> |
| #include <stdlib.h> |
| #include <string.h> |
| #include <limits.h> |
| #include <math.h> |
| #include "basearith.h" |
| #include "bits.h" |
| #include "convolute.h" |
| #include "crt.h" |
| #include "memory.h" |
| #include "typearith.h" |
| #include "umodarith.h" |
| |
| #ifdef PPRO |
| #if defined(_MSC_VER) |
| #include <float.h> |
| #pragma fenv_access(on) |
| #elif !defined(__OpenBSD__) && !defined(__NetBSD__) |
| /* C99 */ |
| #include <fenv.h> |
| #pragma STDC FENV_ACCESS ON |
| #endif |
| #endif |
| |
| |
| #if defined(_MSC_VER) |
| #define ALWAYS_INLINE __forceinline |
| #elif defined(LEGACY_COMPILER) |
| #define ALWAYS_INLINE |
| #undef inline |
| #define inline |
| #else |
| #ifdef TEST_COVERAGE |
| #define ALWAYS_INLINE |
| #else |
| #define ALWAYS_INLINE inline __attribute__ ((always_inline)) |
| #endif |
| #endif |
| |
| |
| #define MPD_NEWTONDIV_CUTOFF 1024L |
| |
| #define MPD_NEW_STATIC(name, flags, exp, digits, len) \ |
| mpd_uint_t name##_data[MPD_MINALLOC_MAX]; \ |
| mpd_t name = {flags|MPD_STATIC|MPD_STATIC_DATA, exp, digits, \ |
| len, MPD_MINALLOC_MAX, name##_data} |
| |
| #define MPD_NEW_CONST(name, flags, exp, digits, len, alloc, initval) \ |
| mpd_uint_t name##_data[alloc] = {initval}; \ |
| mpd_t name = {flags|MPD_STATIC|MPD_CONST_DATA, exp, digits, \ |
| len, alloc, name##_data} |
| |
| #define MPD_NEW_SHARED(name, a) \ |
| mpd_t name = {(a->flags&~MPD_DATAFLAGS)|MPD_STATIC|MPD_SHARED_DATA, \ |
| a->exp, a->digits, a->len, a->alloc, a->data} |
| |
| |
| static mpd_uint_t data_one[1] = {1}; |
| static mpd_uint_t data_zero[1] = {0}; |
| static const mpd_t one = {MPD_STATIC|MPD_CONST_DATA, 0, 1, 1, 1, data_one}; |
| static const mpd_t minus_one = {MPD_NEG|MPD_STATIC|MPD_CONST_DATA, 0, 1, 1, 1, |
| data_one}; |
| static const mpd_t zero = {MPD_STATIC|MPD_CONST_DATA, 0, 1, 1, 1, data_zero}; |
| |
| static inline void _mpd_check_exp(mpd_t *dec, const mpd_context_t *ctx, |
| uint32_t *status); |
| static void _settriple(mpd_t *result, uint8_t sign, mpd_uint_t a, |
| mpd_ssize_t exp); |
| static inline mpd_ssize_t _mpd_real_size(mpd_uint_t *data, mpd_ssize_t size); |
| |
| static void _mpd_qadd(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status); |
| static inline void _mpd_qmul(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status); |
| static void _mpd_base_ndivmod(mpd_t *q, mpd_t *r, const mpd_t *a, |
| const mpd_t *b, uint32_t *status); |
| static inline void _mpd_qpow_uint(mpd_t *result, const mpd_t *base, |
| mpd_uint_t exp, uint8_t resultsign, |
| const mpd_context_t *ctx, uint32_t *status); |
| |
| static mpd_uint_t mpd_qsshiftr(mpd_t *result, const mpd_t *a, mpd_ssize_t n); |
| |
| |
| /******************************************************************************/ |
| /* Performance critical inline functions */ |
| /******************************************************************************/ |
| |
| #ifdef CONFIG_64 |
| /* Digits in a word, primarily useful for the most significant word. */ |
| ALWAYS_INLINE int |
| mpd_word_digits(mpd_uint_t word) |
| { |
| if (word < mpd_pow10[9]) { |
| if (word < mpd_pow10[4]) { |
| if (word < mpd_pow10[2]) { |
| return (word < mpd_pow10[1]) ? 1 : 2; |
| } |
| return (word < mpd_pow10[3]) ? 3 : 4; |
| } |
| if (word < mpd_pow10[6]) { |
| return (word < mpd_pow10[5]) ? 5 : 6; |
| } |
| if (word < mpd_pow10[8]) { |
| return (word < mpd_pow10[7]) ? 7 : 8; |
| } |
| return 9; |
| } |
| if (word < mpd_pow10[14]) { |
| if (word < mpd_pow10[11]) { |
| return (word < mpd_pow10[10]) ? 10 : 11; |
| } |
| if (word < mpd_pow10[13]) { |
| return (word < mpd_pow10[12]) ? 12 : 13; |
| } |
| return 14; |
| } |
| if (word < mpd_pow10[18]) { |
| if (word < mpd_pow10[16]) { |
| return (word < mpd_pow10[15]) ? 15 : 16; |
| } |
| return (word < mpd_pow10[17]) ? 17 : 18; |
| } |
| |
| return (word < mpd_pow10[19]) ? 19 : 20; |
| } |
| #else |
| ALWAYS_INLINE int |
| mpd_word_digits(mpd_uint_t word) |
| { |
| if (word < mpd_pow10[4]) { |
| if (word < mpd_pow10[2]) { |
| return (word < mpd_pow10[1]) ? 1 : 2; |
| } |
| return (word < mpd_pow10[3]) ? 3 : 4; |
| } |
| if (word < mpd_pow10[6]) { |
| return (word < mpd_pow10[5]) ? 5 : 6; |
| } |
| if (word < mpd_pow10[8]) { |
| return (word < mpd_pow10[7]) ? 7 : 8; |
| } |
| |
| return (word < mpd_pow10[9]) ? 9 : 10; |
| } |
| #endif |
| |
| |
| /* Adjusted exponent */ |
| ALWAYS_INLINE mpd_ssize_t |
| mpd_adjexp(const mpd_t *dec) |
| { |
| return (dec->exp + dec->digits) - 1; |
| } |
| |
| /* Etiny */ |
| ALWAYS_INLINE mpd_ssize_t |
| mpd_etiny(const mpd_context_t *ctx) |
| { |
| return ctx->emin - (ctx->prec - 1); |
| } |
| |
| /* Etop: used for folding down in IEEE clamping */ |
| ALWAYS_INLINE mpd_ssize_t |
| mpd_etop(const mpd_context_t *ctx) |
| { |
| return ctx->emax - (ctx->prec - 1); |
| } |
| |
| /* Most significant word */ |
| ALWAYS_INLINE mpd_uint_t |
| mpd_msword(const mpd_t *dec) |
| { |
| assert(dec->len > 0); |
| return dec->data[dec->len-1]; |
| } |
| |
| /* Most significant digit of a word */ |
| inline mpd_uint_t |
| mpd_msd(mpd_uint_t word) |
| { |
| int n; |
| |
| n = mpd_word_digits(word); |
| return word / mpd_pow10[n-1]; |
| } |
| |
| /* Least significant digit of a word */ |
| ALWAYS_INLINE mpd_uint_t |
| mpd_lsd(mpd_uint_t word) |
| { |
| return word % 10; |
| } |
| |
| /* Coefficient size needed to store 'digits' */ |
| ALWAYS_INLINE mpd_ssize_t |
| mpd_digits_to_size(mpd_ssize_t digits) |
| { |
| mpd_ssize_t q, r; |
| |
| _mpd_idiv_word(&q, &r, digits, MPD_RDIGITS); |
| return (r == 0) ? q : q+1; |
| } |
| |
| /* Number of digits in the exponent. Not defined for MPD_SSIZE_MIN. */ |
| inline int |
| mpd_exp_digits(mpd_ssize_t exp) |
| { |
| exp = (exp < 0) ? -exp : exp; |
| return mpd_word_digits(exp); |
| } |
| |
| /* Canonical */ |
| ALWAYS_INLINE int |
| mpd_iscanonical(const mpd_t *dec UNUSED) |
| { |
| return 1; |
| } |
| |
| /* Finite */ |
| ALWAYS_INLINE int |
| mpd_isfinite(const mpd_t *dec) |
| { |
| return !(dec->flags & MPD_SPECIAL); |
| } |
| |
| /* Infinite */ |
| ALWAYS_INLINE int |
| mpd_isinfinite(const mpd_t *dec) |
| { |
| return dec->flags & MPD_INF; |
| } |
| |
| /* NaN */ |
| ALWAYS_INLINE int |
| mpd_isnan(const mpd_t *dec) |
| { |
| return dec->flags & (MPD_NAN|MPD_SNAN); |
| } |
| |
| /* Negative */ |
| ALWAYS_INLINE int |
| mpd_isnegative(const mpd_t *dec) |
| { |
| return dec->flags & MPD_NEG; |
| } |
| |
| /* Positive */ |
| ALWAYS_INLINE int |
| mpd_ispositive(const mpd_t *dec) |
| { |
| return !(dec->flags & MPD_NEG); |
| } |
| |
| /* qNaN */ |
| ALWAYS_INLINE int |
| mpd_isqnan(const mpd_t *dec) |
| { |
| return dec->flags & MPD_NAN; |
| } |
| |
| /* Signed */ |
| ALWAYS_INLINE int |
| mpd_issigned(const mpd_t *dec) |
| { |
| return dec->flags & MPD_NEG; |
| } |
| |
| /* sNaN */ |
| ALWAYS_INLINE int |
| mpd_issnan(const mpd_t *dec) |
| { |
| return dec->flags & MPD_SNAN; |
| } |
| |
| /* Special */ |
| ALWAYS_INLINE int |
| mpd_isspecial(const mpd_t *dec) |
| { |
| return dec->flags & MPD_SPECIAL; |
| } |
| |
| /* Zero */ |
| ALWAYS_INLINE int |
| mpd_iszero(const mpd_t *dec) |
| { |
| return !mpd_isspecial(dec) && mpd_msword(dec) == 0; |
| } |
| |
| /* Test for zero when specials have been ruled out already */ |
| ALWAYS_INLINE int |
| mpd_iszerocoeff(const mpd_t *dec) |
| { |
| return mpd_msword(dec) == 0; |
| } |
| |
| /* Normal */ |
| inline int |
| mpd_isnormal(const mpd_t *dec, const mpd_context_t *ctx) |
| { |
| if (mpd_isspecial(dec)) return 0; |
| if (mpd_iszerocoeff(dec)) return 0; |
| |
| return mpd_adjexp(dec) >= ctx->emin; |
| } |
| |
| /* Subnormal */ |
| inline int |
| mpd_issubnormal(const mpd_t *dec, const mpd_context_t *ctx) |
| { |
| if (mpd_isspecial(dec)) return 0; |
| if (mpd_iszerocoeff(dec)) return 0; |
| |
| return mpd_adjexp(dec) < ctx->emin; |
| } |
| |
| /* Odd word */ |
| ALWAYS_INLINE int |
| mpd_isoddword(mpd_uint_t word) |
| { |
| return word & 1; |
| } |
| |
| /* Odd coefficient */ |
| ALWAYS_INLINE int |
| mpd_isoddcoeff(const mpd_t *dec) |
| { |
| return mpd_isoddword(dec->data[0]); |
| } |
| |
| /* 0 if dec is positive, 1 if dec is negative */ |
| ALWAYS_INLINE uint8_t |
| mpd_sign(const mpd_t *dec) |
| { |
| return dec->flags & MPD_NEG; |
| } |
| |
| /* 1 if dec is positive, -1 if dec is negative */ |
| ALWAYS_INLINE int |
| mpd_arith_sign(const mpd_t *dec) |
| { |
| return 1 - 2 * mpd_isnegative(dec); |
| } |
| |
| /* Radix */ |
| ALWAYS_INLINE long |
| mpd_radix(void) |
| { |
| return 10; |
| } |
| |
| /* Dynamic decimal */ |
| ALWAYS_INLINE int |
| mpd_isdynamic(mpd_t *dec) |
| { |
| return !(dec->flags & MPD_STATIC); |
| } |
| |
| /* Static decimal */ |
| ALWAYS_INLINE int |
| mpd_isstatic(mpd_t *dec) |
| { |
| return dec->flags & MPD_STATIC; |
| } |
| |
| /* Data of decimal is dynamic */ |
| ALWAYS_INLINE int |
| mpd_isdynamic_data(mpd_t *dec) |
| { |
| return !(dec->flags & MPD_DATAFLAGS); |
| } |
| |
| /* Data of decimal is static */ |
| ALWAYS_INLINE int |
| mpd_isstatic_data(mpd_t *dec) |
| { |
| return dec->flags & MPD_STATIC_DATA; |
| } |
| |
| /* Data of decimal is shared */ |
| ALWAYS_INLINE int |
| mpd_isshared_data(mpd_t *dec) |
| { |
| return dec->flags & MPD_SHARED_DATA; |
| } |
| |
| /* Data of decimal is const */ |
| ALWAYS_INLINE int |
| mpd_isconst_data(mpd_t *dec) |
| { |
| return dec->flags & MPD_CONST_DATA; |
| } |
| |
| |
| /******************************************************************************/ |
| /* Inline memory handling */ |
| /******************************************************************************/ |
| |
| /* Fill destination with zeros */ |
| ALWAYS_INLINE void |
| mpd_uint_zero(mpd_uint_t *dest, mpd_size_t len) |
| { |
| mpd_size_t i; |
| |
| for (i = 0; i < len; i++) { |
| dest[i] = 0; |
| } |
| } |
| |
| /* Free a decimal */ |
| ALWAYS_INLINE void |
| mpd_del(mpd_t *dec) |
| { |
| if (mpd_isdynamic_data(dec)) { |
| mpd_free(dec->data); |
| } |
| if (mpd_isdynamic(dec)) { |
| mpd_free(dec); |
| } |
| } |
| |
| /* |
| * Resize the coefficient. Existing data up to 'nwords' is left untouched. |
| * Return 1 on success, 0 otherwise. |
| * |
| * Input invariant: MPD_MINALLOC <= result->alloc. |
| * |
| * Case nwords == result->alloc: |
| * 'result' is unchanged. Return 1. |
| * |
| * Case nwords > result->alloc: |
| * Case realloc success: |
| * The value of 'result' does not change. Return 1. |
| * Case realloc failure: |
| * 'result' is NaN, status is updated with MPD_Malloc_error. Return 0. |
| * |
| * Case nwords < result->alloc: |
| * Case is_static_data or realloc failure [1]: |
| * 'result' is unchanged. Return 1. |
| * Case realloc success: |
| * The value of result is undefined (expected). Return 1. |
| * |
| * |
| * [1] In that case the old (now oversized) area is still valid. |
| */ |
| ALWAYS_INLINE int |
| mpd_qresize(mpd_t *result, mpd_ssize_t nwords, uint32_t *status) |
| { |
| assert(!mpd_isconst_data(result)); /* illegal operation for a const */ |
| assert(!mpd_isshared_data(result)); /* illegal operation for a shared */ |
| assert(MPD_MINALLOC <= result->alloc); |
| |
| nwords = (nwords <= MPD_MINALLOC) ? MPD_MINALLOC : nwords; |
| if (nwords == result->alloc) { |
| return 1; |
| } |
| if (mpd_isstatic_data(result)) { |
| if (nwords > result->alloc) { |
| return mpd_switch_to_dyn(result, nwords, status); |
| } |
| return 1; |
| } |
| |
| return mpd_realloc_dyn(result, nwords, status); |
| } |
| |
| /* Same as mpd_qresize, but the complete coefficient (including the old |
| * memory area!) is initialized to zero. */ |
| ALWAYS_INLINE int |
| mpd_qresize_zero(mpd_t *result, mpd_ssize_t nwords, uint32_t *status) |
| { |
| assert(!mpd_isconst_data(result)); /* illegal operation for a const */ |
| assert(!mpd_isshared_data(result)); /* illegal operation for a shared */ |
| assert(MPD_MINALLOC <= result->alloc); |
| |
| nwords = (nwords <= MPD_MINALLOC) ? MPD_MINALLOC : nwords; |
| if (nwords != result->alloc) { |
| if (mpd_isstatic_data(result)) { |
| if (nwords > result->alloc) { |
| return mpd_switch_to_dyn_zero(result, nwords, status); |
| } |
| } |
| else if (!mpd_realloc_dyn(result, nwords, status)) { |
| return 0; |
| } |
| } |
| |
| mpd_uint_zero(result->data, nwords); |
| return 1; |
| } |
| |
| /* |
| * Reduce memory size for the coefficient to MPD_MINALLOC. In theory, |
| * realloc may fail even when reducing the memory size. But in that case |
| * the old memory area is always big enough, so checking for MPD_Malloc_error |
| * is not imperative. |
| */ |
| ALWAYS_INLINE void |
| mpd_minalloc(mpd_t *result) |
| { |
| assert(!mpd_isconst_data(result)); /* illegal operation for a const */ |
| assert(!mpd_isshared_data(result)); /* illegal operation for a shared */ |
| |
| if (!mpd_isstatic_data(result) && result->alloc > MPD_MINALLOC) { |
| uint8_t err = 0; |
| result->data = mpd_realloc(result->data, MPD_MINALLOC, |
| sizeof *result->data, &err); |
| if (!err) { |
| result->alloc = MPD_MINALLOC; |
| } |
| } |
| } |
| |
| int |
| mpd_resize(mpd_t *result, mpd_ssize_t nwords, mpd_context_t *ctx) |
| { |
| uint32_t status = 0; |
| if (!mpd_qresize(result, nwords, &status)) { |
| mpd_addstatus_raise(ctx, status); |
| return 0; |
| } |
| return 1; |
| } |
| |
| int |
| mpd_resize_zero(mpd_t *result, mpd_ssize_t nwords, mpd_context_t *ctx) |
| { |
| uint32_t status = 0; |
| if (!mpd_qresize_zero(result, nwords, &status)) { |
| mpd_addstatus_raise(ctx, status); |
| return 0; |
| } |
| return 1; |
| } |
| |
| |
| /******************************************************************************/ |
| /* Set attributes of a decimal */ |
| /******************************************************************************/ |
| |
| /* Set digits. Assumption: result->len is initialized and > 0. */ |
| inline void |
| mpd_setdigits(mpd_t *result) |
| { |
| mpd_ssize_t wdigits = mpd_word_digits(mpd_msword(result)); |
| result->digits = wdigits + (result->len-1) * MPD_RDIGITS; |
| } |
| |
| /* Set sign */ |
| ALWAYS_INLINE void |
| mpd_set_sign(mpd_t *result, uint8_t sign) |
| { |
| result->flags &= ~MPD_NEG; |
| result->flags |= sign; |
| } |
| |
| /* Copy sign from another decimal */ |
| ALWAYS_INLINE void |
| mpd_signcpy(mpd_t *result, mpd_t *a) |
| { |
| uint8_t sign = a->flags&MPD_NEG; |
| |
| result->flags &= ~MPD_NEG; |
| result->flags |= sign; |
| } |
| |
| /* Set infinity */ |
| ALWAYS_INLINE void |
| mpd_set_infinity(mpd_t *result) |
| { |
| result->flags &= ~MPD_SPECIAL; |
| result->flags |= MPD_INF; |
| } |
| |
| /* Set qNaN */ |
| ALWAYS_INLINE void |
| mpd_set_qnan(mpd_t *result) |
| { |
| result->flags &= ~MPD_SPECIAL; |
| result->flags |= MPD_NAN; |
| } |
| |
| /* Set sNaN */ |
| ALWAYS_INLINE void |
| mpd_set_snan(mpd_t *result) |
| { |
| result->flags &= ~MPD_SPECIAL; |
| result->flags |= MPD_SNAN; |
| } |
| |
| /* Set to negative */ |
| ALWAYS_INLINE void |
| mpd_set_negative(mpd_t *result) |
| { |
| result->flags |= MPD_NEG; |
| } |
| |
| /* Set to positive */ |
| ALWAYS_INLINE void |
| mpd_set_positive(mpd_t *result) |
| { |
| result->flags &= ~MPD_NEG; |
| } |
| |
| /* Set to dynamic */ |
| ALWAYS_INLINE void |
| mpd_set_dynamic(mpd_t *result) |
| { |
| result->flags &= ~MPD_STATIC; |
| } |
| |
| /* Set to static */ |
| ALWAYS_INLINE void |
| mpd_set_static(mpd_t *result) |
| { |
| result->flags |= MPD_STATIC; |
| } |
| |
| /* Set data to dynamic */ |
| ALWAYS_INLINE void |
| mpd_set_dynamic_data(mpd_t *result) |
| { |
| result->flags &= ~MPD_DATAFLAGS; |
| } |
| |
| /* Set data to static */ |
| ALWAYS_INLINE void |
| mpd_set_static_data(mpd_t *result) |
| { |
| result->flags &= ~MPD_DATAFLAGS; |
| result->flags |= MPD_STATIC_DATA; |
| } |
| |
| /* Set data to shared */ |
| ALWAYS_INLINE void |
| mpd_set_shared_data(mpd_t *result) |
| { |
| result->flags &= ~MPD_DATAFLAGS; |
| result->flags |= MPD_SHARED_DATA; |
| } |
| |
| /* Set data to const */ |
| ALWAYS_INLINE void |
| mpd_set_const_data(mpd_t *result) |
| { |
| result->flags &= ~MPD_DATAFLAGS; |
| result->flags |= MPD_CONST_DATA; |
| } |
| |
| /* Clear flags, preserving memory attributes. */ |
| ALWAYS_INLINE void |
| mpd_clear_flags(mpd_t *result) |
| { |
| result->flags &= (MPD_STATIC|MPD_DATAFLAGS); |
| } |
| |
| /* Set flags, preserving memory attributes. */ |
| ALWAYS_INLINE void |
| mpd_set_flags(mpd_t *result, uint8_t flags) |
| { |
| result->flags &= (MPD_STATIC|MPD_DATAFLAGS); |
| result->flags |= flags; |
| } |
| |
| /* Copy flags, preserving memory attributes of result. */ |
| ALWAYS_INLINE void |
| mpd_copy_flags(mpd_t *result, const mpd_t *a) |
| { |
| uint8_t aflags = a->flags; |
| result->flags &= (MPD_STATIC|MPD_DATAFLAGS); |
| result->flags |= (aflags & ~(MPD_STATIC|MPD_DATAFLAGS)); |
| } |
| |
| /* Initialize a workcontext from ctx. Set traps, flags and newtrap to 0. */ |
| static inline void |
| mpd_workcontext(mpd_context_t *workctx, const mpd_context_t *ctx) |
| { |
| workctx->prec = ctx->prec; |
| workctx->emax = ctx->emax; |
| workctx->emin = ctx->emin; |
| workctx->round = ctx->round; |
| workctx->traps = 0; |
| workctx->status = 0; |
| workctx->newtrap = 0; |
| workctx->clamp = ctx->clamp; |
| workctx->allcr = ctx->allcr; |
| } |
| |
| |
| /******************************************************************************/ |
| /* Getting and setting parts of decimals */ |
| /******************************************************************************/ |
| |
| /* Flip the sign of a decimal */ |
| static inline void |
| _mpd_negate(mpd_t *dec) |
| { |
| dec->flags ^= MPD_NEG; |
| } |
| |
| /* Set coefficient to zero */ |
| void |
| mpd_zerocoeff(mpd_t *result) |
| { |
| mpd_minalloc(result); |
| result->digits = 1; |
| result->len = 1; |
| result->data[0] = 0; |
| } |
| |
| /* Set the coefficient to all nines. */ |
| void |
| mpd_qmaxcoeff(mpd_t *result, const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_ssize_t len, r; |
| |
| _mpd_idiv_word(&len, &r, ctx->prec, MPD_RDIGITS); |
| len = (r == 0) ? len : len+1; |
| |
| if (!mpd_qresize(result, len, status)) { |
| return; |
| } |
| |
| result->len = len; |
| result->digits = ctx->prec; |
| |
| --len; |
| if (r > 0) { |
| result->data[len--] = mpd_pow10[r]-1; |
| } |
| for (; len >= 0; --len) { |
| result->data[len] = MPD_RADIX-1; |
| } |
| } |
| |
| /* |
| * Cut off the most significant digits so that the rest fits in ctx->prec. |
| * Cannot fail. |
| */ |
| static void |
| _mpd_cap(mpd_t *result, const mpd_context_t *ctx) |
| { |
| uint32_t dummy; |
| mpd_ssize_t len, r; |
| |
| if (result->len > 0 && result->digits > ctx->prec) { |
| _mpd_idiv_word(&len, &r, ctx->prec, MPD_RDIGITS); |
| len = (r == 0) ? len : len+1; |
| |
| if (r != 0) { |
| result->data[len-1] %= mpd_pow10[r]; |
| } |
| |
| len = _mpd_real_size(result->data, len); |
| /* resize to fewer words cannot fail */ |
| mpd_qresize(result, len, &dummy); |
| result->len = len; |
| mpd_setdigits(result); |
| } |
| if (mpd_iszero(result)) { |
| _settriple(result, mpd_sign(result), 0, result->exp); |
| } |
| } |
| |
| /* |
| * Cut off the most significant digits of a NaN payload so that the rest |
| * fits in ctx->prec - ctx->clamp. Cannot fail. |
| */ |
| static void |
| _mpd_fix_nan(mpd_t *result, const mpd_context_t *ctx) |
| { |
| uint32_t dummy; |
| mpd_ssize_t prec; |
| mpd_ssize_t len, r; |
| |
| prec = ctx->prec - ctx->clamp; |
| if (result->len > 0 && result->digits > prec) { |
| if (prec == 0) { |
| mpd_minalloc(result); |
| result->len = result->digits = 0; |
| } |
| else { |
| _mpd_idiv_word(&len, &r, prec, MPD_RDIGITS); |
| len = (r == 0) ? len : len+1; |
| |
| if (r != 0) { |
| result->data[len-1] %= mpd_pow10[r]; |
| } |
| |
| len = _mpd_real_size(result->data, len); |
| /* resize to fewer words cannot fail */ |
| mpd_qresize(result, len, &dummy); |
| result->len = len; |
| mpd_setdigits(result); |
| if (mpd_iszerocoeff(result)) { |
| /* NaN0 is not a valid representation */ |
| result->len = result->digits = 0; |
| } |
| } |
| } |
| } |
| |
| /* |
| * Get n most significant digits from a decimal, where 0 < n <= MPD_UINT_DIGITS. |
| * Assumes MPD_UINT_DIGITS == MPD_RDIGITS+1, which is true for 32 and 64 bit |
| * machines. |
| * |
| * The result of the operation will be in lo. If the operation is impossible, |
| * hi will be nonzero. This is used to indicate an error. |
| */ |
| static inline void |
| _mpd_get_msdigits(mpd_uint_t *hi, mpd_uint_t *lo, const mpd_t *dec, |
| unsigned int n) |
| { |
| mpd_uint_t r, tmp; |
| |
| assert(0 < n && n <= MPD_RDIGITS+1); |
| |
| _mpd_div_word(&tmp, &r, dec->digits, MPD_RDIGITS); |
| r = (r == 0) ? MPD_RDIGITS : r; /* digits in the most significant word */ |
| |
| *hi = 0; |
| *lo = dec->data[dec->len-1]; |
| if (n <= r) { |
| *lo /= mpd_pow10[r-n]; |
| } |
| else if (dec->len > 1) { |
| /* at this point 1 <= r < n <= MPD_RDIGITS+1 */ |
| _mpd_mul_words(hi, lo, *lo, mpd_pow10[n-r]); |
| tmp = dec->data[dec->len-2] / mpd_pow10[MPD_RDIGITS-(n-r)]; |
| *lo = *lo + tmp; |
| if (*lo < tmp) (*hi)++; |
| } |
| } |
| |
| |
| /******************************************************************************/ |
| /* Gathering information about a decimal */ |
| /******************************************************************************/ |
| |
| /* The real size of the coefficient without leading zero words. */ |
| static inline mpd_ssize_t |
| _mpd_real_size(mpd_uint_t *data, mpd_ssize_t size) |
| { |
| while (size > 1 && data[size-1] == 0) { |
| size--; |
| } |
| |
| return size; |
| } |
| |
| /* Return number of trailing zeros. No errors are possible. */ |
| mpd_ssize_t |
| mpd_trail_zeros(const mpd_t *dec) |
| { |
| mpd_uint_t word; |
| mpd_ssize_t i, tz = 0; |
| |
| for (i=0; i < dec->len; ++i) { |
| if (dec->data[i] != 0) { |
| word = dec->data[i]; |
| tz = i * MPD_RDIGITS; |
| while (word % 10 == 0) { |
| word /= 10; |
| tz++; |
| } |
| break; |
| } |
| } |
| |
| return tz; |
| } |
| |
| /* Integer: Undefined for specials */ |
| static int |
| _mpd_isint(const mpd_t *dec) |
| { |
| mpd_ssize_t tz; |
| |
| if (mpd_iszerocoeff(dec)) { |
| return 1; |
| } |
| |
| tz = mpd_trail_zeros(dec); |
| return (dec->exp + tz >= 0); |
| } |
| |
| /* Integer */ |
| int |
| mpd_isinteger(const mpd_t *dec) |
| { |
| if (mpd_isspecial(dec)) { |
| return 0; |
| } |
| return _mpd_isint(dec); |
| } |
| |
| /* Word is a power of 10 */ |
| static int |
| mpd_word_ispow10(mpd_uint_t word) |
| { |
| int n; |
| |
| n = mpd_word_digits(word); |
| if (word == mpd_pow10[n-1]) { |
| return 1; |
| } |
| |
| return 0; |
| } |
| |
| /* Coefficient is a power of 10 */ |
| static int |
| mpd_coeff_ispow10(const mpd_t *dec) |
| { |
| if (mpd_word_ispow10(mpd_msword(dec))) { |
| if (_mpd_isallzero(dec->data, dec->len-1)) { |
| return 1; |
| } |
| } |
| |
| return 0; |
| } |
| |
| /* All digits of a word are nines */ |
| static int |
| mpd_word_isallnine(mpd_uint_t word) |
| { |
| int n; |
| |
| n = mpd_word_digits(word); |
| if (word == mpd_pow10[n]-1) { |
| return 1; |
| } |
| |
| return 0; |
| } |
| |
| /* All digits of the coefficient are nines */ |
| static int |
| mpd_coeff_isallnine(const mpd_t *dec) |
| { |
| if (mpd_word_isallnine(mpd_msword(dec))) { |
| if (_mpd_isallnine(dec->data, dec->len-1)) { |
| return 1; |
| } |
| } |
| |
| return 0; |
| } |
| |
| /* Odd decimal: Undefined for non-integers! */ |
| int |
| mpd_isodd(const mpd_t *dec) |
| { |
| mpd_uint_t q, r; |
| assert(mpd_isinteger(dec)); |
| if (mpd_iszerocoeff(dec)) return 0; |
| if (dec->exp < 0) { |
| _mpd_div_word(&q, &r, -dec->exp, MPD_RDIGITS); |
| q = dec->data[q] / mpd_pow10[r]; |
| return mpd_isoddword(q); |
| } |
| return dec->exp == 0 && mpd_isoddword(dec->data[0]); |
| } |
| |
| /* Even: Undefined for non-integers! */ |
| int |
| mpd_iseven(const mpd_t *dec) |
| { |
| return !mpd_isodd(dec); |
| } |
| |
| /******************************************************************************/ |
| /* Getting and setting decimals */ |
| /******************************************************************************/ |
| |
| /* Internal function: Set a static decimal from a triple, no error checking. */ |
| static void |
| _ssettriple(mpd_t *result, uint8_t sign, mpd_uint_t a, mpd_ssize_t exp) |
| { |
| mpd_set_flags(result, sign); |
| result->exp = exp; |
| _mpd_div_word(&result->data[1], &result->data[0], a, MPD_RADIX); |
| result->len = (result->data[1] == 0) ? 1 : 2; |
| mpd_setdigits(result); |
| } |
| |
| /* Internal function: Set a decimal from a triple, no error checking. */ |
| static void |
| _settriple(mpd_t *result, uint8_t sign, mpd_uint_t a, mpd_ssize_t exp) |
| { |
| mpd_minalloc(result); |
| mpd_set_flags(result, sign); |
| result->exp = exp; |
| _mpd_div_word(&result->data[1], &result->data[0], a, MPD_RADIX); |
| result->len = (result->data[1] == 0) ? 1 : 2; |
| mpd_setdigits(result); |
| } |
| |
| /* Set a special number from a triple */ |
| void |
| mpd_setspecial(mpd_t *result, uint8_t sign, uint8_t type) |
| { |
| mpd_minalloc(result); |
| result->flags &= ~(MPD_NEG|MPD_SPECIAL); |
| result->flags |= (sign|type); |
| result->exp = result->digits = result->len = 0; |
| } |
| |
| /* Set result of NaN with an error status */ |
| void |
| mpd_seterror(mpd_t *result, uint32_t flags, uint32_t *status) |
| { |
| mpd_minalloc(result); |
| mpd_set_qnan(result); |
| mpd_set_positive(result); |
| result->exp = result->digits = result->len = 0; |
| *status |= flags; |
| } |
| |
| /* quietly set a static decimal from an mpd_ssize_t */ |
| void |
| mpd_qsset_ssize(mpd_t *result, mpd_ssize_t a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_uint_t u; |
| uint8_t sign = MPD_POS; |
| |
| if (a < 0) { |
| if (a == MPD_SSIZE_MIN) { |
| u = (mpd_uint_t)MPD_SSIZE_MAX + |
| (-(MPD_SSIZE_MIN+MPD_SSIZE_MAX)); |
| } |
| else { |
| u = -a; |
| } |
| sign = MPD_NEG; |
| } |
| else { |
| u = a; |
| } |
| _ssettriple(result, sign, u, 0); |
| mpd_qfinalize(result, ctx, status); |
| } |
| |
| /* quietly set a static decimal from an mpd_uint_t */ |
| void |
| mpd_qsset_uint(mpd_t *result, mpd_uint_t a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| _ssettriple(result, MPD_POS, a, 0); |
| mpd_qfinalize(result, ctx, status); |
| } |
| |
| /* quietly set a static decimal from an int32_t */ |
| void |
| mpd_qsset_i32(mpd_t *result, int32_t a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_qsset_ssize(result, a, ctx, status); |
| } |
| |
| /* quietly set a static decimal from a uint32_t */ |
| void |
| mpd_qsset_u32(mpd_t *result, uint32_t a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_qsset_uint(result, a, ctx, status); |
| } |
| |
| #ifdef CONFIG_64 |
| /* quietly set a static decimal from an int64_t */ |
| void |
| mpd_qsset_i64(mpd_t *result, int64_t a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_qsset_ssize(result, a, ctx, status); |
| } |
| |
| /* quietly set a static decimal from a uint64_t */ |
| void |
| mpd_qsset_u64(mpd_t *result, uint64_t a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_qsset_uint(result, a, ctx, status); |
| } |
| #endif |
| |
| /* quietly set a decimal from an mpd_ssize_t */ |
| void |
| mpd_qset_ssize(mpd_t *result, mpd_ssize_t a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_minalloc(result); |
| mpd_qsset_ssize(result, a, ctx, status); |
| } |
| |
| /* quietly set a decimal from an mpd_uint_t */ |
| void |
| mpd_qset_uint(mpd_t *result, mpd_uint_t a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| _settriple(result, MPD_POS, a, 0); |
| mpd_qfinalize(result, ctx, status); |
| } |
| |
| /* quietly set a decimal from an int32_t */ |
| void |
| mpd_qset_i32(mpd_t *result, int32_t a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_qset_ssize(result, a, ctx, status); |
| } |
| |
| /* quietly set a decimal from a uint32_t */ |
| void |
| mpd_qset_u32(mpd_t *result, uint32_t a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_qset_uint(result, a, ctx, status); |
| } |
| |
| #if defined(CONFIG_32) && !defined(LEGACY_COMPILER) |
| /* set a decimal from a uint64_t */ |
| static void |
| _c32setu64(mpd_t *result, uint64_t u, uint8_t sign, uint32_t *status) |
| { |
| mpd_uint_t w[3]; |
| uint64_t q; |
| int i, len; |
| |
| len = 0; |
| do { |
| q = u / MPD_RADIX; |
| w[len] = (mpd_uint_t)(u - q * MPD_RADIX); |
| u = q; len++; |
| } while (u != 0); |
| |
| if (!mpd_qresize(result, len, status)) { |
| return; |
| } |
| for (i = 0; i < len; i++) { |
| result->data[i] = w[i]; |
| } |
| |
| mpd_set_sign(result, sign); |
| result->exp = 0; |
| result->len = len; |
| mpd_setdigits(result); |
| } |
| |
| static void |
| _c32_qset_u64(mpd_t *result, uint64_t a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| _c32setu64(result, a, MPD_POS, status); |
| mpd_qfinalize(result, ctx, status); |
| } |
| |
| /* set a decimal from an int64_t */ |
| static void |
| _c32_qset_i64(mpd_t *result, int64_t a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| uint64_t u; |
| uint8_t sign = MPD_POS; |
| |
| if (a < 0) { |
| if (a == INT64_MIN) { |
| u = (uint64_t)INT64_MAX + (-(INT64_MIN+INT64_MAX)); |
| } |
| else { |
| u = -a; |
| } |
| sign = MPD_NEG; |
| } |
| else { |
| u = a; |
| } |
| _c32setu64(result, u, sign, status); |
| mpd_qfinalize(result, ctx, status); |
| } |
| #endif /* CONFIG_32 && !LEGACY_COMPILER */ |
| |
| #ifndef LEGACY_COMPILER |
| /* quietly set a decimal from an int64_t */ |
| void |
| mpd_qset_i64(mpd_t *result, int64_t a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| #ifdef CONFIG_64 |
| mpd_qset_ssize(result, a, ctx, status); |
| #else |
| _c32_qset_i64(result, a, ctx, status); |
| #endif |
| } |
| |
| /* quietly set a decimal from a uint64_t */ |
| void |
| mpd_qset_u64(mpd_t *result, uint64_t a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| #ifdef CONFIG_64 |
| mpd_qset_uint(result, a, ctx, status); |
| #else |
| _c32_qset_u64(result, a, ctx, status); |
| #endif |
| } |
| #endif /* !LEGACY_COMPILER */ |
| |
| |
| /* |
| * Quietly get an mpd_uint_t from a decimal. Assumes |
| * MPD_UINT_DIGITS == MPD_RDIGITS+1, which is true for |
| * 32 and 64 bit machines. |
| * |
| * If the operation is impossible, MPD_Invalid_operation is set. |
| */ |
| static mpd_uint_t |
| _mpd_qget_uint(int use_sign, const mpd_t *a, uint32_t *status) |
| { |
| mpd_t tmp; |
| mpd_uint_t tmp_data[2]; |
| mpd_uint_t lo, hi; |
| |
| if (mpd_isspecial(a)) { |
| *status |= MPD_Invalid_operation; |
| return MPD_UINT_MAX; |
| } |
| if (mpd_iszero(a)) { |
| return 0; |
| } |
| if (use_sign && mpd_isnegative(a)) { |
| *status |= MPD_Invalid_operation; |
| return MPD_UINT_MAX; |
| } |
| |
| if (a->digits+a->exp > MPD_RDIGITS+1) { |
| *status |= MPD_Invalid_operation; |
| return MPD_UINT_MAX; |
| } |
| |
| if (a->exp < 0) { |
| if (!_mpd_isint(a)) { |
| *status |= MPD_Invalid_operation; |
| return MPD_UINT_MAX; |
| } |
| /* At this point a->digits+a->exp <= MPD_RDIGITS+1, |
| * so the shift fits. */ |
| tmp.data = tmp_data; |
| tmp.flags = MPD_STATIC|MPD_STATIC_DATA; |
| tmp.alloc = 2; |
| mpd_qsshiftr(&tmp, a, -a->exp); |
| tmp.exp = 0; |
| a = &tmp; |
| } |
| |
| _mpd_get_msdigits(&hi, &lo, a, MPD_RDIGITS+1); |
| if (hi) { |
| *status |= MPD_Invalid_operation; |
| return MPD_UINT_MAX; |
| } |
| |
| if (a->exp > 0) { |
| _mpd_mul_words(&hi, &lo, lo, mpd_pow10[a->exp]); |
| if (hi) { |
| *status |= MPD_Invalid_operation; |
| return MPD_UINT_MAX; |
| } |
| } |
| |
| return lo; |
| } |
| |
| /* |
| * Sets Invalid_operation for: |
| * - specials |
| * - negative numbers (except negative zero) |
| * - non-integers |
| * - overflow |
| */ |
| mpd_uint_t |
| mpd_qget_uint(const mpd_t *a, uint32_t *status) |
| { |
| return _mpd_qget_uint(1, a, status); |
| } |
| |
| /* Same as above, but gets the absolute value, i.e. the sign is ignored. */ |
| mpd_uint_t |
| mpd_qabs_uint(const mpd_t *a, uint32_t *status) |
| { |
| return _mpd_qget_uint(0, a, status); |
| } |
| |
| /* quietly get an mpd_ssize_t from a decimal */ |
| mpd_ssize_t |
| mpd_qget_ssize(const mpd_t *a, uint32_t *status) |
| { |
| mpd_uint_t u; |
| int isneg; |
| |
| u = mpd_qabs_uint(a, status); |
| if (*status&MPD_Invalid_operation) { |
| return MPD_SSIZE_MAX; |
| } |
| |
| isneg = mpd_isnegative(a); |
| if (u <= MPD_SSIZE_MAX) { |
| return isneg ? -((mpd_ssize_t)u) : (mpd_ssize_t)u; |
| } |
| else if (isneg && u+(MPD_SSIZE_MIN+MPD_SSIZE_MAX) == MPD_SSIZE_MAX) { |
| return MPD_SSIZE_MIN; |
| } |
| |
| *status |= MPD_Invalid_operation; |
| return MPD_SSIZE_MAX; |
| } |
| |
| #ifdef CONFIG_64 |
| /* quietly get a uint64_t from a decimal */ |
| uint64_t |
| mpd_qget_u64(const mpd_t *a, uint32_t *status) |
| { |
| return mpd_qget_uint(a, status); |
| } |
| |
| /* quietly get an int64_t from a decimal */ |
| int64_t |
| mpd_qget_i64(const mpd_t *a, uint32_t *status) |
| { |
| return mpd_qget_ssize(a, status); |
| } |
| #else |
| /* quietly get a uint32_t from a decimal */ |
| uint32_t |
| mpd_qget_u32(const mpd_t *a, uint32_t *status) |
| { |
| return mpd_qget_uint(a, status); |
| } |
| |
| /* quietly get an int32_t from a decimal */ |
| int32_t |
| mpd_qget_i32(const mpd_t *a, uint32_t *status) |
| { |
| return mpd_qget_ssize(a, status); |
| } |
| #endif |
| |
| |
| /******************************************************************************/ |
| /* Filtering input of functions, finalizing output of functions */ |
| /******************************************************************************/ |
| |
| /* |
| * Check if the operand is NaN, copy to result and return 1 if this is |
| * the case. Copying can fail since NaNs are allowed to have a payload that |
| * does not fit in MPD_MINALLOC. |
| */ |
| int |
| mpd_qcheck_nan(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| if (mpd_isnan(a)) { |
| *status |= mpd_issnan(a) ? MPD_Invalid_operation : 0; |
| mpd_qcopy(result, a, status); |
| mpd_set_qnan(result); |
| _mpd_fix_nan(result, ctx); |
| return 1; |
| } |
| return 0; |
| } |
| |
| /* |
| * Check if either operand is NaN, copy to result and return 1 if this |
| * is the case. Copying can fail since NaNs are allowed to have a payload |
| * that does not fit in MPD_MINALLOC. |
| */ |
| int |
| mpd_qcheck_nans(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| if ((a->flags|b->flags)&(MPD_NAN|MPD_SNAN)) { |
| const mpd_t *choice = b; |
| if (mpd_issnan(a)) { |
| choice = a; |
| *status |= MPD_Invalid_operation; |
| } |
| else if (mpd_issnan(b)) { |
| *status |= MPD_Invalid_operation; |
| } |
| else if (mpd_isqnan(a)) { |
| choice = a; |
| } |
| mpd_qcopy(result, choice, status); |
| mpd_set_qnan(result); |
| _mpd_fix_nan(result, ctx); |
| return 1; |
| } |
| return 0; |
| } |
| |
| /* |
| * Check if one of the operands is NaN, copy to result and return 1 if this |
| * is the case. Copying can fail since NaNs are allowed to have a payload |
| * that does not fit in MPD_MINALLOC. |
| */ |
| static int |
| mpd_qcheck_3nans(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_t *c, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| if ((a->flags|b->flags|c->flags)&(MPD_NAN|MPD_SNAN)) { |
| const mpd_t *choice = c; |
| if (mpd_issnan(a)) { |
| choice = a; |
| *status |= MPD_Invalid_operation; |
| } |
| else if (mpd_issnan(b)) { |
| choice = b; |
| *status |= MPD_Invalid_operation; |
| } |
| else if (mpd_issnan(c)) { |
| *status |= MPD_Invalid_operation; |
| } |
| else if (mpd_isqnan(a)) { |
| choice = a; |
| } |
| else if (mpd_isqnan(b)) { |
| choice = b; |
| } |
| mpd_qcopy(result, choice, status); |
| mpd_set_qnan(result); |
| _mpd_fix_nan(result, ctx); |
| return 1; |
| } |
| return 0; |
| } |
| |
| /* Check if rounding digit 'rnd' leads to an increment. */ |
| static inline int |
| _mpd_rnd_incr(const mpd_t *dec, mpd_uint_t rnd, const mpd_context_t *ctx) |
| { |
| int ld; |
| |
| switch (ctx->round) { |
| case MPD_ROUND_DOWN: case MPD_ROUND_TRUNC: |
| return 0; |
| case MPD_ROUND_HALF_UP: |
| return (rnd >= 5); |
| case MPD_ROUND_HALF_EVEN: |
| return (rnd > 5) || ((rnd == 5) && mpd_isoddcoeff(dec)); |
| case MPD_ROUND_CEILING: |
| return !(rnd == 0 || mpd_isnegative(dec)); |
| case MPD_ROUND_FLOOR: |
| return !(rnd == 0 || mpd_ispositive(dec)); |
| case MPD_ROUND_HALF_DOWN: |
| return (rnd > 5); |
| case MPD_ROUND_UP: |
| return !(rnd == 0); |
| case MPD_ROUND_05UP: |
| ld = (int)mpd_lsd(dec->data[0]); |
| return (!(rnd == 0) && (ld == 0 || ld == 5)); |
| default: |
| /* Without a valid context, further results will be undefined. */ |
| return 0; /* GCOV_NOT_REACHED */ |
| } |
| } |
| |
| /* |
| * Apply rounding to a decimal that has been right-shifted into a full |
| * precision decimal. If an increment leads to an overflow of the precision, |
| * adjust the coefficient and the exponent and check the new exponent for |
| * overflow. |
| */ |
| static inline void |
| _mpd_apply_round(mpd_t *dec, mpd_uint_t rnd, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| if (_mpd_rnd_incr(dec, rnd, ctx)) { |
| /* We have a number with exactly ctx->prec digits. The increment |
| * can only lead to an overflow if the decimal is all nines. In |
| * that case, the result is a power of ten with prec+1 digits. |
| * |
| * If the precision is a multiple of MPD_RDIGITS, this situation is |
| * detected by _mpd_baseincr returning a carry. |
| * If the precision is not a multiple of MPD_RDIGITS, we have to |
| * check if the result has one digit too many. |
| */ |
| mpd_uint_t carry = _mpd_baseincr(dec->data, dec->len); |
| if (carry) { |
| dec->data[dec->len-1] = mpd_pow10[MPD_RDIGITS-1]; |
| dec->exp += 1; |
| _mpd_check_exp(dec, ctx, status); |
| return; |
| } |
| mpd_setdigits(dec); |
| if (dec->digits > ctx->prec) { |
| mpd_qshiftr_inplace(dec, 1); |
| dec->exp += 1; |
| dec->digits = ctx->prec; |
| _mpd_check_exp(dec, ctx, status); |
| } |
| } |
| } |
| |
| /* |
| * Apply rounding to a decimal. Allow overflow of the precision. |
| */ |
| static inline void |
| _mpd_apply_round_excess(mpd_t *dec, mpd_uint_t rnd, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| if (_mpd_rnd_incr(dec, rnd, ctx)) { |
| mpd_uint_t carry = _mpd_baseincr(dec->data, dec->len); |
| if (carry) { |
| if (!mpd_qresize(dec, dec->len+1, status)) { |
| return; |
| } |
| dec->data[dec->len] = 1; |
| dec->len += 1; |
| } |
| mpd_setdigits(dec); |
| } |
| } |
| |
| /* |
| * Apply rounding to a decimal that has been right-shifted into a decimal |
| * with full precision or less. Return failure if an increment would |
| * overflow the precision. |
| */ |
| static inline int |
| _mpd_apply_round_fit(mpd_t *dec, mpd_uint_t rnd, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| if (_mpd_rnd_incr(dec, rnd, ctx)) { |
| mpd_uint_t carry = _mpd_baseincr(dec->data, dec->len); |
| if (carry) { |
| if (!mpd_qresize(dec, dec->len+1, status)) { |
| return 0; |
| } |
| dec->data[dec->len] = 1; |
| dec->len += 1; |
| } |
| mpd_setdigits(dec); |
| if (dec->digits > ctx->prec) { |
| mpd_seterror(dec, MPD_Invalid_operation, status); |
| return 0; |
| } |
| } |
| return 1; |
| } |
| |
| /* Check a normal number for overflow, underflow, clamping. If the operand |
| is modified, it will be zero, special or (sub)normal with a coefficient |
| that fits into the current context precision. */ |
| static inline void |
| _mpd_check_exp(mpd_t *dec, const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_ssize_t adjexp, etiny, shift; |
| int rnd; |
| |
| adjexp = mpd_adjexp(dec); |
| if (adjexp > ctx->emax) { |
| |
| if (mpd_iszerocoeff(dec)) { |
| dec->exp = ctx->emax; |
| if (ctx->clamp) { |
| dec->exp -= (ctx->prec-1); |
| } |
| mpd_zerocoeff(dec); |
| *status |= MPD_Clamped; |
| return; |
| } |
| |
| switch (ctx->round) { |
| case MPD_ROUND_HALF_UP: case MPD_ROUND_HALF_EVEN: |
| case MPD_ROUND_HALF_DOWN: case MPD_ROUND_UP: |
| case MPD_ROUND_TRUNC: |
| mpd_setspecial(dec, mpd_sign(dec), MPD_INF); |
| break; |
| case MPD_ROUND_DOWN: case MPD_ROUND_05UP: |
| mpd_qmaxcoeff(dec, ctx, status); |
| dec->exp = ctx->emax - ctx->prec + 1; |
| break; |
| case MPD_ROUND_CEILING: |
| if (mpd_isnegative(dec)) { |
| mpd_qmaxcoeff(dec, ctx, status); |
| dec->exp = ctx->emax - ctx->prec + 1; |
| } |
| else { |
| mpd_setspecial(dec, MPD_POS, MPD_INF); |
| } |
| break; |
| case MPD_ROUND_FLOOR: |
| if (mpd_ispositive(dec)) { |
| mpd_qmaxcoeff(dec, ctx, status); |
| dec->exp = ctx->emax - ctx->prec + 1; |
| } |
| else { |
| mpd_setspecial(dec, MPD_NEG, MPD_INF); |
| } |
| break; |
| default: /* debug */ |
| abort(); /* GCOV_NOT_REACHED */ |
| } |
| |
| *status |= MPD_Overflow|MPD_Inexact|MPD_Rounded; |
| |
| } /* fold down */ |
| else if (ctx->clamp && dec->exp > mpd_etop(ctx)) { |
| /* At this point adjexp=exp+digits-1 <= emax and exp > etop=emax-prec+1: |
| * (1) shift = exp -emax+prec-1 > 0 |
| * (2) digits+shift = exp+digits-1 - emax + prec <= prec */ |
| shift = dec->exp - mpd_etop(ctx); |
| if (!mpd_qshiftl(dec, dec, shift, status)) { |
| return; |
| } |
| dec->exp -= shift; |
| *status |= MPD_Clamped; |
| if (!mpd_iszerocoeff(dec) && adjexp < ctx->emin) { |
| /* Underflow is impossible, since exp < etiny=emin-prec+1 |
| * and exp > etop=emax-prec+1 would imply emax < emin. */ |
| *status |= MPD_Subnormal; |
| } |
| } |
| else if (adjexp < ctx->emin) { |
| |
| etiny = mpd_etiny(ctx); |
| |
| if (mpd_iszerocoeff(dec)) { |
| if (dec->exp < etiny) { |
| dec->exp = etiny; |
| mpd_zerocoeff(dec); |
| *status |= MPD_Clamped; |
| } |
| return; |
| } |
| |
| *status |= MPD_Subnormal; |
| if (dec->exp < etiny) { |
| /* At this point adjexp=exp+digits-1 < emin and exp < etiny=emin-prec+1: |
| * (1) shift = emin-prec+1 - exp > 0 |
| * (2) digits-shift = exp+digits-1 - emin + prec < prec */ |
| shift = etiny - dec->exp; |
| rnd = (int)mpd_qshiftr_inplace(dec, shift); |
| dec->exp = etiny; |
| /* We always have a spare digit in case of an increment. */ |
| _mpd_apply_round_excess(dec, rnd, ctx, status); |
| *status |= MPD_Rounded; |
| if (rnd) { |
| *status |= (MPD_Inexact|MPD_Underflow); |
| if (mpd_iszerocoeff(dec)) { |
| mpd_zerocoeff(dec); |
| *status |= MPD_Clamped; |
| } |
| } |
| } |
| /* Case exp >= etiny=emin-prec+1: |
| * (1) adjexp=exp+digits-1 < emin |
| * (2) digits < emin-exp+1 <= prec */ |
| } |
| } |
| |
| /* Transcendental functions do not always set Underflow reliably, |
| * since they only use as much precision as is necessary for correct |
| * rounding. If a result like 1.0000000000e-101 is finalized, there |
| * is no rounding digit that would trigger Underflow. But we can |
| * assume Inexact, so a short check suffices. */ |
| static inline void |
| mpd_check_underflow(mpd_t *dec, const mpd_context_t *ctx, uint32_t *status) |
| { |
| if (mpd_adjexp(dec) < ctx->emin && !mpd_iszero(dec) && |
| dec->exp < mpd_etiny(ctx)) { |
| *status |= MPD_Underflow; |
| } |
| } |
| |
| /* Check if a normal number must be rounded after the exponent has been checked. */ |
| static inline void |
| _mpd_check_round(mpd_t *dec, const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_uint_t rnd; |
| mpd_ssize_t shift; |
| |
| /* must handle specials: _mpd_check_exp() can produce infinities or NaNs */ |
| if (mpd_isspecial(dec)) { |
| return; |
| } |
| |
| if (dec->digits > ctx->prec) { |
| shift = dec->digits - ctx->prec; |
| rnd = mpd_qshiftr_inplace(dec, shift); |
| dec->exp += shift; |
| _mpd_apply_round(dec, rnd, ctx, status); |
| *status |= MPD_Rounded; |
| if (rnd) { |
| *status |= MPD_Inexact; |
| } |
| } |
| } |
| |
| /* Finalize all operations. */ |
| void |
| mpd_qfinalize(mpd_t *result, const mpd_context_t *ctx, uint32_t *status) |
| { |
| if (mpd_isspecial(result)) { |
| if (mpd_isnan(result)) { |
| _mpd_fix_nan(result, ctx); |
| } |
| return; |
| } |
| |
| _mpd_check_exp(result, ctx, status); |
| _mpd_check_round(result, ctx, status); |
| } |
| |
| |
| /******************************************************************************/ |
| /* Copying */ |
| /******************************************************************************/ |
| |
| /* Internal function: Copy a decimal, share data with src: USE WITH CARE! */ |
| static inline void |
| _mpd_copy_shared(mpd_t *dest, const mpd_t *src) |
| { |
| dest->flags = src->flags; |
| dest->exp = src->exp; |
| dest->digits = src->digits; |
| dest->len = src->len; |
| dest->alloc = src->alloc; |
| dest->data = src->data; |
| |
| mpd_set_shared_data(dest); |
| } |
| |
| /* |
| * Copy a decimal. In case of an error, status is set to MPD_Malloc_error. |
| */ |
| int |
| mpd_qcopy(mpd_t *result, const mpd_t *a, uint32_t *status) |
| { |
| if (result == a) return 1; |
| |
| if (!mpd_qresize(result, a->len, status)) { |
| return 0; |
| } |
| |
| mpd_copy_flags(result, a); |
| result->exp = a->exp; |
| result->digits = a->digits; |
| result->len = a->len; |
| memcpy(result->data, a->data, a->len * (sizeof *result->data)); |
| |
| return 1; |
| } |
| |
| /* |
| * Copy to a decimal with a static buffer. The caller has to make sure that |
| * the buffer is big enough. Cannot fail. |
| */ |
| static void |
| mpd_qcopy_static(mpd_t *result, const mpd_t *a) |
| { |
| if (result == a) return; |
| |
| memcpy(result->data, a->data, a->len * (sizeof *result->data)); |
| |
| mpd_copy_flags(result, a); |
| result->exp = a->exp; |
| result->digits = a->digits; |
| result->len = a->len; |
| } |
| |
| /* |
| * Return a newly allocated copy of the operand. In case of an error, |
| * status is set to MPD_Malloc_error and the return value is NULL. |
| */ |
| mpd_t * |
| mpd_qncopy(const mpd_t *a) |
| { |
| mpd_t *result; |
| |
| if ((result = mpd_qnew_size(a->len)) == NULL) { |
| return NULL; |
| } |
| memcpy(result->data, a->data, a->len * (sizeof *result->data)); |
| mpd_copy_flags(result, a); |
| result->exp = a->exp; |
| result->digits = a->digits; |
| result->len = a->len; |
| |
| return result; |
| } |
| |
| /* |
| * Copy a decimal and set the sign to positive. In case of an error, the |
| * status is set to MPD_Malloc_error. |
| */ |
| int |
| mpd_qcopy_abs(mpd_t *result, const mpd_t *a, uint32_t *status) |
| { |
| if (!mpd_qcopy(result, a, status)) { |
| return 0; |
| } |
| mpd_set_positive(result); |
| return 1; |
| } |
| |
| /* |
| * Copy a decimal and negate the sign. In case of an error, the |
| * status is set to MPD_Malloc_error. |
| */ |
| int |
| mpd_qcopy_negate(mpd_t *result, const mpd_t *a, uint32_t *status) |
| { |
| if (!mpd_qcopy(result, a, status)) { |
| return 0; |
| } |
| _mpd_negate(result); |
| return 1; |
| } |
| |
| /* |
| * Copy a decimal, setting the sign of the first operand to the sign of the |
| * second operand. In case of an error, the status is set to MPD_Malloc_error. |
| */ |
| int |
| mpd_qcopy_sign(mpd_t *result, const mpd_t *a, const mpd_t *b, uint32_t *status) |
| { |
| uint8_t sign_b = mpd_sign(b); /* result may equal b! */ |
| |
| if (!mpd_qcopy(result, a, status)) { |
| return 0; |
| } |
| mpd_set_sign(result, sign_b); |
| return 1; |
| } |
| |
| |
| /******************************************************************************/ |
| /* Comparisons */ |
| /******************************************************************************/ |
| |
| /* |
| * For all functions that compare two operands and return an int the usual |
| * convention applies to the return value: |
| * |
| * -1 if op1 < op2 |
| * 0 if op1 == op2 |
| * 1 if op1 > op2 |
| * |
| * INT_MAX for error |
| */ |
| |
| |
| /* Convenience macro. If a and b are not equal, return from the calling |
| * function with the correct comparison value. */ |
| #define CMP_EQUAL_OR_RETURN(a, b) \ |
| if (a != b) { \ |
| if (a < b) { \ |
| return -1; \ |
| } \ |
| return 1; \ |
| } |
| |
| /* |
| * Compare the data of big and small. This function does the equivalent |
| * of first shifting small to the left and then comparing the data of |
| * big and small, except that no allocation for the left shift is needed. |
| */ |
| static int |
| _mpd_basecmp(mpd_uint_t *big, mpd_uint_t *small, mpd_size_t n, mpd_size_t m, |
| mpd_size_t shift) |
| { |
| #if defined(__GNUC__) && !defined(__INTEL_COMPILER) && !defined(__clang__) |
| /* spurious uninitialized warnings */ |
| mpd_uint_t l=l, lprev=lprev, h=h; |
| #else |
| mpd_uint_t l, lprev, h; |
| #endif |
| mpd_uint_t q, r; |
| mpd_uint_t ph, x; |
| |
| assert(m > 0 && n >= m && shift > 0); |
| |
| _mpd_div_word(&q, &r, (mpd_uint_t)shift, MPD_RDIGITS); |
| |
| if (r != 0) { |
| |
| ph = mpd_pow10[r]; |
| |
| --m; --n; |
| _mpd_divmod_pow10(&h, &lprev, small[m--], MPD_RDIGITS-r); |
| if (h != 0) { |
| CMP_EQUAL_OR_RETURN(big[n], h) |
| --n; |
| } |
| for (; m != MPD_SIZE_MAX; m--,n--) { |
| _mpd_divmod_pow10(&h, &l, small[m], MPD_RDIGITS-r); |
| x = ph * lprev + h; |
| CMP_EQUAL_OR_RETURN(big[n], x) |
| lprev = l; |
| } |
| x = ph * lprev; |
| CMP_EQUAL_OR_RETURN(big[q], x) |
| } |
| else { |
| while (--m != MPD_SIZE_MAX) { |
| CMP_EQUAL_OR_RETURN(big[m+q], small[m]) |
| } |
| } |
| |
| return !_mpd_isallzero(big, q); |
| } |
| |
| /* Compare two decimals with the same adjusted exponent. */ |
| static int |
| _mpd_cmp_same_adjexp(const mpd_t *a, const mpd_t *b) |
| { |
| mpd_ssize_t shift, i; |
| |
| if (a->exp != b->exp) { |
| /* Cannot wrap: a->exp + a->digits = b->exp + b->digits, so |
| * a->exp - b->exp = b->digits - a->digits. */ |
| shift = a->exp - b->exp; |
| if (shift > 0) { |
| return -1 * _mpd_basecmp(b->data, a->data, b->len, a->len, shift); |
| } |
| else { |
| return _mpd_basecmp(a->data, b->data, a->len, b->len, -shift); |
| } |
| } |
| |
| /* |
| * At this point adjexp(a) == adjexp(b) and a->exp == b->exp, |
| * so a->digits == b->digits, therefore a->len == b->len. |
| */ |
| for (i = a->len-1; i >= 0; --i) { |
| CMP_EQUAL_OR_RETURN(a->data[i], b->data[i]) |
| } |
| |
| return 0; |
| } |
| |
| /* Compare two numerical values. */ |
| static int |
| _mpd_cmp(const mpd_t *a, const mpd_t *b) |
| { |
| mpd_ssize_t adjexp_a, adjexp_b; |
| |
| /* equal pointers */ |
| if (a == b) { |
| return 0; |
| } |
| |
| /* infinities */ |
| if (mpd_isinfinite(a)) { |
| if (mpd_isinfinite(b)) { |
| return mpd_isnegative(b) - mpd_isnegative(a); |
| } |
| return mpd_arith_sign(a); |
| } |
| if (mpd_isinfinite(b)) { |
| return -mpd_arith_sign(b); |
| } |
| |
| /* zeros */ |
| if (mpd_iszerocoeff(a)) { |
| if (mpd_iszerocoeff(b)) { |
| return 0; |
| } |
| return -mpd_arith_sign(b); |
| } |
| if (mpd_iszerocoeff(b)) { |
| return mpd_arith_sign(a); |
| } |
| |
| /* different signs */ |
| if (mpd_sign(a) != mpd_sign(b)) { |
| return mpd_sign(b) - mpd_sign(a); |
| } |
| |
| /* different adjusted exponents */ |
| adjexp_a = mpd_adjexp(a); |
| adjexp_b = mpd_adjexp(b); |
| if (adjexp_a != adjexp_b) { |
| if (adjexp_a < adjexp_b) { |
| return -1 * mpd_arith_sign(a); |
| } |
| return mpd_arith_sign(a); |
| } |
| |
| /* same adjusted exponents */ |
| return _mpd_cmp_same_adjexp(a, b) * mpd_arith_sign(a); |
| } |
| |
| /* Compare the absolutes of two numerical values. */ |
| static int |
| _mpd_cmp_abs(const mpd_t *a, const mpd_t *b) |
| { |
| mpd_ssize_t adjexp_a, adjexp_b; |
| |
| /* equal pointers */ |
| if (a == b) { |
| return 0; |
| } |
| |
| /* infinities */ |
| if (mpd_isinfinite(a)) { |
| if (mpd_isinfinite(b)) { |
| return 0; |
| } |
| return 1; |
| } |
| if (mpd_isinfinite(b)) { |
| return -1; |
| } |
| |
| /* zeros */ |
| if (mpd_iszerocoeff(a)) { |
| if (mpd_iszerocoeff(b)) { |
| return 0; |
| } |
| return -1; |
| } |
| if (mpd_iszerocoeff(b)) { |
| return 1; |
| } |
| |
| /* different adjusted exponents */ |
| adjexp_a = mpd_adjexp(a); |
| adjexp_b = mpd_adjexp(b); |
| if (adjexp_a != adjexp_b) { |
| if (adjexp_a < adjexp_b) { |
| return -1; |
| } |
| return 1; |
| } |
| |
| /* same adjusted exponents */ |
| return _mpd_cmp_same_adjexp(a, b); |
| } |
| |
| /* Compare two values and return an integer result. */ |
| int |
| mpd_qcmp(const mpd_t *a, const mpd_t *b, uint32_t *status) |
| { |
| if (mpd_isspecial(a) || mpd_isspecial(b)) { |
| if (mpd_isnan(a) || mpd_isnan(b)) { |
| *status |= MPD_Invalid_operation; |
| return INT_MAX; |
| } |
| } |
| |
| return _mpd_cmp(a, b); |
| } |
| |
| /* |
| * Compare a and b, convert the usual integer result to a decimal and |
| * store it in 'result'. For convenience, the integer result of the comparison |
| * is returned. Comparisons involving NaNs return NaN/INT_MAX. |
| */ |
| int |
| mpd_qcompare(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| int c; |
| |
| if (mpd_isspecial(a) || mpd_isspecial(b)) { |
| if (mpd_qcheck_nans(result, a, b, ctx, status)) { |
| return INT_MAX; |
| } |
| } |
| |
| c = _mpd_cmp(a, b); |
| _settriple(result, (c < 0), (c != 0), 0); |
| return c; |
| } |
| |
| /* Same as mpd_compare(), but signal for all NaNs, i.e. also for quiet NaNs. */ |
| int |
| mpd_qcompare_signal(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| int c; |
| |
| if (mpd_isspecial(a) || mpd_isspecial(b)) { |
| if (mpd_qcheck_nans(result, a, b, ctx, status)) { |
| *status |= MPD_Invalid_operation; |
| return INT_MAX; |
| } |
| } |
| |
| c = _mpd_cmp(a, b); |
| _settriple(result, (c < 0), (c != 0), 0); |
| return c; |
| } |
| |
| /* Compare the operands using a total order. */ |
| int |
| mpd_cmp_total(const mpd_t *a, const mpd_t *b) |
| { |
| mpd_t aa, bb; |
| int nan_a, nan_b; |
| int c; |
| |
| if (mpd_sign(a) != mpd_sign(b)) { |
| return mpd_sign(b) - mpd_sign(a); |
| } |
| |
| |
| if (mpd_isnan(a)) { |
| c = 1; |
| if (mpd_isnan(b)) { |
| nan_a = (mpd_isqnan(a)) ? 1 : 0; |
| nan_b = (mpd_isqnan(b)) ? 1 : 0; |
| if (nan_b == nan_a) { |
| if (a->len > 0 && b->len > 0) { |
| _mpd_copy_shared(&aa, a); |
| _mpd_copy_shared(&bb, b); |
| aa.exp = bb.exp = 0; |
| /* compare payload */ |
| c = _mpd_cmp_abs(&aa, &bb); |
| } |
| else { |
| c = (a->len > 0) - (b->len > 0); |
| } |
| } |
| else { |
| c = nan_a - nan_b; |
| } |
| } |
| } |
| else if (mpd_isnan(b)) { |
| c = -1; |
| } |
| else { |
| c = _mpd_cmp_abs(a, b); |
| if (c == 0 && a->exp != b->exp) { |
| c = (a->exp < b->exp) ? -1 : 1; |
| } |
| } |
| |
| return c * mpd_arith_sign(a); |
| } |
| |
| /* |
| * Compare a and b according to a total order, convert the usual integer result |
| * to a decimal and store it in 'result'. For convenience, the integer result |
| * of the comparison is returned. |
| */ |
| int |
| mpd_compare_total(mpd_t *result, const mpd_t *a, const mpd_t *b) |
| { |
| int c; |
| |
| c = mpd_cmp_total(a, b); |
| _settriple(result, (c < 0), (c != 0), 0); |
| return c; |
| } |
| |
| /* Compare the magnitude of the operands using a total order. */ |
| int |
| mpd_cmp_total_mag(const mpd_t *a, const mpd_t *b) |
| { |
| mpd_t aa, bb; |
| |
| _mpd_copy_shared(&aa, a); |
| _mpd_copy_shared(&bb, b); |
| |
| mpd_set_positive(&aa); |
| mpd_set_positive(&bb); |
| |
| return mpd_cmp_total(&aa, &bb); |
| } |
| |
| /* |
| * Compare the magnitude of a and b according to a total order, convert the |
| * the usual integer result to a decimal and store it in 'result'. |
| * For convenience, the integer result of the comparison is returned. |
| */ |
| int |
| mpd_compare_total_mag(mpd_t *result, const mpd_t *a, const mpd_t *b) |
| { |
| int c; |
| |
| c = mpd_cmp_total_mag(a, b); |
| _settriple(result, (c < 0), (c != 0), 0); |
| return c; |
| } |
| |
| /* Determine an ordering for operands that are numerically equal. */ |
| static inline int |
| _mpd_cmp_numequal(const mpd_t *a, const mpd_t *b) |
| { |
| int sign_a, sign_b; |
| int c; |
| |
| sign_a = mpd_sign(a); |
| sign_b = mpd_sign(b); |
| if (sign_a != sign_b) { |
| c = sign_b - sign_a; |
| } |
| else { |
| c = (a->exp < b->exp) ? -1 : 1; |
| c *= mpd_arith_sign(a); |
| } |
| |
| return c; |
| } |
| |
| |
| /******************************************************************************/ |
| /* Shifting the coefficient */ |
| /******************************************************************************/ |
| |
| /* |
| * Shift the coefficient of the operand to the left, no check for specials. |
| * Both operands may be the same pointer. If the result length has to be |
| * increased, mpd_qresize() might fail with MPD_Malloc_error. |
| */ |
| int |
| mpd_qshiftl(mpd_t *result, const mpd_t *a, mpd_ssize_t n, uint32_t *status) |
| { |
| mpd_ssize_t size; |
| |
| assert(!mpd_isspecial(a)); |
| assert(n >= 0); |
| |
| if (mpd_iszerocoeff(a) || n == 0) { |
| return mpd_qcopy(result, a, status); |
| } |
| |
| size = mpd_digits_to_size(a->digits+n); |
| if (!mpd_qresize(result, size, status)) { |
| return 0; /* result is NaN */ |
| } |
| |
| _mpd_baseshiftl(result->data, a->data, size, a->len, n); |
| |
| mpd_copy_flags(result, a); |
| result->exp = a->exp; |
| result->digits = a->digits+n; |
| result->len = size; |
| |
| return 1; |
| } |
| |
| /* Determine the rounding indicator if all digits of the coefficient are shifted |
| * out of the picture. */ |
| static mpd_uint_t |
| _mpd_get_rnd(const mpd_uint_t *data, mpd_ssize_t len, int use_msd) |
| { |
| mpd_uint_t rnd = 0, rest = 0, word; |
| |
| word = data[len-1]; |
| /* special treatment for the most significant digit if shift == digits */ |
| if (use_msd) { |
| _mpd_divmod_pow10(&rnd, &rest, word, mpd_word_digits(word)-1); |
| if (len > 1 && rest == 0) { |
| rest = !_mpd_isallzero(data, len-1); |
| } |
| } |
| else { |
| rest = !_mpd_isallzero(data, len); |
| } |
| |
| return (rnd == 0 || rnd == 5) ? rnd + !!rest : rnd; |
| } |
| |
| /* |
| * Same as mpd_qshiftr(), but 'result' is an mpd_t with a static coefficient. |
| * It is the caller's responsibility to ensure that the coefficient is big |
| * enough. The function cannot fail. |
| */ |
| static mpd_uint_t |
| mpd_qsshiftr(mpd_t *result, const mpd_t *a, mpd_ssize_t n) |
| { |
| mpd_uint_t rnd; |
| mpd_ssize_t size; |
| |
| assert(!mpd_isspecial(a)); |
| assert(n >= 0); |
| |
| if (mpd_iszerocoeff(a) || n == 0) { |
| mpd_qcopy_static(result, a); |
| return 0; |
| } |
| |
| if (n >= a->digits) { |
| rnd = _mpd_get_rnd(a->data, a->len, (n==a->digits)); |
| mpd_zerocoeff(result); |
| } |
| else { |
| result->digits = a->digits-n; |
| size = mpd_digits_to_size(result->digits); |
| rnd = _mpd_baseshiftr(result->data, a->data, a->len, n); |
| result->len = size; |
| } |
| |
| mpd_copy_flags(result, a); |
| result->exp = a->exp; |
| |
| return rnd; |
| } |
| |
| /* |
| * Inplace shift of the coefficient to the right, no check for specials. |
| * Returns the rounding indicator for mpd_rnd_incr(). |
| * The function cannot fail. |
| */ |
| mpd_uint_t |
| mpd_qshiftr_inplace(mpd_t *result, mpd_ssize_t n) |
| { |
| uint32_t dummy; |
| mpd_uint_t rnd; |
| mpd_ssize_t size; |
| |
| assert(!mpd_isspecial(result)); |
| assert(n >= 0); |
| |
| if (mpd_iszerocoeff(result) || n == 0) { |
| return 0; |
| } |
| |
| if (n >= result->digits) { |
| rnd = _mpd_get_rnd(result->data, result->len, (n==result->digits)); |
| mpd_zerocoeff(result); |
| } |
| else { |
| rnd = _mpd_baseshiftr(result->data, result->data, result->len, n); |
| result->digits -= n; |
| size = mpd_digits_to_size(result->digits); |
| /* reducing the size cannot fail */ |
| mpd_qresize(result, size, &dummy); |
| result->len = size; |
| } |
| |
| return rnd; |
| } |
| |
| /* |
| * Shift the coefficient of the operand to the right, no check for specials. |
| * Both operands may be the same pointer. Returns the rounding indicator to |
| * be used by mpd_rnd_incr(). If the result length has to be increased, |
| * mpd_qcopy() or mpd_qresize() might fail with MPD_Malloc_error. In those |
| * cases, MPD_UINT_MAX is returned. |
| */ |
| mpd_uint_t |
| mpd_qshiftr(mpd_t *result, const mpd_t *a, mpd_ssize_t n, uint32_t *status) |
| { |
| mpd_uint_t rnd; |
| mpd_ssize_t size; |
| |
| assert(!mpd_isspecial(a)); |
| assert(n >= 0); |
| |
| if (mpd_iszerocoeff(a) || n == 0) { |
| if (!mpd_qcopy(result, a, status)) { |
| return MPD_UINT_MAX; |
| } |
| return 0; |
| } |
| |
| if (n >= a->digits) { |
| rnd = _mpd_get_rnd(a->data, a->len, (n==a->digits)); |
| mpd_zerocoeff(result); |
| } |
| else { |
| result->digits = a->digits-n; |
| size = mpd_digits_to_size(result->digits); |
| if (result == a) { |
| rnd = _mpd_baseshiftr(result->data, a->data, a->len, n); |
| /* reducing the size cannot fail */ |
| mpd_qresize(result, size, status); |
| } |
| else { |
| if (!mpd_qresize(result, size, status)) { |
| return MPD_UINT_MAX; |
| } |
| rnd = _mpd_baseshiftr(result->data, a->data, a->len, n); |
| } |
| result->len = size; |
| } |
| |
| mpd_copy_flags(result, a); |
| result->exp = a->exp; |
| |
| return rnd; |
| } |
| |
| |
| /******************************************************************************/ |
| /* Miscellaneous operations */ |
| /******************************************************************************/ |
| |
| /* Logical And */ |
| void |
| mpd_qand(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| const mpd_t *big = a, *small = b; |
| mpd_uint_t x, y, z, xbit, ybit; |
| int k, mswdigits; |
| mpd_ssize_t i; |
| |
| if (mpd_isspecial(a) || mpd_isspecial(b) || |
| mpd_isnegative(a) || mpd_isnegative(b) || |
| a->exp != 0 || b->exp != 0) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| if (b->digits > a->digits) { |
| big = b; |
| small = a; |
| } |
| if (!mpd_qresize(result, big->len, status)) { |
| return; |
| } |
| |
| |
| /* full words */ |
| for (i = 0; i < small->len-1; i++) { |
| x = small->data[i]; |
| y = big->data[i]; |
| z = 0; |
| for (k = 0; k < MPD_RDIGITS; k++) { |
| xbit = x % 10; |
| x /= 10; |
| ybit = y % 10; |
| y /= 10; |
| if (xbit > 1 || ybit > 1) { |
| goto invalid_operation; |
| } |
| z += (xbit&ybit) ? mpd_pow10[k] : 0; |
| } |
| result->data[i] = z; |
| } |
| /* most significant word of small */ |
| x = small->data[i]; |
| y = big->data[i]; |
| z = 0; |
| mswdigits = mpd_word_digits(x); |
| for (k = 0; k < mswdigits; k++) { |
| xbit = x % 10; |
| x /= 10; |
| ybit = y % 10; |
| y /= 10; |
| if (xbit > 1 || ybit > 1) { |
| goto invalid_operation; |
| } |
| z += (xbit&ybit) ? mpd_pow10[k] : 0; |
| } |
| result->data[i++] = z; |
| |
| /* scan the rest of y for digits > 1 */ |
| for (; k < MPD_RDIGITS; k++) { |
| ybit = y % 10; |
| y /= 10; |
| if (ybit > 1) { |
| goto invalid_operation; |
| } |
| } |
| /* scan the rest of big for digits > 1 */ |
| for (; i < big->len; i++) { |
| y = big->data[i]; |
| for (k = 0; k < MPD_RDIGITS; k++) { |
| ybit = y % 10; |
| y /= 10; |
| if (ybit > 1) { |
| goto invalid_operation; |
| } |
| } |
| } |
| |
| mpd_clear_flags(result); |
| result->exp = 0; |
| result->len = _mpd_real_size(result->data, small->len); |
| mpd_qresize(result, result->len, status); |
| mpd_setdigits(result); |
| _mpd_cap(result, ctx); |
| return; |
| |
| invalid_operation: |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| } |
| |
| /* Class of an operand. Returns a pointer to the constant name. */ |
| const char * |
| mpd_class(const mpd_t *a, const mpd_context_t *ctx) |
| { |
| if (mpd_isnan(a)) { |
| if (mpd_isqnan(a)) |
| return "NaN"; |
| else |
| return "sNaN"; |
| } |
| else if (mpd_ispositive(a)) { |
| if (mpd_isinfinite(a)) |
| return "+Infinity"; |
| else if (mpd_iszero(a)) |
| return "+Zero"; |
| else if (mpd_isnormal(a, ctx)) |
| return "+Normal"; |
| else |
| return "+Subnormal"; |
| } |
| else { |
| if (mpd_isinfinite(a)) |
| return "-Infinity"; |
| else if (mpd_iszero(a)) |
| return "-Zero"; |
| else if (mpd_isnormal(a, ctx)) |
| return "-Normal"; |
| else |
| return "-Subnormal"; |
| } |
| } |
| |
| /* Logical Xor */ |
| void |
| mpd_qinvert(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_uint_t x, z, xbit; |
| mpd_ssize_t i, digits, len; |
| mpd_ssize_t q, r; |
| int k; |
| |
| if (mpd_isspecial(a) || mpd_isnegative(a) || a->exp != 0) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| |
| digits = (a->digits < ctx->prec) ? ctx->prec : a->digits; |
| _mpd_idiv_word(&q, &r, digits, MPD_RDIGITS); |
| len = (r == 0) ? q : q+1; |
| if (!mpd_qresize(result, len, status)) { |
| return; |
| } |
| |
| for (i = 0; i < len; i++) { |
| x = (i < a->len) ? a->data[i] : 0; |
| z = 0; |
| for (k = 0; k < MPD_RDIGITS; k++) { |
| xbit = x % 10; |
| x /= 10; |
| if (xbit > 1) { |
| goto invalid_operation; |
| } |
| z += !xbit ? mpd_pow10[k] : 0; |
| } |
| result->data[i] = z; |
| } |
| |
| mpd_clear_flags(result); |
| result->exp = 0; |
| result->len = _mpd_real_size(result->data, len); |
| mpd_qresize(result, result->len, status); |
| mpd_setdigits(result); |
| _mpd_cap(result, ctx); |
| return; |
| |
| invalid_operation: |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| } |
| |
| /* Exponent of the magnitude of the most significant digit of the operand. */ |
| void |
| mpd_qlogb(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| if (mpd_isspecial(a)) { |
| if (mpd_qcheck_nan(result, a, ctx, status)) { |
| return; |
| } |
| mpd_setspecial(result, MPD_POS, MPD_INF); |
| } |
| else if (mpd_iszerocoeff(a)) { |
| mpd_setspecial(result, MPD_NEG, MPD_INF); |
| *status |= MPD_Division_by_zero; |
| } |
| else { |
| mpd_qset_ssize(result, mpd_adjexp(a), ctx, status); |
| } |
| } |
| |
| /* Logical Or */ |
| void |
| mpd_qor(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| const mpd_t *big = a, *small = b; |
| mpd_uint_t x, y, z, xbit, ybit; |
| int k, mswdigits; |
| mpd_ssize_t i; |
| |
| if (mpd_isspecial(a) || mpd_isspecial(b) || |
| mpd_isnegative(a) || mpd_isnegative(b) || |
| a->exp != 0 || b->exp != 0) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| if (b->digits > a->digits) { |
| big = b; |
| small = a; |
| } |
| if (!mpd_qresize(result, big->len, status)) { |
| return; |
| } |
| |
| |
| /* full words */ |
| for (i = 0; i < small->len-1; i++) { |
| x = small->data[i]; |
| y = big->data[i]; |
| z = 0; |
| for (k = 0; k < MPD_RDIGITS; k++) { |
| xbit = x % 10; |
| x /= 10; |
| ybit = y % 10; |
| y /= 10; |
| if (xbit > 1 || ybit > 1) { |
| goto invalid_operation; |
| } |
| z += (xbit|ybit) ? mpd_pow10[k] : 0; |
| } |
| result->data[i] = z; |
| } |
| /* most significant word of small */ |
| x = small->data[i]; |
| y = big->data[i]; |
| z = 0; |
| mswdigits = mpd_word_digits(x); |
| for (k = 0; k < mswdigits; k++) { |
| xbit = x % 10; |
| x /= 10; |
| ybit = y % 10; |
| y /= 10; |
| if (xbit > 1 || ybit > 1) { |
| goto invalid_operation; |
| } |
| z += (xbit|ybit) ? mpd_pow10[k] : 0; |
| } |
| |
| /* scan for digits > 1 and copy the rest of y */ |
| for (; k < MPD_RDIGITS; k++) { |
| ybit = y % 10; |
| y /= 10; |
| if (ybit > 1) { |
| goto invalid_operation; |
| } |
| z += ybit*mpd_pow10[k]; |
| } |
| result->data[i++] = z; |
| /* scan for digits > 1 and copy the rest of big */ |
| for (; i < big->len; i++) { |
| y = big->data[i]; |
| for (k = 0; k < MPD_RDIGITS; k++) { |
| ybit = y % 10; |
| y /= 10; |
| if (ybit > 1) { |
| goto invalid_operation; |
| } |
| } |
| result->data[i] = big->data[i]; |
| } |
| |
| mpd_clear_flags(result); |
| result->exp = 0; |
| result->len = _mpd_real_size(result->data, big->len); |
| mpd_qresize(result, result->len, status); |
| mpd_setdigits(result); |
| _mpd_cap(result, ctx); |
| return; |
| |
| invalid_operation: |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| } |
| |
| /* |
| * Rotate the coefficient of 'a' by 'b' digits. 'b' must be an integer with |
| * exponent 0. |
| */ |
| void |
| mpd_qrotate(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| uint32_t workstatus = 0; |
| MPD_NEW_STATIC(tmp,0,0,0,0); |
| MPD_NEW_STATIC(big,0,0,0,0); |
| MPD_NEW_STATIC(small,0,0,0,0); |
| mpd_ssize_t n, lshift, rshift; |
| |
| if (mpd_isspecial(a) || mpd_isspecial(b)) { |
| if (mpd_qcheck_nans(result, a, b, ctx, status)) { |
| return; |
| } |
| } |
| if (b->exp != 0 || mpd_isinfinite(b)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| |
| n = mpd_qget_ssize(b, &workstatus); |
| if (workstatus&MPD_Invalid_operation) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| if (n > ctx->prec || n < -ctx->prec) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| if (mpd_isinfinite(a)) { |
| mpd_qcopy(result, a, status); |
| return; |
| } |
| |
| if (n >= 0) { |
| lshift = n; |
| rshift = ctx->prec-n; |
| } |
| else { |
| lshift = ctx->prec+n; |
| rshift = -n; |
| } |
| |
| if (a->digits > ctx->prec) { |
| if (!mpd_qcopy(&tmp, a, status)) { |
| mpd_seterror(result, MPD_Malloc_error, status); |
| goto finish; |
| } |
| _mpd_cap(&tmp, ctx); |
| a = &tmp; |
| } |
| |
| if (!mpd_qshiftl(&big, a, lshift, status)) { |
| mpd_seterror(result, MPD_Malloc_error, status); |
| goto finish; |
| } |
| _mpd_cap(&big, ctx); |
| |
| if (mpd_qshiftr(&small, a, rshift, status) == MPD_UINT_MAX) { |
| mpd_seterror(result, MPD_Malloc_error, status); |
| goto finish; |
| } |
| _mpd_qadd(result, &big, &small, ctx, status); |
| |
| |
| finish: |
| mpd_del(&tmp); |
| mpd_del(&big); |
| mpd_del(&small); |
| } |
| |
| /* |
| * b must be an integer with exponent 0 and in the range +-2*(emax + prec). |
| * XXX: In my opinion +-(2*emax + prec) would be more sensible. |
| * The result is a with the value of b added to its exponent. |
| */ |
| void |
| mpd_qscaleb(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| uint32_t workstatus = 0; |
| mpd_uint_t n, maxjump; |
| #ifndef LEGACY_COMPILER |
| int64_t exp; |
| #else |
| mpd_uint_t x; |
| int x_sign, n_sign; |
| mpd_ssize_t exp; |
| #endif |
| |
| if (mpd_isspecial(a) || mpd_isspecial(b)) { |
| if (mpd_qcheck_nans(result, a, b, ctx, status)) { |
| return; |
| } |
| } |
| if (b->exp != 0 || mpd_isinfinite(b)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| |
| n = mpd_qabs_uint(b, &workstatus); |
| /* the spec demands this */ |
| maxjump = 2 * (mpd_uint_t)(ctx->emax + ctx->prec); |
| |
| if (n > maxjump || workstatus&MPD_Invalid_operation) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| if (mpd_isinfinite(a)) { |
| mpd_qcopy(result, a, status); |
| return; |
| } |
| |
| #ifndef LEGACY_COMPILER |
| exp = a->exp + (int64_t)n * mpd_arith_sign(b); |
| exp = (exp > MPD_EXP_INF) ? MPD_EXP_INF : exp; |
| exp = (exp < MPD_EXP_CLAMP) ? MPD_EXP_CLAMP : exp; |
| #else |
| x = (a->exp < 0) ? -a->exp : a->exp; |
| x_sign = (a->exp < 0) ? 1 : 0; |
| n_sign = mpd_isnegative(b) ? 1 : 0; |
| |
| if (x_sign == n_sign) { |
| x = x + n; |
| if (x < n) x = MPD_UINT_MAX; |
| } |
| else { |
| x_sign = (x >= n) ? x_sign : n_sign; |
| x = (x >= n) ? x - n : n - x; |
| } |
| if (!x_sign && x > MPD_EXP_INF) x = MPD_EXP_INF; |
| if (x_sign && x > -MPD_EXP_CLAMP) x = -MPD_EXP_CLAMP; |
| exp = x_sign ? -((mpd_ssize_t)x) : (mpd_ssize_t)x; |
| #endif |
| |
| mpd_qcopy(result, a, status); |
| result->exp = (mpd_ssize_t)exp; |
| |
| mpd_qfinalize(result, ctx, status); |
| } |
| |
| /* |
| * Shift the coefficient by n digits, positive n is a left shift. In the case |
| * of a left shift, the result is decapitated to fit the context precision. If |
| * you don't want that, use mpd_shiftl(). |
| */ |
| void |
| mpd_qshiftn(mpd_t *result, const mpd_t *a, mpd_ssize_t n, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| if (mpd_isspecial(a)) { |
| if (mpd_qcheck_nan(result, a, ctx, status)) { |
| return; |
| } |
| mpd_qcopy(result, a, status); |
| return; |
| } |
| |
| if (n >= 0 && n <= ctx->prec) { |
| mpd_qshiftl(result, a, n, status); |
| _mpd_cap(result, ctx); |
| } |
| else if (n < 0 && n >= -ctx->prec) { |
| if (!mpd_qcopy(result, a, status)) { |
| return; |
| } |
| _mpd_cap(result, ctx); |
| mpd_qshiftr_inplace(result, -n); |
| } |
| else { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| } |
| } |
| |
| /* |
| * Same as mpd_shiftn(), but the shift is specified by the decimal b, which |
| * must be an integer with a zero exponent. Infinities remain infinities. |
| */ |
| void |
| mpd_qshift(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| uint32_t workstatus = 0; |
| mpd_ssize_t n; |
| |
| if (mpd_isspecial(a) || mpd_isspecial(b)) { |
| if (mpd_qcheck_nans(result, a, b, ctx, status)) { |
| return; |
| } |
| } |
| if (b->exp != 0 || mpd_isinfinite(b)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| |
| n = mpd_qget_ssize(b, &workstatus); |
| if (workstatus&MPD_Invalid_operation) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| if (n > ctx->prec || n < -ctx->prec) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| if (mpd_isinfinite(a)) { |
| mpd_qcopy(result, a, status); |
| return; |
| } |
| |
| if (n >= 0) { |
| mpd_qshiftl(result, a, n, status); |
| _mpd_cap(result, ctx); |
| } |
| else { |
| if (!mpd_qcopy(result, a, status)) { |
| return; |
| } |
| _mpd_cap(result, ctx); |
| mpd_qshiftr_inplace(result, -n); |
| } |
| } |
| |
| /* Logical Xor */ |
| void |
| mpd_qxor(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| const mpd_t *big = a, *small = b; |
| mpd_uint_t x, y, z, xbit, ybit; |
| int k, mswdigits; |
| mpd_ssize_t i; |
| |
| if (mpd_isspecial(a) || mpd_isspecial(b) || |
| mpd_isnegative(a) || mpd_isnegative(b) || |
| a->exp != 0 || b->exp != 0) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| if (b->digits > a->digits) { |
| big = b; |
| small = a; |
| } |
| if (!mpd_qresize(result, big->len, status)) { |
| return; |
| } |
| |
| |
| /* full words */ |
| for (i = 0; i < small->len-1; i++) { |
| x = small->data[i]; |
| y = big->data[i]; |
| z = 0; |
| for (k = 0; k < MPD_RDIGITS; k++) { |
| xbit = x % 10; |
| x /= 10; |
| ybit = y % 10; |
| y /= 10; |
| if (xbit > 1 || ybit > 1) { |
| goto invalid_operation; |
| } |
| z += (xbit^ybit) ? mpd_pow10[k] : 0; |
| } |
| result->data[i] = z; |
| } |
| /* most significant word of small */ |
| x = small->data[i]; |
| y = big->data[i]; |
| z = 0; |
| mswdigits = mpd_word_digits(x); |
| for (k = 0; k < mswdigits; k++) { |
| xbit = x % 10; |
| x /= 10; |
| ybit = y % 10; |
| y /= 10; |
| if (xbit > 1 || ybit > 1) { |
| goto invalid_operation; |
| } |
| z += (xbit^ybit) ? mpd_pow10[k] : 0; |
| } |
| |
| /* scan for digits > 1 and copy the rest of y */ |
| for (; k < MPD_RDIGITS; k++) { |
| ybit = y % 10; |
| y /= 10; |
| if (ybit > 1) { |
| goto invalid_operation; |
| } |
| z += ybit*mpd_pow10[k]; |
| } |
| result->data[i++] = z; |
| /* scan for digits > 1 and copy the rest of big */ |
| for (; i < big->len; i++) { |
| y = big->data[i]; |
| for (k = 0; k < MPD_RDIGITS; k++) { |
| ybit = y % 10; |
| y /= 10; |
| if (ybit > 1) { |
| goto invalid_operation; |
| } |
| } |
| result->data[i] = big->data[i]; |
| } |
| |
| mpd_clear_flags(result); |
| result->exp = 0; |
| result->len = _mpd_real_size(result->data, big->len); |
| mpd_qresize(result, result->len, status); |
| mpd_setdigits(result); |
| _mpd_cap(result, ctx); |
| return; |
| |
| invalid_operation: |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| } |
| |
| |
| /******************************************************************************/ |
| /* Arithmetic operations */ |
| /******************************************************************************/ |
| |
| /* |
| * The absolute value of a. If a is negative, the result is the same |
| * as the result of the minus operation. Otherwise, the result is the |
| * result of the plus operation. |
| */ |
| void |
| mpd_qabs(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| if (mpd_isspecial(a)) { |
| if (mpd_qcheck_nan(result, a, ctx, status)) { |
| return; |
| } |
| } |
| |
| if (mpd_isnegative(a)) { |
| mpd_qminus(result, a, ctx, status); |
| } |
| else { |
| mpd_qplus(result, a, ctx, status); |
| } |
| } |
| |
| static inline void |
| _mpd_ptrswap(mpd_t **a, mpd_t **b) |
| { |
| mpd_t *t = *a; |
| *a = *b; |
| *b = t; |
| } |
| |
| /* Add or subtract infinities. */ |
| static void |
| _mpd_qaddsub_inf(mpd_t *result, const mpd_t *a, const mpd_t *b, uint8_t sign_b, |
| uint32_t *status) |
| { |
| if (mpd_isinfinite(a)) { |
| if (mpd_sign(a) != sign_b && mpd_isinfinite(b)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| } |
| else { |
| mpd_setspecial(result, mpd_sign(a), MPD_INF); |
| } |
| return; |
| } |
| assert(mpd_isinfinite(b)); |
| mpd_setspecial(result, sign_b, MPD_INF); |
| } |
| |
| /* Add or subtract non-special numbers. */ |
| static void |
| _mpd_qaddsub(mpd_t *result, const mpd_t *a, const mpd_t *b, uint8_t sign_b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_t *big, *small; |
| MPD_NEW_STATIC(big_aligned,0,0,0,0); |
| MPD_NEW_CONST(tiny,0,0,1,1,1,1); |
| mpd_uint_t carry; |
| mpd_ssize_t newsize, shift; |
| mpd_ssize_t exp, i; |
| int swap = 0; |
| |
| |
| /* compare exponents */ |
| big = (mpd_t *)a; small = (mpd_t *)b; |
| if (big->exp != small->exp) { |
| if (small->exp > big->exp) { |
| _mpd_ptrswap(&big, &small); |
| swap++; |
| } |
| /* align the coefficients */ |
| if (!mpd_iszerocoeff(big)) { |
| exp = big->exp - 1; |
| exp += (big->digits > ctx->prec) ? 0 : big->digits-ctx->prec-1; |
| if (mpd_adjexp(small) < exp) { |
| /* |
| * Avoid huge shifts by substituting a value for small that is |
| * guaranteed to produce the same results. |
| * |
| * adjexp(small) < exp if and only if: |
| * |
| * bdigits <= prec AND |
| * bdigits+shift >= prec+2+sdigits AND |
| * exp = bexp+bdigits-prec-2 |
| * |
| * 1234567000000000 -> bdigits + shift |
| * ----------XX1234 -> sdigits |
| * ----------X1 -> tiny-digits |
| * |- prec -| |
| * |
| * OR |
| * |
| * bdigits > prec AND |
| * shift > sdigits AND |
| * exp = bexp-1 |
| * |
| * 1234567892100000 -> bdigits + shift |
| * ----------XX1234 -> sdigits |
| * ----------X1 -> tiny-digits |
| * |- prec -| |
| * |
| * If tiny is zero, adding or subtracting is a no-op. |
| * Otherwise, adding tiny generates a non-zero digit either |
| * below the rounding digit or the least significant digit |
| * of big. When subtracting, tiny is in the same position as |
| * the carry that would be generated by subtracting sdigits. |
| */ |
| mpd_copy_flags(&tiny, small); |
| tiny.exp = exp; |
| tiny.digits = 1; |
| tiny.len = 1; |
| tiny.data[0] = mpd_iszerocoeff(small) ? 0 : 1; |
| small = &tiny; |
| } |
| /* This cannot wrap: the difference is positive and <= maxprec */ |
| shift = big->exp - small->exp; |
| if (!mpd_qshiftl(&big_aligned, big, shift, status)) { |
| mpd_seterror(result, MPD_Malloc_error, status); |
| goto finish; |
| } |
| big = &big_aligned; |
| } |
| } |
| result->exp = small->exp; |
| |
| |
| /* compare length of coefficients */ |
| if (big->len < small->len) { |
| _mpd_ptrswap(&big, &small); |
| swap++; |
| } |
| |
| newsize = big->len; |
| if (!mpd_qresize(result, newsize, status)) { |
| goto finish; |
| } |
| |
| if (mpd_sign(a) == sign_b) { |
| |
| carry = _mpd_baseadd(result->data, big->data, small->data, |
| big->len, small->len); |
| |
| if (carry) { |
| newsize = big->len + 1; |
| if (!mpd_qresize(result, newsize, status)) { |
| goto finish; |
| } |
| result->data[newsize-1] = carry; |
| } |
| |
| result->len = newsize; |
| mpd_set_flags(result, sign_b); |
| } |
| else { |
| if (big->len == small->len) { |
| for (i=big->len-1; i >= 0; --i) { |
| if (big->data[i] != small->data[i]) { |
| if (big->data[i] < small->data[i]) { |
| _mpd_ptrswap(&big, &small); |
| swap++; |
| } |
| break; |
| } |
| } |
| } |
| |
| _mpd_basesub(result->data, big->data, small->data, |
| big->len, small->len); |
| newsize = _mpd_real_size(result->data, big->len); |
| /* resize to smaller cannot fail */ |
| (void)mpd_qresize(result, newsize, status); |
| |
| result->len = newsize; |
| sign_b = (swap & 1) ? sign_b : mpd_sign(a); |
| mpd_set_flags(result, sign_b); |
| |
| if (mpd_iszerocoeff(result)) { |
| mpd_set_positive(result); |
| if (ctx->round == MPD_ROUND_FLOOR) { |
| mpd_set_negative(result); |
| } |
| } |
| } |
| |
| mpd_setdigits(result); |
| |
| finish: |
| mpd_del(&big_aligned); |
| } |
| |
| /* Add a and b. No specials, no finalizing. */ |
| static void |
| _mpd_qadd(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| _mpd_qaddsub(result, a, b, mpd_sign(b), ctx, status); |
| } |
| |
| /* Subtract b from a. No specials, no finalizing. */ |
| static void |
| _mpd_qsub(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| _mpd_qaddsub(result, a, b, !mpd_sign(b), ctx, status); |
| } |
| |
| /* Add a and b. */ |
| void |
| mpd_qadd(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| if (mpd_isspecial(a) || mpd_isspecial(b)) { |
| if (mpd_qcheck_nans(result, a, b, ctx, status)) { |
| return; |
| } |
| _mpd_qaddsub_inf(result, a, b, mpd_sign(b), status); |
| return; |
| } |
| |
| _mpd_qaddsub(result, a, b, mpd_sign(b), ctx, status); |
| mpd_qfinalize(result, ctx, status); |
| } |
| |
| /* Add a and b. Set NaN/Invalid_operation if the result is inexact. */ |
| static void |
| _mpd_qadd_exact(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| uint32_t workstatus = 0; |
| |
| mpd_qadd(result, a, b, ctx, &workstatus); |
| *status |= workstatus; |
| if (workstatus & (MPD_Inexact|MPD_Rounded|MPD_Clamped)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| } |
| } |
| |
| /* Subtract b from a. */ |
| void |
| mpd_qsub(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| if (mpd_isspecial(a) || mpd_isspecial(b)) { |
| if (mpd_qcheck_nans(result, a, b, ctx, status)) { |
| return; |
| } |
| _mpd_qaddsub_inf(result, a, b, !mpd_sign(b), status); |
| return; |
| } |
| |
| _mpd_qaddsub(result, a, b, !mpd_sign(b), ctx, status); |
| mpd_qfinalize(result, ctx, status); |
| } |
| |
| /* Subtract b from a. Set NaN/Invalid_operation if the result is inexact. */ |
| static void |
| _mpd_qsub_exact(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| uint32_t workstatus = 0; |
| |
| mpd_qsub(result, a, b, ctx, &workstatus); |
| *status |= workstatus; |
| if (workstatus & (MPD_Inexact|MPD_Rounded|MPD_Clamped)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| } |
| } |
| |
| /* Add decimal and mpd_ssize_t. */ |
| void |
| mpd_qadd_ssize(mpd_t *result, const mpd_t *a, mpd_ssize_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_context_t maxcontext; |
| MPD_NEW_STATIC(bb,0,0,0,0); |
| |
| mpd_maxcontext(&maxcontext); |
| mpd_qsset_ssize(&bb, b, &maxcontext, status); |
| mpd_qadd(result, a, &bb, ctx, status); |
| mpd_del(&bb); |
| } |
| |
| /* Add decimal and mpd_uint_t. */ |
| void |
| mpd_qadd_uint(mpd_t *result, const mpd_t *a, mpd_uint_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_context_t maxcontext; |
| MPD_NEW_STATIC(bb,0,0,0,0); |
| |
| mpd_maxcontext(&maxcontext); |
| mpd_qsset_uint(&bb, b, &maxcontext, status); |
| mpd_qadd(result, a, &bb, ctx, status); |
| mpd_del(&bb); |
| } |
| |
| /* Subtract mpd_ssize_t from decimal. */ |
| void |
| mpd_qsub_ssize(mpd_t *result, const mpd_t *a, mpd_ssize_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_context_t maxcontext; |
| MPD_NEW_STATIC(bb,0,0,0,0); |
| |
| mpd_maxcontext(&maxcontext); |
| mpd_qsset_ssize(&bb, b, &maxcontext, status); |
| mpd_qsub(result, a, &bb, ctx, status); |
| mpd_del(&bb); |
| } |
| |
| /* Subtract mpd_uint_t from decimal. */ |
| void |
| mpd_qsub_uint(mpd_t *result, const mpd_t *a, mpd_uint_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_context_t maxcontext; |
| MPD_NEW_STATIC(bb,0,0,0,0); |
| |
| mpd_maxcontext(&maxcontext); |
| mpd_qsset_uint(&bb, b, &maxcontext, status); |
| mpd_qsub(result, a, &bb, ctx, status); |
| mpd_del(&bb); |
| } |
| |
| /* Add decimal and int32_t. */ |
| void |
| mpd_qadd_i32(mpd_t *result, const mpd_t *a, int32_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_qadd_ssize(result, a, b, ctx, status); |
| } |
| |
| /* Add decimal and uint32_t. */ |
| void |
| mpd_qadd_u32(mpd_t *result, const mpd_t *a, uint32_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_qadd_uint(result, a, b, ctx, status); |
| } |
| |
| #ifdef CONFIG_64 |
| /* Add decimal and int64_t. */ |
| void |
| mpd_qadd_i64(mpd_t *result, const mpd_t *a, int64_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_qadd_ssize(result, a, b, ctx, status); |
| } |
| |
| /* Add decimal and uint64_t. */ |
| void |
| mpd_qadd_u64(mpd_t *result, const mpd_t *a, uint64_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_qadd_uint(result, a, b, ctx, status); |
| } |
| #endif |
| |
| /* Subtract int32_t from decimal. */ |
| void |
| mpd_qsub_i32(mpd_t *result, const mpd_t *a, int32_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_qsub_ssize(result, a, b, ctx, status); |
| } |
| |
| /* Subtract uint32_t from decimal. */ |
| void |
| mpd_qsub_u32(mpd_t *result, const mpd_t *a, uint32_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_qsub_uint(result, a, b, ctx, status); |
| } |
| |
| #ifdef CONFIG_64 |
| /* Subtract int64_t from decimal. */ |
| void |
| mpd_qsub_i64(mpd_t *result, const mpd_t *a, int64_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_qsub_ssize(result, a, b, ctx, status); |
| } |
| |
| /* Subtract uint64_t from decimal. */ |
| void |
| mpd_qsub_u64(mpd_t *result, const mpd_t *a, uint64_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_qsub_uint(result, a, b, ctx, status); |
| } |
| #endif |
| |
| |
| /* Divide infinities. */ |
| static void |
| _mpd_qdiv_inf(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| if (mpd_isinfinite(a)) { |
| if (mpd_isinfinite(b)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| mpd_setspecial(result, mpd_sign(a)^mpd_sign(b), MPD_INF); |
| return; |
| } |
| assert(mpd_isinfinite(b)); |
| _settriple(result, mpd_sign(a)^mpd_sign(b), 0, mpd_etiny(ctx)); |
| *status |= MPD_Clamped; |
| } |
| |
| enum {NO_IDEAL_EXP, SET_IDEAL_EXP}; |
| /* Divide a by b. */ |
| static void |
| _mpd_qdiv(int action, mpd_t *q, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| MPD_NEW_STATIC(aligned,0,0,0,0); |
| mpd_uint_t ld; |
| mpd_ssize_t shift, exp, tz; |
| mpd_ssize_t newsize; |
| mpd_ssize_t ideal_exp; |
| mpd_uint_t rem; |
| uint8_t sign_a = mpd_sign(a); |
| uint8_t sign_b = mpd_sign(b); |
| |
| |
| if (mpd_isspecial(a) || mpd_isspecial(b)) { |
| if (mpd_qcheck_nans(q, a, b, ctx, status)) { |
| return; |
| } |
| _mpd_qdiv_inf(q, a, b, ctx, status); |
| return; |
| } |
| if (mpd_iszerocoeff(b)) { |
| if (mpd_iszerocoeff(a)) { |
| mpd_seterror(q, MPD_Division_undefined, status); |
| } |
| else { |
| mpd_setspecial(q, sign_a^sign_b, MPD_INF); |
| *status |= MPD_Division_by_zero; |
| } |
| return; |
| } |
| if (mpd_iszerocoeff(a)) { |
| exp = a->exp - b->exp; |
| _settriple(q, sign_a^sign_b, 0, exp); |
| mpd_qfinalize(q, ctx, status); |
| return; |
| } |
| |
| shift = (b->digits - a->digits) + ctx->prec + 1; |
| ideal_exp = a->exp - b->exp; |
| exp = ideal_exp - shift; |
| if (shift > 0) { |
| if (!mpd_qshiftl(&aligned, a, shift, status)) { |
| mpd_seterror(q, MPD_Malloc_error, status); |
| goto finish; |
| } |
| a = &aligned; |
| } |
| else if (shift < 0) { |
| shift = -shift; |
| if (!mpd_qshiftl(&aligned, b, shift, status)) { |
| mpd_seterror(q, MPD_Malloc_error, status); |
| goto finish; |
| } |
| b = &aligned; |
| } |
| |
| |
| newsize = a->len - b->len + 1; |
| if ((q != b && q != a) || (q == b && newsize > b->len)) { |
| if (!mpd_qresize(q, newsize, status)) { |
| mpd_seterror(q, MPD_Malloc_error, status); |
| goto finish; |
| } |
| } |
| |
| |
| if (b->len == 1) { |
| rem = _mpd_shortdiv(q->data, a->data, a->len, b->data[0]); |
| } |
| else if (b->len <= MPD_NEWTONDIV_CUTOFF) { |
| int ret = _mpd_basedivmod(q->data, NULL, a->data, b->data, |
| a->len, b->len); |
| if (ret < 0) { |
| mpd_seterror(q, MPD_Malloc_error, status); |
| goto finish; |
| } |
| rem = ret; |
| } |
| else { |
| MPD_NEW_STATIC(r,0,0,0,0); |
| _mpd_base_ndivmod(q, &r, a, b, status); |
| if (mpd_isspecial(q) || mpd_isspecial(&r)) { |
| mpd_setspecial(q, MPD_POS, MPD_NAN); |
| mpd_del(&r); |
| goto finish; |
| } |
| rem = !mpd_iszerocoeff(&r); |
| mpd_del(&r); |
| newsize = q->len; |
| } |
| |
| newsize = _mpd_real_size(q->data, newsize); |
| /* resize to smaller cannot fail */ |
| mpd_qresize(q, newsize, status); |
| mpd_set_flags(q, sign_a^sign_b); |
| q->len = newsize; |
| mpd_setdigits(q); |
| |
| shift = ideal_exp - exp; |
| if (rem) { |
| ld = mpd_lsd(q->data[0]); |
| if (ld == 0 || ld == 5) { |
| q->data[0] += 1; |
| } |
| } |
| else if (action == SET_IDEAL_EXP && shift > 0) { |
| tz = mpd_trail_zeros(q); |
| shift = (tz > shift) ? shift : tz; |
| mpd_qshiftr_inplace(q, shift); |
| exp += shift; |
| } |
| |
| q->exp = exp; |
| |
| |
| finish: |
| mpd_del(&aligned); |
| mpd_qfinalize(q, ctx, status); |
| } |
| |
| /* Divide a by b. */ |
| void |
| mpd_qdiv(mpd_t *q, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| _mpd_qdiv(SET_IDEAL_EXP, q, a, b, ctx, status); |
| } |
| |
| /* Internal function. */ |
| static void |
| _mpd_qdivmod(mpd_t *q, mpd_t *r, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| MPD_NEW_STATIC(aligned,0,0,0,0); |
| mpd_ssize_t qsize, rsize; |
| mpd_ssize_t ideal_exp, expdiff, shift; |
| uint8_t sign_a = mpd_sign(a); |
| uint8_t sign_ab = mpd_sign(a)^mpd_sign(b); |
| |
| |
| ideal_exp = (a->exp > b->exp) ? b->exp : a->exp; |
| if (mpd_iszerocoeff(a)) { |
| if (!mpd_qcopy(r, a, status)) { |
| goto nanresult; /* GCOV_NOT_REACHED */ |
| } |
| r->exp = ideal_exp; |
| _settriple(q, sign_ab, 0, 0); |
| return; |
| } |
| |
| expdiff = mpd_adjexp(a) - mpd_adjexp(b); |
| if (expdiff < 0) { |
| if (a->exp > b->exp) { |
| /* positive and less than b->digits - a->digits */ |
| shift = a->exp - b->exp; |
| if (!mpd_qshiftl(r, a, shift, status)) { |
| goto nanresult; |
| } |
| r->exp = ideal_exp; |
| } |
| else { |
| if (!mpd_qcopy(r, a, status)) { |
| goto nanresult; |
| } |
| } |
| _settriple(q, sign_ab, 0, 0); |
| return; |
| } |
| if (expdiff > ctx->prec) { |
| *status |= MPD_Division_impossible; |
| goto nanresult; |
| } |
| |
| |
| /* |
| * At this point we have: |
| * (1) 0 <= a->exp + a->digits - b->exp - b->digits <= prec |
| * (2) a->exp - b->exp >= b->digits - a->digits |
| * (3) a->exp - b->exp <= prec + b->digits - a->digits |
| */ |
| if (a->exp != b->exp) { |
| shift = a->exp - b->exp; |
| if (shift > 0) { |
| /* by (3), after the shift a->digits <= prec + b->digits */ |
| if (!mpd_qshiftl(&aligned, a, shift, status)) { |
| goto nanresult; |
| } |
| a = &aligned; |
| } |
| else { |
| shift = -shift; |
| /* by (2), after the shift b->digits <= a->digits */ |
| if (!mpd_qshiftl(&aligned, b, shift, status)) { |
| goto nanresult; |
| } |
| b = &aligned; |
| } |
| } |
| |
| |
| qsize = a->len - b->len + 1; |
| if (!(q == a && qsize < a->len) && !(q == b && qsize < b->len)) { |
| if (!mpd_qresize(q, qsize, status)) { |
| goto nanresult; |
| } |
| } |
| |
| rsize = b->len; |
| if (!(r == a && rsize < a->len)) { |
| if (!mpd_qresize(r, rsize, status)) { |
| goto nanresult; |
| } |
| } |
| |
| if (b->len == 1) { |
| if (a->len == 1) { |
| _mpd_div_word(&q->data[0], &r->data[0], a->data[0], b->data[0]); |
| } |
| else { |
| r->data[0] = _mpd_shortdiv(q->data, a->data, a->len, b->data[0]); |
| } |
| } |
| else if (b->len <= MPD_NEWTONDIV_CUTOFF) { |
| int ret; |
| ret = _mpd_basedivmod(q->data, r->data, a->data, b->data, |
| a->len, b->len); |
| if (ret == -1) { |
| *status |= MPD_Malloc_error; |
| goto nanresult; |
| } |
| } |
| else { |
| _mpd_base_ndivmod(q, r, a, b, status); |
| if (mpd_isspecial(q) || mpd_isspecial(r)) { |
| goto nanresult; |
| } |
| qsize = q->len; |
| rsize = r->len; |
| } |
| |
| qsize = _mpd_real_size(q->data, qsize); |
| /* resize to smaller cannot fail */ |
| mpd_qresize(q, qsize, status); |
| q->len = qsize; |
| mpd_setdigits(q); |
| mpd_set_flags(q, sign_ab); |
| q->exp = 0; |
| if (q->digits > ctx->prec) { |
| *status |= MPD_Division_impossible; |
| goto nanresult; |
| } |
| |
| rsize = _mpd_real_size(r->data, rsize); |
| /* resize to smaller cannot fail */ |
| mpd_qresize(r, rsize, status); |
| r->len = rsize; |
| mpd_setdigits(r); |
| mpd_set_flags(r, sign_a); |
| r->exp = ideal_exp; |
| |
| out: |
| mpd_del(&aligned); |
| return; |
| |
| nanresult: |
| mpd_setspecial(q, MPD_POS, MPD_NAN); |
| mpd_setspecial(r, MPD_POS, MPD_NAN); |
| goto out; |
| } |
| |
| /* Integer division with remainder. */ |
| void |
| mpd_qdivmod(mpd_t *q, mpd_t *r, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| uint8_t sign = mpd_sign(a)^mpd_sign(b); |
| |
| if (mpd_isspecial(a) || mpd_isspecial(b)) { |
| if (mpd_qcheck_nans(q, a, b, ctx, status)) { |
| mpd_qcopy(r, q, status); |
| return; |
| } |
| if (mpd_isinfinite(a)) { |
| if (mpd_isinfinite(b)) { |
| mpd_setspecial(q, MPD_POS, MPD_NAN); |
| } |
| else { |
| mpd_setspecial(q, sign, MPD_INF); |
| } |
| mpd_setspecial(r, MPD_POS, MPD_NAN); |
| *status |= MPD_Invalid_operation; |
| return; |
| } |
| if (mpd_isinfinite(b)) { |
| if (!mpd_qcopy(r, a, status)) { |
| mpd_seterror(q, MPD_Malloc_error, status); |
| return; |
| } |
| mpd_qfinalize(r, ctx, status); |
| _settriple(q, sign, 0, 0); |
| return; |
| } |
| /* debug */ |
| abort(); /* GCOV_NOT_REACHED */ |
| } |
| if (mpd_iszerocoeff(b)) { |
| if (mpd_iszerocoeff(a)) { |
| mpd_setspecial(q, MPD_POS, MPD_NAN); |
| mpd_setspecial(r, MPD_POS, MPD_NAN); |
| *status |= MPD_Division_undefined; |
| } |
| else { |
| mpd_setspecial(q, sign, MPD_INF); |
| mpd_setspecial(r, MPD_POS, MPD_NAN); |
| *status |= (MPD_Division_by_zero|MPD_Invalid_operation); |
| } |
| return; |
| } |
| |
| _mpd_qdivmod(q, r, a, b, ctx, status); |
| mpd_qfinalize(q, ctx, status); |
| mpd_qfinalize(r, ctx, status); |
| } |
| |
| void |
| mpd_qdivint(mpd_t *q, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| MPD_NEW_STATIC(r,0,0,0,0); |
| uint8_t sign = mpd_sign(a)^mpd_sign(b); |
| |
| if (mpd_isspecial(a) || mpd_isspecial(b)) { |
| if (mpd_qcheck_nans(q, a, b, ctx, status)) { |
| return; |
| } |
| if (mpd_isinfinite(a) && mpd_isinfinite(b)) { |
| mpd_seterror(q, MPD_Invalid_operation, status); |
| return; |
| } |
| if (mpd_isinfinite(a)) { |
| mpd_setspecial(q, sign, MPD_INF); |
| return; |
| } |
| if (mpd_isinfinite(b)) { |
| _settriple(q, sign, 0, 0); |
| return; |
| } |
| /* debug */ |
| abort(); /* GCOV_NOT_REACHED */ |
| } |
| if (mpd_iszerocoeff(b)) { |
| if (mpd_iszerocoeff(a)) { |
| mpd_seterror(q, MPD_Division_undefined, status); |
| } |
| else { |
| mpd_setspecial(q, sign, MPD_INF); |
| *status |= MPD_Division_by_zero; |
| } |
| return; |
| } |
| |
| |
| _mpd_qdivmod(q, &r, a, b, ctx, status); |
| mpd_del(&r); |
| mpd_qfinalize(q, ctx, status); |
| } |
| |
| /* Divide decimal by mpd_ssize_t. */ |
| void |
| mpd_qdiv_ssize(mpd_t *result, const mpd_t *a, mpd_ssize_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_context_t maxcontext; |
| MPD_NEW_STATIC(bb,0,0,0,0); |
| |
| mpd_maxcontext(&maxcontext); |
| mpd_qsset_ssize(&bb, b, &maxcontext, status); |
| mpd_qdiv(result, a, &bb, ctx, status); |
| mpd_del(&bb); |
| } |
| |
| /* Divide decimal by mpd_uint_t. */ |
| void |
| mpd_qdiv_uint(mpd_t *result, const mpd_t *a, mpd_uint_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_context_t maxcontext; |
| MPD_NEW_STATIC(bb,0,0,0,0); |
| |
| mpd_maxcontext(&maxcontext); |
| mpd_qsset_uint(&bb, b, &maxcontext, status); |
| mpd_qdiv(result, a, &bb, ctx, status); |
| mpd_del(&bb); |
| } |
| |
| /* Divide decimal by int32_t. */ |
| void |
| mpd_qdiv_i32(mpd_t *result, const mpd_t *a, int32_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_qdiv_ssize(result, a, b, ctx, status); |
| } |
| |
| /* Divide decimal by uint32_t. */ |
| void |
| mpd_qdiv_u32(mpd_t *result, const mpd_t *a, uint32_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_qdiv_uint(result, a, b, ctx, status); |
| } |
| |
| #ifdef CONFIG_64 |
| /* Divide decimal by int64_t. */ |
| void |
| mpd_qdiv_i64(mpd_t *result, const mpd_t *a, int64_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_qdiv_ssize(result, a, b, ctx, status); |
| } |
| |
| /* Divide decimal by uint64_t. */ |
| void |
| mpd_qdiv_u64(mpd_t *result, const mpd_t *a, uint64_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_qdiv_uint(result, a, b, ctx, status); |
| } |
| #endif |
| |
| /* Pad the result with trailing zeros if it has fewer digits than prec. */ |
| static void |
| _mpd_zeropad(mpd_t *result, const mpd_context_t *ctx, uint32_t *status) |
| { |
| if (!mpd_isspecial(result) && !mpd_iszero(result) && |
| result->digits < ctx->prec) { |
| mpd_ssize_t shift = ctx->prec - result->digits; |
| mpd_qshiftl(result, result, shift, status); |
| result->exp -= shift; |
| } |
| } |
| |
| /* Check if the result is guaranteed to be one. */ |
| static int |
| _mpd_qexp_check_one(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| MPD_NEW_CONST(lim,0,-(ctx->prec+1),1,1,1,9); |
| MPD_NEW_SHARED(aa, a); |
| |
| mpd_set_positive(&aa); |
| |
| /* abs(a) <= 9 * 10**(-prec-1) */ |
| if (_mpd_cmp(&aa, &lim) <= 0) { |
| _settriple(result, 0, 1, 0); |
| *status |= MPD_Rounded|MPD_Inexact; |
| return 1; |
| } |
| |
| return 0; |
| } |
| |
| /* |
| * Get the number of iterations for the Horner scheme in _mpd_qexp(). |
| */ |
| static inline mpd_ssize_t |
| _mpd_get_exp_iterations(const mpd_t *r, mpd_ssize_t p) |
| { |
| mpd_ssize_t log10pbyr; /* lower bound for log10(p / abs(r)) */ |
| mpd_ssize_t n; |
| |
| assert(p >= 10); |
| assert(!mpd_iszero(r)); |
| assert(-p < mpd_adjexp(r) && mpd_adjexp(r) <= -1); |
| |
| #ifdef CONFIG_64 |
| if (p > (mpd_ssize_t)(1ULL<<52)) { |
| return MPD_SSIZE_MAX; |
| } |
| #endif |
| |
| /* |
| * Lower bound for log10(p / abs(r)): adjexp(p) - (adjexp(r) + 1) |
| * At this point (for CONFIG_64, CONFIG_32 is not problematic): |
| * 1) 10 <= p <= 2**52 |
| * 2) -p < adjexp(r) <= -1 |
| * 3) 1 <= log10pbyr <= 2**52 + 14 |
| */ |
| log10pbyr = (mpd_word_digits(p)-1) - (mpd_adjexp(r)+1); |
| |
| /* |
| * The numerator in the paper is 1.435 * p - 1.182, calculated |
| * exactly. We compensate for rounding errors by using 1.43503. |
| * ACL2 proofs: |
| * 1) exp-iter-approx-lower-bound: The term below evaluated |
| * in 53-bit floating point arithmetic is greater than or |
| * equal to the exact term used in the paper. |
| * 2) exp-iter-approx-upper-bound: The term below is less than |
| * or equal to 3/2 * p <= 3/2 * 2**52. |
| */ |
| n = (mpd_ssize_t)ceil((1.43503*(double)p - 1.182) / (double)log10pbyr); |
| return n >= 3 ? n : 3; |
| } |
| |
| /* |
| * Internal function, specials have been dealt with. Apart from Overflow |
| * and Underflow, two cases must be considered for the error of the result: |
| * |
| * 1) abs(a) <= 9 * 10**(-prec-1) ==> result == 1 |
| * |
| * Absolute error: abs(1 - e**x) < 10**(-prec) |
| * ------------------------------------------- |
| * |
| * 2) abs(a) > 9 * 10**(-prec-1) |
| * |
| * Relative error: abs(result - e**x) < 0.5 * 10**(-prec) * e**x |
| * ------------------------------------------------------------- |
| * |
| * The algorithm is from Hull&Abrham, Variable Precision Exponential Function, |
| * ACM Transactions on Mathematical Software, Vol. 12, No. 2, June 1986. |
| * |
| * Main differences: |
| * |
| * - The number of iterations for the Horner scheme is calculated using |
| * 53-bit floating point arithmetic. |
| * |
| * - In the error analysis for ER (relative error accumulated in the |
| * evaluation of the truncated series) the reduced operand r may |
| * have any number of digits. |
| * ACL2 proof: exponent-relative-error |
| * |
| * - The analysis for early abortion has been adapted for the mpd_t |
| * ranges. |
| */ |
| static void |
| _mpd_qexp(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_context_t workctx; |
| MPD_NEW_STATIC(tmp,0,0,0,0); |
| MPD_NEW_STATIC(sum,0,0,0,0); |
| MPD_NEW_CONST(word,0,0,1,1,1,1); |
| mpd_ssize_t j, n, t; |
| |
| assert(!mpd_isspecial(a)); |
| |
| if (mpd_iszerocoeff(a)) { |
| _settriple(result, MPD_POS, 1, 0); |
| return; |
| } |
| |
| /* |
| * We are calculating e^x = e^(r*10^t) = (e^r)^(10^t), where abs(r) < 1 and t >= 0. |
| * |
| * If t > 0, we have: |
| * |
| * (1) 0.1 <= r < 1, so e^0.1 <= e^r. If t > MAX_T, overflow occurs: |
| * |
| * MAX-EMAX+1 < log10(e^(0.1*10*t)) <= log10(e^(r*10^t)) < adjexp(e^(r*10^t))+1 |
| * |
| * (2) -1 < r <= -0.1, so e^r <= e^-0.1. If t > MAX_T, underflow occurs: |
| * |
| * adjexp(e^(r*10^t)) <= log10(e^(r*10^t)) <= log10(e^(-0.1*10^t)) < MIN-ETINY |
| */ |
| #if defined(CONFIG_64) |
| #define MPD_EXP_MAX_T 19 |
| #elif defined(CONFIG_32) |
| #define MPD_EXP_MAX_T 10 |
| #endif |
| t = a->digits + a->exp; |
| t = (t > 0) ? t : 0; |
| if (t > MPD_EXP_MAX_T) { |
| if (mpd_ispositive(a)) { |
| mpd_setspecial(result, MPD_POS, MPD_INF); |
| *status |= MPD_Overflow|MPD_Inexact|MPD_Rounded; |
| } |
| else { |
| _settriple(result, MPD_POS, 0, mpd_etiny(ctx)); |
| *status |= (MPD_Inexact|MPD_Rounded|MPD_Subnormal| |
| MPD_Underflow|MPD_Clamped); |
| } |
| return; |
| } |
| |
| /* abs(a) <= 9 * 10**(-prec-1) */ |
| if (_mpd_qexp_check_one(result, a, ctx, status)) { |
| return; |
| } |
| |
| mpd_maxcontext(&workctx); |
| workctx.prec = ctx->prec + t + 2; |
| workctx.prec = (workctx.prec < 10) ? 10 : workctx.prec; |
| workctx.round = MPD_ROUND_HALF_EVEN; |
| |
| if (!mpd_qcopy(result, a, status)) { |
| return; |
| } |
| result->exp -= t; |
| |
| /* |
| * At this point: |
| * 1) 9 * 10**(-prec-1) < abs(a) |
| * 2) 9 * 10**(-prec-t-1) < abs(r) |
| * 3) log10(9) - prec - t - 1 < log10(abs(r)) < adjexp(abs(r)) + 1 |
| * 4) - prec - t - 2 < adjexp(abs(r)) <= -1 |
| */ |
| n = _mpd_get_exp_iterations(result, workctx.prec); |
| if (n == MPD_SSIZE_MAX) { |
| mpd_seterror(result, MPD_Invalid_operation, status); /* GCOV_UNLIKELY */ |
| return; /* GCOV_UNLIKELY */ |
| } |
| |
| _settriple(&sum, MPD_POS, 1, 0); |
| |
| for (j = n-1; j >= 1; j--) { |
| word.data[0] = j; |
| mpd_setdigits(&word); |
| mpd_qdiv(&tmp, result, &word, &workctx, &workctx.status); |
| mpd_qfma(&sum, &sum, &tmp, &one, &workctx, &workctx.status); |
| } |
| |
| #ifdef CONFIG_64 |
| _mpd_qpow_uint(result, &sum, mpd_pow10[t], MPD_POS, &workctx, status); |
| #else |
| if (t <= MPD_MAX_POW10) { |
| _mpd_qpow_uint(result, &sum, mpd_pow10[t], MPD_POS, &workctx, status); |
| } |
| else { |
| t -= MPD_MAX_POW10; |
| _mpd_qpow_uint(&tmp, &sum, mpd_pow10[MPD_MAX_POW10], MPD_POS, |
| &workctx, status); |
| _mpd_qpow_uint(result, &tmp, mpd_pow10[t], MPD_POS, &workctx, status); |
| } |
| #endif |
| |
| mpd_del(&tmp); |
| mpd_del(&sum); |
| *status |= (workctx.status&MPD_Errors); |
| *status |= (MPD_Inexact|MPD_Rounded); |
| } |
| |
| /* exp(a) */ |
| void |
| mpd_qexp(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_context_t workctx; |
| |
| if (mpd_isspecial(a)) { |
| if (mpd_qcheck_nan(result, a, ctx, status)) { |
| return; |
| } |
| if (mpd_isnegative(a)) { |
| _settriple(result, MPD_POS, 0, 0); |
| } |
| else { |
| mpd_setspecial(result, MPD_POS, MPD_INF); |
| } |
| return; |
| } |
| if (mpd_iszerocoeff(a)) { |
| _settriple(result, MPD_POS, 1, 0); |
| return; |
| } |
| |
| workctx = *ctx; |
| workctx.round = MPD_ROUND_HALF_EVEN; |
| |
| if (ctx->allcr) { |
| MPD_NEW_STATIC(t1, 0,0,0,0); |
| MPD_NEW_STATIC(t2, 0,0,0,0); |
| MPD_NEW_STATIC(ulp, 0,0,0,0); |
| MPD_NEW_STATIC(aa, 0,0,0,0); |
| mpd_ssize_t prec; |
| mpd_ssize_t ulpexp; |
| uint32_t workstatus; |
| |
| if (result == a) { |
| if (!mpd_qcopy(&aa, a, status)) { |
| mpd_seterror(result, MPD_Malloc_error, status); |
| return; |
| } |
| a = &aa; |
| } |
| |
| workctx.clamp = 0; |
| prec = ctx->prec + 3; |
| while (1) { |
| workctx.prec = prec; |
| workstatus = 0; |
| |
| _mpd_qexp(result, a, &workctx, &workstatus); |
| *status |= workstatus; |
| |
| ulpexp = result->exp + result->digits - workctx.prec; |
| if (workstatus & MPD_Underflow) { |
| /* The effective work precision is result->digits. */ |
| ulpexp = result->exp; |
| } |
| _ssettriple(&ulp, MPD_POS, 1, ulpexp); |
| |
| /* |
| * At this point [1]: |
| * 1) abs(result - e**x) < 0.5 * 10**(-prec) * e**x |
| * 2) result - ulp < e**x < result + ulp |
| * 3) result - ulp < result < result + ulp |
| * |
| * If round(result-ulp)==round(result+ulp), then |
| * round(result)==round(e**x). Therefore the result |
| * is correctly rounded. |
| * |
| * [1] If abs(a) <= 9 * 10**(-prec-1), use the absolute |
| * error for a similar argument. |
| */ |
| workctx.prec = ctx->prec; |
| mpd_qadd(&t1, result, &ulp, &workctx, &workctx.status); |
| mpd_qsub(&t2, result, &ulp, &workctx, &workctx.status); |
| if (mpd_isspecial(result) || mpd_iszerocoeff(result) || |
| mpd_qcmp(&t1, &t2, status) == 0) { |
| workctx.clamp = ctx->clamp; |
| _mpd_zeropad(result, &workctx, status); |
| mpd_check_underflow(result, &workctx, status); |
| mpd_qfinalize(result, &workctx, status); |
| break; |
| } |
| prec += MPD_RDIGITS; |
| } |
| mpd_del(&t1); |
| mpd_del(&t2); |
| mpd_del(&ulp); |
| mpd_del(&aa); |
| } |
| else { |
| _mpd_qexp(result, a, &workctx, status); |
| _mpd_zeropad(result, &workctx, status); |
| mpd_check_underflow(result, &workctx, status); |
| mpd_qfinalize(result, &workctx, status); |
| } |
| } |
| |
| /* Fused multiply-add: (a * b) + c, with a single final rounding. */ |
| void |
| mpd_qfma(mpd_t *result, const mpd_t *a, const mpd_t *b, const mpd_t *c, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| uint32_t workstatus = 0; |
| mpd_t *cc = (mpd_t *)c; |
| |
| if (result == c) { |
| if ((cc = mpd_qncopy(c)) == NULL) { |
| mpd_seterror(result, MPD_Malloc_error, status); |
| return; |
| } |
| } |
| |
| _mpd_qmul(result, a, b, ctx, &workstatus); |
| if (!(workstatus&MPD_Invalid_operation)) { |
| mpd_qadd(result, result, cc, ctx, &workstatus); |
| } |
| |
| if (cc != c) mpd_del(cc); |
| *status |= workstatus; |
| } |
| |
| /* |
| * Schedule the optimal precision increase for the Newton iteration. |
| * v := input operand |
| * z_0 := initial approximation |
| * initprec := natural number such that abs(log(v) - z_0) < 10**-initprec |
| * maxprec := target precision |
| * |
| * For convenience the output klist contains the elements in reverse order: |
| * klist := [k_n-1, ..., k_0], where |
| * 1) k_0 <= initprec and |
| * 2) abs(log(v) - result) < 10**(-2*k_n-1 + 1) <= 10**-maxprec. |
| */ |
| static inline int |
| ln_schedule_prec(mpd_ssize_t klist[MPD_MAX_PREC_LOG2], mpd_ssize_t maxprec, |
| mpd_ssize_t initprec) |
| { |
| mpd_ssize_t k; |
| int i; |
| |
| assert(maxprec >= 2 && initprec >= 2); |
| if (maxprec <= initprec) return -1; |
| |
| i = 0; k = maxprec; |
| do { |
| k = (k+2) / 2; |
| klist[i++] = k; |
| } while (k > initprec); |
| |
| return i-1; |
| } |
| |
| /* The constants have been verified with both decimal.py and mpfr. */ |
| #ifdef CONFIG_64 |
| #if MPD_RDIGITS != 19 |
| #error "mpdecimal.c: MPD_RDIGITS must be 19." |
| #endif |
| static const mpd_uint_t mpd_ln10_data[MPD_MINALLOC_MAX] = { |
| 6983716328982174407ULL, 9089704281976336583ULL, 1515961135648465461ULL, |
| 4416816335727555703ULL, 2900988039194170265ULL, 2307925037472986509ULL, |
| 107598438319191292ULL, 3466624107184669231ULL, 4450099781311469159ULL, |
| 9807828059751193854ULL, 7713456862091670584ULL, 1492198849978748873ULL, |
| 6528728696511086257ULL, 2385392051446341972ULL, 8692180205189339507ULL, |
| 6518769751037497088ULL, 2375253577097505395ULL, 9095610299291824318ULL, |
| 982748238504564801ULL, 5438635917781170543ULL, 7547331541421808427ULL, |
| 752371033310119785ULL, 3171643095059950878ULL, 9785265383207606726ULL, |
| 2932258279850258550ULL, 5497347726624257094ULL, 2976979522110718264ULL, |
| 9221477656763693866ULL, 1979650047149510504ULL, 6674183485704422507ULL, |
| 9702766860595249671ULL, 9278096762712757753ULL, 9314848524948644871ULL, |
| 6826928280848118428ULL, 754403708474699401ULL, 230105703089634572ULL, |
| 1929203337658714166ULL, 7589402567763113569ULL, 4208241314695689016ULL, |
| 2922455440575892572ULL, 9356734206705811364ULL, 2684916746550586856ULL, |
| 644507064800027750ULL, 9476834636167921018ULL, 5659121373450747856ULL, |
| 2835522011480466371ULL, 6470806855677432162ULL, 7141748003688084012ULL, |
| 9619404400222105101ULL, 5504893431493939147ULL, 6674744042432743651ULL, |
| 2287698219886746543ULL, 7773262884616336622ULL, 1985283935053089653ULL, |
| 4680843799894826233ULL, 8168948290720832555ULL, 8067566662873690987ULL, |
| 6248633409525465082ULL, 9829834196778404228ULL, 3524802359972050895ULL, |
| 3327900967572609677ULL, 110148862877297603ULL, 179914546843642076ULL, |
| 2302585092994045684ULL |
| }; |
| #else |
| #if MPD_RDIGITS != 9 |
| #error "mpdecimal.c: MPD_RDIGITS must be 9." |
| #endif |
| static const mpd_uint_t mpd_ln10_data[MPD_MINALLOC_MAX] = { |
| 401682692UL, 708474699UL, 720754403UL, 30896345UL, 602301057UL, 765871416UL, |
| 192920333UL, 763113569UL, 589402567UL, 956890167UL, 82413146UL, 589257242UL, |
| 245544057UL, 811364292UL, 734206705UL, 868569356UL, 167465505UL, 775026849UL, |
| 706480002UL, 18064450UL, 636167921UL, 569476834UL, 734507478UL, 156591213UL, |
| 148046637UL, 283552201UL, 677432162UL, 470806855UL, 880840126UL, 417480036UL, |
| 210510171UL, 940440022UL, 939147961UL, 893431493UL, 436515504UL, 440424327UL, |
| 654366747UL, 821988674UL, 622228769UL, 884616336UL, 537773262UL, 350530896UL, |
| 319852839UL, 989482623UL, 468084379UL, 720832555UL, 168948290UL, 736909878UL, |
| 675666628UL, 546508280UL, 863340952UL, 404228624UL, 834196778UL, 508959829UL, |
| 23599720UL, 967735248UL, 96757260UL, 603332790UL, 862877297UL, 760110148UL, |
| 468436420UL, 401799145UL, 299404568UL, 230258509UL |
| }; |
| #endif |
| /* _mpd_ln10 is used directly for precisions smaller than MINALLOC_MAX*RDIGITS. |
| Otherwise, it serves as the initial approximation for calculating ln(10). */ |
| static const mpd_t _mpd_ln10 = { |
| MPD_STATIC|MPD_CONST_DATA, -(MPD_MINALLOC_MAX*MPD_RDIGITS-1), |
| MPD_MINALLOC_MAX*MPD_RDIGITS, MPD_MINALLOC_MAX, MPD_MINALLOC_MAX, |
| (mpd_uint_t *)mpd_ln10_data |
| }; |
| |
| /* |
| * Set 'result' to log(10). |
| * Ulp error: abs(result - log(10)) < ulp(log(10)) |
| * Relative error: abs(result - log(10)) < 5 * 10**-prec * log(10) |
| * |
| * NOTE: The relative error is not derived from the ulp error, but |
| * calculated separately using the fact that 23/10 < log(10) < 24/10. |
| */ |
| void |
| mpd_qln10(mpd_t *result, mpd_ssize_t prec, uint32_t *status) |
| { |
| mpd_context_t varcontext, maxcontext; |
| MPD_NEW_STATIC(tmp, 0,0,0,0); |
| MPD_NEW_CONST(static10, 0,0,2,1,1,10); |
| mpd_ssize_t klist[MPD_MAX_PREC_LOG2]; |
| mpd_uint_t rnd; |
| mpd_ssize_t shift; |
| int i; |
| |
| assert(prec >= 1); |
| |
| shift = MPD_MINALLOC_MAX*MPD_RDIGITS-prec; |
| shift = shift < 0 ? 0 : shift; |
| |
| rnd = mpd_qshiftr(result, &_mpd_ln10, shift, status); |
| if (rnd == MPD_UINT_MAX) { |
| mpd_seterror(result, MPD_Malloc_error, status); |
| return; |
| } |
| result->exp = -(result->digits-1); |
| |
| mpd_maxcontext(&maxcontext); |
| if (prec < MPD_MINALLOC_MAX*MPD_RDIGITS) { |
| maxcontext.prec = prec; |
| _mpd_apply_round_excess(result, rnd, &maxcontext, status); |
| *status |= (MPD_Inexact|MPD_Rounded); |
| return; |
| } |
| |
| mpd_maxcontext(&varcontext); |
| varcontext.round = MPD_ROUND_TRUNC; |
| |
| i = ln_schedule_prec(klist, prec+2, -result->exp); |
| for (; i >= 0; i--) { |
| varcontext.prec = 2*klist[i]+3; |
| result->flags ^= MPD_NEG; |
| _mpd_qexp(&tmp, result, &varcontext, status); |
| result->flags ^= MPD_NEG; |
| mpd_qmul(&tmp, &static10, &tmp, &varcontext, status); |
| mpd_qsub(&tmp, &tmp, &one, &maxcontext, status); |
| mpd_qadd(result, result, &tmp, &maxcontext, status); |
| if (mpd_isspecial(result)) { |
| break; |
| } |
| } |
| |
| mpd_del(&tmp); |
| maxcontext.prec = prec; |
| mpd_qfinalize(result, &maxcontext, status); |
| } |
| |
| /* |
| * Initial approximations for the ln() iteration. The values have the |
| * following properties (established with both decimal.py and mpfr): |
| * |
| * Index 0 - 400, logarithms of x in [1.00, 5.00]: |
| * abs(lnapprox[i] * 10**-3 - log((i+100)/100)) < 10**-2 |
| * abs(lnapprox[i] * 10**-3 - log((i+1+100)/100)) < 10**-2 |
| * |
| * Index 401 - 899, logarithms of x in (0.500, 0.999]: |
| * abs(-lnapprox[i] * 10**-3 - log((i+100)/1000)) < 10**-2 |
| * abs(-lnapprox[i] * 10**-3 - log((i+1+100)/1000)) < 10**-2 |
| */ |
| static const uint16_t lnapprox[900] = { |
| /* index 0 - 400: log((i+100)/100) * 1000 */ |
| 0, 10, 20, 30, 39, 49, 58, 68, 77, 86, 95, 104, 113, 122, 131, 140, 148, 157, |
| 166, 174, 182, 191, 199, 207, 215, 223, 231, 239, 247, 255, 262, 270, 278, |
| 285, 293, 300, 308, 315, 322, 329, 336, 344, 351, 358, 365, 372, 378, 385, |
| 392, 399, 406, 412, 419, 425, 432, 438, 445, 451, 457, 464, 470, 476, 482, |
| 489, 495, 501, 507, 513, 519, 525, 531, 536, 542, 548, 554, 560, 565, 571, |
| 577, 582, 588, 593, 599, 604, 610, 615, 621, 626, 631, 637, 642, 647, 652, |
| 658, 663, 668, 673, 678, 683, 688, 693, 698, 703, 708, 713, 718, 723, 728, |
| 732, 737, 742, 747, 751, 756, 761, 766, 770, 775, 779, 784, 788, 793, 798, |
| 802, 806, 811, 815, 820, 824, 829, 833, 837, 842, 846, 850, 854, 859, 863, |
| 867, 871, 876, 880, 884, 888, 892, 896, 900, 904, 908, 912, 916, 920, 924, |
| 928, 932, 936, 940, 944, 948, 952, 956, 959, 963, 967, 971, 975, 978, 982, |
| 986, 990, 993, 997, 1001, 1004, 1008, 1012, 1015, 1019, 1022, 1026, 1030, |
| 1033, 1037, 1040, 1044, 1047, 1051, 1054, 1058, 1061, 1065, 1068, 1072, 1075, |
| 1078, 1082, 1085, 1089, 1092, 1095, 1099, 1102, 1105, 1109, 1112, 1115, 1118, |
| 1122, 1125, 1128, 1131, 1135, 1138, 1141, 1144, 1147, 1151, 1154, 1157, 1160, |
| 1163, 1166, 1169, 1172, 1176, 1179, 1182, 1185, 1188, 1191, 1194, 1197, 1200, |
| 1203, 1206, 1209, 1212, 1215, 1218, 1221, 1224, 1227, 1230, 1233, 1235, 1238, |
| 1241, 1244, 1247, 1250, 1253, 1256, 1258, 1261, 1264, 1267, 1270, 1273, 1275, |
| 1278, 1281, 1284, 1286, 1289, 1292, 1295, 1297, 1300, 1303, 1306, 1308, 1311, |
| 1314, 1316, 1319, 1322, 1324, 1327, 1330, 1332, 1335, 1338, 1340, 1343, 1345, |
| 1348, 1351, 1353, 1356, 1358, 1361, 1364, 1366, 1369, 1371, 1374, 1376, 1379, |
| 1381, 1384, 1386, 1389, 1391, 1394, 1396, 1399, 1401, 1404, 1406, 1409, 1411, |
| 1413, 1416, 1418, 1421, 1423, 1426, 1428, 1430, 1433, 1435, 1437, 1440, 1442, |
| 1445, 1447, 1449, 1452, 1454, 1456, 1459, 1461, 1463, 1466, 1468, 1470, 1472, |
| 1475, 1477, 1479, 1482, 1484, 1486, 1488, 1491, 1493, 1495, 1497, 1500, 1502, |
| 1504, 1506, 1509, 1511, 1513, 1515, 1517, 1520, 1522, 1524, 1526, 1528, 1530, |
| 1533, 1535, 1537, 1539, 1541, 1543, 1545, 1548, 1550, 1552, 1554, 1556, 1558, |
| 1560, 1562, 1564, 1567, 1569, 1571, 1573, 1575, 1577, 1579, 1581, 1583, 1585, |
| 1587, 1589, 1591, 1593, 1595, 1597, 1599, 1601, 1603, 1605, 1607, 1609, |
| /* index 401 - 899: -log((i+100)/1000) * 1000 */ |
| 691, 689, 687, 685, 683, 681, 679, 677, 675, 673, 671, 669, 668, 666, 664, |
| 662, 660, 658, 656, 654, 652, 650, 648, 646, 644, 642, 641, 639, 637, 635, |
| 633, 631, 629, 627, 626, 624, 622, 620, 618, 616, 614, 612, 611, 609, 607, |
| 605, 603, 602, 600, 598, 596, 594, 592, 591, 589, 587, 585, 583, 582, 580, |
| 578, 576, 574, 573, 571, 569, 567, 566, 564, 562, 560, 559, 557, 555, 553, |
| 552, 550, 548, 546, 545, 543, 541, 540, 538, 536, 534, 533, 531, 529, 528, |
| 526, 524, 523, 521, 519, 518, 516, 514, 512, 511, 509, 508, 506, 504, 502, |
| 501, 499, 498, 496, 494, 493, 491, 489, 488, 486, 484, 483, 481, 480, 478, |
| 476, 475, 473, 472, 470, 468, 467, 465, 464, 462, 460, 459, 457, 456, 454, |
| 453, 451, 449, 448, 446, 445, 443, 442, 440, 438, 437, 435, 434, 432, 431, |
| 429, 428, 426, 425, 423, 422, 420, 419, 417, 416, 414, 412, 411, 410, 408, |
| 406, 405, 404, 402, 400, 399, 398, 396, 394, 393, 392, 390, 389, 387, 386, |
| 384, 383, 381, 380, 378, 377, 375, 374, 372, 371, 370, 368, 367, 365, 364, |
| 362, 361, 360, 358, 357, 355, 354, 352, 351, 350, 348, 347, 345, 344, 342, |
| 341, 340, 338, 337, 336, 334, 333, 331, 330, 328, 327, 326, 324, 323, 322, |
| 320, 319, 318, 316, 315, 313, 312, 311, 309, 308, 306, 305, 304, 302, 301, |
| 300, 298, 297, 296, 294, 293, 292, 290, 289, 288, 286, 285, 284, 282, 281, |
| 280, 278, 277, 276, 274, 273, 272, 270, 269, 268, 267, 265, 264, 263, 261, |
| 260, 259, 258, 256, 255, 254, 252, 251, 250, 248, 247, 246, 245, 243, 242, |
| 241, 240, 238, 237, 236, 234, 233, 232, 231, 229, 228, 227, 226, 224, 223, |
| 222, 221, 219, 218, 217, 216, 214, 213, 212, 211, 210, 208, 207, 206, 205, |
| 203, 202, 201, 200, 198, 197, 196, 195, 194, 192, 191, 190, 189, 188, 186, |
| 185, 184, 183, 182, 180, 179, 178, 177, 176, 174, 173, 172, 171, 170, 168, |
| 167, 166, 165, 164, 162, 161, 160, 159, 158, 157, 156, 154, 153, 152, 151, |
| 150, 148, 147, 146, 145, 144, 143, 142, 140, 139, 138, 137, 136, 135, 134, |
| 132, 131, 130, 129, 128, 127, 126, 124, 123, 122, 121, 120, 119, 118, 116, |
| 115, 114, 113, 112, 111, 110, 109, 108, 106, 105, 104, 103, 102, 101, 100, |
| 99, 98, 97, 95, 94, 93, 92, 91, 90, 89, 88, 87, 86, 84, 83, 82, 81, 80, 79, |
| 78, 77, 76, 75, 74, 73, 72, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, |
| 58, 57, 56, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, |
| 38, 37, 36, 35, 34, 33, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, |
| 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 |
| }; |
| |
| /* |
| * Internal ln() function that does not check for specials, zero or one. |
| * Relative error: abs(result - log(a)) < 0.1 * 10**-prec * abs(log(a)) |
| */ |
| static void |
| _mpd_qln(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_context_t varcontext, maxcontext; |
| mpd_t *z = (mpd_t *) result; |
| MPD_NEW_STATIC(v,0,0,0,0); |
| MPD_NEW_STATIC(vtmp,0,0,0,0); |
| MPD_NEW_STATIC(tmp,0,0,0,0); |
| mpd_ssize_t klist[MPD_MAX_PREC_LOG2]; |
| mpd_ssize_t maxprec, shift, t; |
| mpd_ssize_t a_digits, a_exp; |
| mpd_uint_t dummy, x; |
| int i; |
| |
| assert(!mpd_isspecial(a) && !mpd_iszerocoeff(a)); |
| |
| /* |
| * We are calculating ln(a) = ln(v * 10^t) = ln(v) + t*ln(10), |
| * where 0.5 < v <= 5. |
| */ |
| if (!mpd_qcopy(&v, a, status)) { |
| mpd_seterror(result, MPD_Malloc_error, status); |
| goto finish; |
| } |
| |
| /* Initial approximation: we have at least one non-zero digit */ |
| _mpd_get_msdigits(&dummy, &x, &v, 3); |
| if (x < 10) x *= 10; |
| if (x < 100) x *= 10; |
| x -= 100; |
| |
| /* a may equal z */ |
| a_digits = a->digits; |
| a_exp = a->exp; |
| |
| mpd_minalloc(z); |
| mpd_clear_flags(z); |
| z->data[0] = lnapprox[x]; |
| z->len = 1; |
| z->exp = -3; |
| mpd_setdigits(z); |
| |
| if (x <= 400) { |
| /* Reduce the input operand to 1.00 <= v <= 5.00. Let y = x + 100, |
| * so 100 <= y <= 500. Since y contains the most significant digits |
| * of v, y/100 <= v < (y+1)/100 and abs(z - log(v)) < 10**-2. */ |
| v.exp = -(a_digits - 1); |
| t = a_exp + a_digits - 1; |
| } |
| else { |
| /* Reduce the input operand to 0.500 < v <= 0.999. Let y = x + 100, |
| * so 500 < y <= 999. Since y contains the most significant digits |
| * of v, y/1000 <= v < (y+1)/1000 and abs(z - log(v)) < 10**-2. */ |
| v.exp = -a_digits; |
| t = a_exp + a_digits; |
| mpd_set_negative(z); |
| } |
| |
| mpd_maxcontext(&maxcontext); |
| mpd_maxcontext(&varcontext); |
| varcontext.round = MPD_ROUND_TRUNC; |
| |
| maxprec = ctx->prec + 2; |
| if (t == 0 && (x <= 15 || x >= 800)) { |
| /* 0.900 <= v <= 1.15: Estimate the magnitude of the logarithm. |
| * If ln(v) will underflow, skip the loop. Otherwise, adjust the |
| * precision upwards in order to obtain a sufficient number of |
| * significant digits. |
| * |
| * Case v > 1: |
| * abs((v-1)/10) < abs((v-1)/v) < abs(ln(v)) < abs(v-1) |
| * Case v < 1: |
| * abs(v-1) < abs(ln(v)) < abs((v-1)/v) < abs((v-1)*10) |
| */ |
| int cmp = _mpd_cmp(&v, &one); |
| |
| /* Upper bound (assume v > 1): abs(v-1), unrounded */ |
| _mpd_qsub(&tmp, &v, &one, &maxcontext, &maxcontext.status); |
| if (maxcontext.status & MPD_Errors) { |
| mpd_seterror(result, MPD_Malloc_error, status); |
| goto finish; |
| } |
| |
| if (cmp < 0) { |
| /* v < 1: abs((v-1)*10) */ |
| tmp.exp += 1; |
| } |
| if (mpd_adjexp(&tmp) < mpd_etiny(ctx)) { |
| /* The upper bound is less than etiny: Underflow to zero */ |
| _settriple(result, (cmp<0), 1, mpd_etiny(ctx)-1); |
| goto finish; |
| } |
| /* Lower bound: abs((v-1)/10) or abs(v-1) */ |
| tmp.exp -= 1; |
| if (mpd_adjexp(&tmp) < 0) { |
| /* Absolute error of the loop: abs(z - log(v)) < 10**-p. If |
| * p = ctx->prec+2-adjexp(lower), then the relative error of |
| * the result is (using 10**adjexp(x) <= abs(x)): |
| * |
| * abs(z - log(v)) / abs(log(v)) < 10**-p / abs(log(v)) |
| * <= 10**(-ctx->prec-2) |
| */ |
| maxprec = maxprec - mpd_adjexp(&tmp); |
| } |
| } |
| |
| i = ln_schedule_prec(klist, maxprec, 2); |
| for (; i >= 0; i--) { |
| varcontext.prec = 2*klist[i]+3; |
| z->flags ^= MPD_NEG; |
| _mpd_qexp(&tmp, z, &varcontext, status); |
| z->flags ^= MPD_NEG; |
| |
| if (v.digits > varcontext.prec) { |
| shift = v.digits - varcontext.prec; |
| mpd_qshiftr(&vtmp, &v, shift, status); |
| vtmp.exp += shift; |
| mpd_qmul(&tmp, &vtmp, &tmp, &varcontext, status); |
| } |
| else { |
| mpd_qmul(&tmp, &v, &tmp, &varcontext, status); |
| } |
| |
| mpd_qsub(&tmp, &tmp, &one, &maxcontext, status); |
| mpd_qadd(z, z, &tmp, &maxcontext, status); |
| if (mpd_isspecial(z)) { |
| break; |
| } |
| } |
| |
| /* |
| * Case t == 0: |
| * t * log(10) == 0, the result does not change and the analysis |
| * above applies. If v < 0.900 or v > 1.15, the relative error is |
| * less than 10**(-ctx.prec-1). |
| * Case t != 0: |
| * z := approx(log(v)) |
| * y := approx(log(10)) |
| * p := maxprec = ctx->prec + 2 |
| * Absolute errors: |
| * 1) abs(z - log(v)) < 10**-p |
| * 2) abs(y - log(10)) < 10**-p |
| * The multiplication is exact, so: |
| * 3) abs(t*y - t*log(10)) < t*10**-p |
| * The sum is exact, so: |
| * 4) abs((z + t*y) - (log(v) + t*log(10))) < (abs(t) + 1) * 10**-p |
| * Bounds for log(v) and log(10): |
| * 5) -7/10 < log(v) < 17/10 |
| * 6) 23/10 < log(10) < 24/10 |
| * Using 4), 5), 6) and t != 0, the relative error is: |
| * |
| * 7) relerr < ((abs(t) + 1)*10**-p) / abs(log(v) + t*log(10)) |
| * < 0.5 * 10**(-p + 1) = 0.5 * 10**(-ctx->prec-1) |
| */ |
| mpd_qln10(&v, maxprec+1, status); |
| mpd_qmul_ssize(&tmp, &v, t, &maxcontext, status); |
| mpd_qadd(result, &tmp, z, &maxcontext, status); |
| |
| |
| finish: |
| *status |= (MPD_Inexact|MPD_Rounded); |
| mpd_del(&v); |
| mpd_del(&vtmp); |
| mpd_del(&tmp); |
| } |
| |
| /* ln(a) */ |
| void |
| mpd_qln(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_context_t workctx; |
| mpd_ssize_t adjexp, t; |
| |
| if (mpd_isspecial(a)) { |
| if (mpd_qcheck_nan(result, a, ctx, status)) { |
| return; |
| } |
| if (mpd_isnegative(a)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| mpd_setspecial(result, MPD_POS, MPD_INF); |
| return; |
| } |
| if (mpd_iszerocoeff(a)) { |
| mpd_setspecial(result, MPD_NEG, MPD_INF); |
| return; |
| } |
| if (mpd_isnegative(a)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| if (_mpd_cmp(a, &one) == 0) { |
| _settriple(result, MPD_POS, 0, 0); |
| return; |
| } |
| /* |
| * Check if the result will overflow (0 < x, x != 1): |
| * 1) log10(x) < 0 iff adjexp(x) < 0 |
| * 2) 0 < x /\ x <= y ==> adjexp(x) <= adjexp(y) |
| * 3) 0 < x /\ x != 1 ==> 2 * abs(log10(x)) < abs(log(x)) |
| * 4) adjexp(x) <= log10(x) < adjexp(x) + 1 |
| * |
| * Case adjexp(x) >= 0: |
| * 5) 2 * adjexp(x) < abs(log(x)) |
| * Case adjexp(x) > 0: |
| * 6) adjexp(2 * adjexp(x)) <= adjexp(abs(log(x))) |
| * Case adjexp(x) == 0: |
| * mpd_exp_digits(t)-1 == 0 <= emax (the shortcut is not triggered) |
| * |
| * Case adjexp(x) < 0: |
| * 7) 2 * (-adjexp(x) - 1) < abs(log(x)) |
| * Case adjexp(x) < -1: |
| * 8) adjexp(2 * (-adjexp(x) - 1)) <= adjexp(abs(log(x))) |
| * Case adjexp(x) == -1: |
| * mpd_exp_digits(t)-1 == 0 <= emax (the shortcut is not triggered) |
| */ |
| adjexp = mpd_adjexp(a); |
| t = (adjexp < 0) ? -adjexp-1 : adjexp; |
| t *= 2; |
| if (mpd_exp_digits(t)-1 > ctx->emax) { |
| *status |= MPD_Overflow|MPD_Inexact|MPD_Rounded; |
| mpd_setspecial(result, (adjexp<0), MPD_INF); |
| return; |
| } |
| |
| workctx = *ctx; |
| workctx.round = MPD_ROUND_HALF_EVEN; |
| |
| if (ctx->allcr) { |
| MPD_NEW_STATIC(t1, 0,0,0,0); |
| MPD_NEW_STATIC(t2, 0,0,0,0); |
| MPD_NEW_STATIC(ulp, 0,0,0,0); |
| MPD_NEW_STATIC(aa, 0,0,0,0); |
| mpd_ssize_t prec; |
| |
| if (result == a) { |
| if (!mpd_qcopy(&aa, a, status)) { |
| mpd_seterror(result, MPD_Malloc_error, status); |
| return; |
| } |
| a = &aa; |
| } |
| |
| workctx.clamp = 0; |
| prec = ctx->prec + 3; |
| while (1) { |
| workctx.prec = prec; |
| _mpd_qln(result, a, &workctx, status); |
| _ssettriple(&ulp, MPD_POS, 1, |
| result->exp + result->digits-workctx.prec); |
| |
| workctx.prec = ctx->prec; |
| mpd_qadd(&t1, result, &ulp, &workctx, &workctx.status); |
| mpd_qsub(&t2, result, &ulp, &workctx, &workctx.status); |
| if (mpd_isspecial(result) || mpd_iszerocoeff(result) || |
| mpd_qcmp(&t1, &t2, status) == 0) { |
| workctx.clamp = ctx->clamp; |
| mpd_check_underflow(result, &workctx, status); |
| mpd_qfinalize(result, &workctx, status); |
| break; |
| } |
| prec += MPD_RDIGITS; |
| } |
| mpd_del(&t1); |
| mpd_del(&t2); |
| mpd_del(&ulp); |
| mpd_del(&aa); |
| } |
| else { |
| _mpd_qln(result, a, &workctx, status); |
| mpd_check_underflow(result, &workctx, status); |
| mpd_qfinalize(result, &workctx, status); |
| } |
| } |
| |
| /* |
| * Internal log10() function that does not check for specials, zero or one. |
| * Case SKIP_FINALIZE: |
| * Relative error: abs(result - log10(a)) < 0.1 * 10**-prec * abs(log10(a)) |
| * Case DO_FINALIZE: |
| * Ulp error: abs(result - log10(a)) < ulp(log10(a)) |
| */ |
| enum {SKIP_FINALIZE, DO_FINALIZE}; |
| static void |
| _mpd_qlog10(int action, mpd_t *result, const mpd_t *a, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_context_t workctx; |
| MPD_NEW_STATIC(ln10,0,0,0,0); |
| |
| mpd_maxcontext(&workctx); |
| workctx.prec = ctx->prec + 3; |
| /* relative error: 0.1 * 10**(-p-3). The specific underflow shortcut |
| * in _mpd_qln() does not change the final result. */ |
| _mpd_qln(result, a, &workctx, status); |
| /* relative error: 5 * 10**(-p-3) */ |
| mpd_qln10(&ln10, workctx.prec, status); |
| |
| if (action == DO_FINALIZE) { |
| workctx = *ctx; |
| workctx.round = MPD_ROUND_HALF_EVEN; |
| } |
| /* SKIP_FINALIZE: relative error: 5 * 10**(-p-3) */ |
| _mpd_qdiv(NO_IDEAL_EXP, result, result, &ln10, &workctx, status); |
| |
| mpd_del(&ln10); |
| } |
| |
| /* log10(a) */ |
| void |
| mpd_qlog10(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_context_t workctx; |
| mpd_ssize_t adjexp, t; |
| |
| workctx = *ctx; |
| workctx.round = MPD_ROUND_HALF_EVEN; |
| |
| if (mpd_isspecial(a)) { |
| if (mpd_qcheck_nan(result, a, ctx, status)) { |
| return; |
| } |
| if (mpd_isnegative(a)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| mpd_setspecial(result, MPD_POS, MPD_INF); |
| return; |
| } |
| if (mpd_iszerocoeff(a)) { |
| mpd_setspecial(result, MPD_NEG, MPD_INF); |
| return; |
| } |
| if (mpd_isnegative(a)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| if (mpd_coeff_ispow10(a)) { |
| uint8_t sign = 0; |
| adjexp = mpd_adjexp(a); |
| if (adjexp < 0) { |
| sign = 1; |
| adjexp = -adjexp; |
| } |
| _settriple(result, sign, adjexp, 0); |
| mpd_qfinalize(result, &workctx, status); |
| return; |
| } |
| /* |
| * Check if the result will overflow (0 < x, x != 1): |
| * 1) log10(x) < 0 iff adjexp(x) < 0 |
| * 2) 0 < x /\ x <= y ==> adjexp(x) <= adjexp(y) |
| * 3) adjexp(x) <= log10(x) < adjexp(x) + 1 |
| * |
| * Case adjexp(x) >= 0: |
| * 4) adjexp(x) <= abs(log10(x)) |
| * Case adjexp(x) > 0: |
| * 5) adjexp(adjexp(x)) <= adjexp(abs(log10(x))) |
| * Case adjexp(x) == 0: |
| * mpd_exp_digits(t)-1 == 0 <= emax (the shortcut is not triggered) |
| * |
| * Case adjexp(x) < 0: |
| * 6) -adjexp(x) - 1 < abs(log10(x)) |
| * Case adjexp(x) < -1: |
| * 7) adjexp(-adjexp(x) - 1) <= adjexp(abs(log(x))) |
| * Case adjexp(x) == -1: |
| * mpd_exp_digits(t)-1 == 0 <= emax (the shortcut is not triggered) |
| */ |
| adjexp = mpd_adjexp(a); |
| t = (adjexp < 0) ? -adjexp-1 : adjexp; |
| if (mpd_exp_digits(t)-1 > ctx->emax) { |
| *status |= MPD_Overflow|MPD_Inexact|MPD_Rounded; |
| mpd_setspecial(result, (adjexp<0), MPD_INF); |
| return; |
| } |
| |
| if (ctx->allcr) { |
| MPD_NEW_STATIC(t1, 0,0,0,0); |
| MPD_NEW_STATIC(t2, 0,0,0,0); |
| MPD_NEW_STATIC(ulp, 0,0,0,0); |
| MPD_NEW_STATIC(aa, 0,0,0,0); |
| mpd_ssize_t prec; |
| |
| if (result == a) { |
| if (!mpd_qcopy(&aa, a, status)) { |
| mpd_seterror(result, MPD_Malloc_error, status); |
| return; |
| } |
| a = &aa; |
| } |
| |
| workctx.clamp = 0; |
| prec = ctx->prec + 3; |
| while (1) { |
| workctx.prec = prec; |
| _mpd_qlog10(SKIP_FINALIZE, result, a, &workctx, status); |
| _ssettriple(&ulp, MPD_POS, 1, |
| result->exp + result->digits-workctx.prec); |
| |
| workctx.prec = ctx->prec; |
| mpd_qadd(&t1, result, &ulp, &workctx, &workctx.status); |
| mpd_qsub(&t2, result, &ulp, &workctx, &workctx.status); |
| if (mpd_isspecial(result) || mpd_iszerocoeff(result) || |
| mpd_qcmp(&t1, &t2, status) == 0) { |
| workctx.clamp = ctx->clamp; |
| mpd_check_underflow(result, &workctx, status); |
| mpd_qfinalize(result, &workctx, status); |
| break; |
| } |
| prec += MPD_RDIGITS; |
| } |
| mpd_del(&t1); |
| mpd_del(&t2); |
| mpd_del(&ulp); |
| mpd_del(&aa); |
| } |
| else { |
| _mpd_qlog10(DO_FINALIZE, result, a, &workctx, status); |
| mpd_check_underflow(result, &workctx, status); |
| } |
| } |
| |
| /* |
| * Maximum of the two operands. Attention: If one operand is a quiet NaN and the |
| * other is numeric, the numeric operand is returned. This may not be what one |
| * expects. |
| */ |
| void |
| mpd_qmax(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| int c; |
| |
| if (mpd_isqnan(a) && !mpd_isnan(b)) { |
| mpd_qcopy(result, b, status); |
| } |
| else if (mpd_isqnan(b) && !mpd_isnan(a)) { |
| mpd_qcopy(result, a, status); |
| } |
| else if (mpd_qcheck_nans(result, a, b, ctx, status)) { |
| return; |
| } |
| else { |
| c = _mpd_cmp(a, b); |
| if (c == 0) { |
| c = _mpd_cmp_numequal(a, b); |
| } |
| |
| if (c < 0) { |
| mpd_qcopy(result, b, status); |
| } |
| else { |
| mpd_qcopy(result, a, status); |
| } |
| } |
| |
| mpd_qfinalize(result, ctx, status); |
| } |
| |
| /* |
| * Maximum magnitude: Same as mpd_max(), but compares the operands with their |
| * sign ignored. |
| */ |
| void |
| mpd_qmax_mag(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| int c; |
| |
| if (mpd_isqnan(a) && !mpd_isnan(b)) { |
| mpd_qcopy(result, b, status); |
| } |
| else if (mpd_isqnan(b) && !mpd_isnan(a)) { |
| mpd_qcopy(result, a, status); |
| } |
| else if (mpd_qcheck_nans(result, a, b, ctx, status)) { |
| return; |
| } |
| else { |
| c = _mpd_cmp_abs(a, b); |
| if (c == 0) { |
| c = _mpd_cmp_numequal(a, b); |
| } |
| |
| if (c < 0) { |
| mpd_qcopy(result, b, status); |
| } |
| else { |
| mpd_qcopy(result, a, status); |
| } |
| } |
| |
| mpd_qfinalize(result, ctx, status); |
| } |
| |
| /* |
| * Minimum of the two operands. Attention: If one operand is a quiet NaN and the |
| * other is numeric, the numeric operand is returned. This may not be what one |
| * expects. |
| */ |
| void |
| mpd_qmin(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| int c; |
| |
| if (mpd_isqnan(a) && !mpd_isnan(b)) { |
| mpd_qcopy(result, b, status); |
| } |
| else if (mpd_isqnan(b) && !mpd_isnan(a)) { |
| mpd_qcopy(result, a, status); |
| } |
| else if (mpd_qcheck_nans(result, a, b, ctx, status)) { |
| return; |
| } |
| else { |
| c = _mpd_cmp(a, b); |
| if (c == 0) { |
| c = _mpd_cmp_numequal(a, b); |
| } |
| |
| if (c < 0) { |
| mpd_qcopy(result, a, status); |
| } |
| else { |
| mpd_qcopy(result, b, status); |
| } |
| } |
| |
| mpd_qfinalize(result, ctx, status); |
| } |
| |
| /* |
| * Minimum magnitude: Same as mpd_min(), but compares the operands with their |
| * sign ignored. |
| */ |
| void |
| mpd_qmin_mag(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| int c; |
| |
| if (mpd_isqnan(a) && !mpd_isnan(b)) { |
| mpd_qcopy(result, b, status); |
| } |
| else if (mpd_isqnan(b) && !mpd_isnan(a)) { |
| mpd_qcopy(result, a, status); |
| } |
| else if (mpd_qcheck_nans(result, a, b, ctx, status)) { |
| return; |
| } |
| else { |
| c = _mpd_cmp_abs(a, b); |
| if (c == 0) { |
| c = _mpd_cmp_numequal(a, b); |
| } |
| |
| if (c < 0) { |
| mpd_qcopy(result, a, status); |
| } |
| else { |
| mpd_qcopy(result, b, status); |
| } |
| } |
| |
| mpd_qfinalize(result, ctx, status); |
| } |
| |
| /* Minimum space needed for the result array in _karatsuba_rec(). */ |
| static inline mpd_size_t |
| _kmul_resultsize(mpd_size_t la, mpd_size_t lb) |
| { |
| mpd_size_t n, m; |
| |
| n = add_size_t(la, lb); |
| n = add_size_t(n, 1); |
| |
| m = (la+1)/2 + 1; |
| m = mul_size_t(m, 3); |
| |
| return (m > n) ? m : n; |
| } |
| |
| /* Work space needed in _karatsuba_rec(). lim >= 4 */ |
| static inline mpd_size_t |
| _kmul_worksize(mpd_size_t n, mpd_size_t lim) |
| { |
| mpd_size_t m; |
| |
| if (n <= lim) { |
| return 0; |
| } |
| |
| m = (n+1)/2 + 1; |
| |
| return add_size_t(mul_size_t(m, 2), _kmul_worksize(m, lim)); |
| } |
| |
| |
| #define MPD_KARATSUBA_BASECASE 16 /* must be >= 4 */ |
| |
| /* |
| * Add the product of a and b to c. |
| * c must be _kmul_resultsize(la, lb) in size. |
| * w is used as a work array and must be _kmul_worksize(a, lim) in size. |
| * Roman E. Maeder, Storage Allocation for the Karatsuba Integer Multiplication |
| * Algorithm. In "Design and implementation of symbolic computation systems", |
| * Springer, 1993, ISBN 354057235X, 9783540572350. |
| */ |
| static void |
| _karatsuba_rec(mpd_uint_t *c, const mpd_uint_t *a, const mpd_uint_t *b, |
| mpd_uint_t *w, mpd_size_t la, mpd_size_t lb) |
| { |
| mpd_size_t m, lt; |
| |
| assert(la >= lb && lb > 0); |
| assert(la <= MPD_KARATSUBA_BASECASE || w != NULL); |
| |
| if (la <= MPD_KARATSUBA_BASECASE) { |
| _mpd_basemul(c, a, b, la, lb); |
| return; |
| } |
| |
| m = (la+1)/2; /* ceil(la/2) */ |
| |
| /* lb <= m < la */ |
| if (lb <= m) { |
| |
| /* lb can now be larger than la-m */ |
| if (lb > la-m) { |
| lt = lb + lb + 1; /* space needed for result array */ |
| mpd_uint_zero(w, lt); /* clear result array */ |
| _karatsuba_rec(w, b, a+m, w+lt, lb, la-m); /* b*ah */ |
| } |
| else { |
| lt = (la-m) + (la-m) + 1; /* space needed for result array */ |
| mpd_uint_zero(w, lt); /* clear result array */ |
| _karatsuba_rec(w, a+m, b, w+lt, la-m, lb); /* ah*b */ |
| } |
| _mpd_baseaddto(c+m, w, (la-m)+lb); /* add ah*b*B**m */ |
| |
| lt = m + m + 1; /* space needed for the result array */ |
| mpd_uint_zero(w, lt); /* clear result array */ |
| _karatsuba_rec(w, a, b, w+lt, m, lb); /* al*b */ |
| _mpd_baseaddto(c, w, m+lb); /* add al*b */ |
| |
| return; |
| } |
| |
| /* la >= lb > m */ |
| memcpy(w, a, m * sizeof *w); |
| w[m] = 0; |
| _mpd_baseaddto(w, a+m, la-m); |
| |
| memcpy(w+(m+1), b, m * sizeof *w); |
| w[m+1+m] = 0; |
| _mpd_baseaddto(w+(m+1), b+m, lb-m); |
| |
| _karatsuba_rec(c+m, w, w+(m+1), w+2*(m+1), m+1, m+1); |
| |
| lt = (la-m) + (la-m) + 1; |
| mpd_uint_zero(w, lt); |
| |
| _karatsuba_rec(w, a+m, b+m, w+lt, la-m, lb-m); |
| |
| _mpd_baseaddto(c+2*m, w, (la-m) + (lb-m)); |
| _mpd_basesubfrom(c+m, w, (la-m) + (lb-m)); |
| |
| lt = m + m + 1; |
| mpd_uint_zero(w, lt); |
| |
| _karatsuba_rec(w, a, b, w+lt, m, m); |
| _mpd_baseaddto(c, w, m+m); |
| _mpd_basesubfrom(c+m, w, m+m); |
| |
| return; |
| } |
| |
| /* |
| * Multiply u and v, using Karatsuba multiplication. Returns a pointer |
| * to the result or NULL in case of failure (malloc error). |
| * Conditions: ulen >= vlen, ulen >= 4 |
| */ |
| static mpd_uint_t * |
| _mpd_kmul(const mpd_uint_t *u, const mpd_uint_t *v, |
| mpd_size_t ulen, mpd_size_t vlen, |
| mpd_size_t *rsize) |
| { |
| mpd_uint_t *result = NULL, *w = NULL; |
| mpd_size_t m; |
| |
| assert(ulen >= 4); |
| assert(ulen >= vlen); |
| |
| *rsize = _kmul_resultsize(ulen, vlen); |
| if ((result = mpd_calloc(*rsize, sizeof *result)) == NULL) { |
| return NULL; |
| } |
| |
| m = _kmul_worksize(ulen, MPD_KARATSUBA_BASECASE); |
| if (m && ((w = mpd_calloc(m, sizeof *w)) == NULL)) { |
| mpd_free(result); |
| return NULL; |
| } |
| |
| _karatsuba_rec(result, u, v, w, ulen, vlen); |
| |
| |
| if (w) mpd_free(w); |
| return result; |
| } |
| |
| |
| /* |
| * Determine the minimum length for the number theoretic transform. Valid |
| * transform lengths are 2**n or 3*2**n, where 2**n <= MPD_MAXTRANSFORM_2N. |
| * The function finds the shortest length m such that rsize <= m. |
| */ |
| static inline mpd_size_t |
| _mpd_get_transform_len(mpd_size_t rsize) |
| { |
| mpd_size_t log2rsize; |
| mpd_size_t x, step; |
| |
| assert(rsize >= 4); |
| log2rsize = mpd_bsr(rsize); |
| |
| if (rsize <= 1024) { |
| /* 2**n is faster in this range. */ |
| x = ((mpd_size_t)1)<<log2rsize; |
| return (rsize == x) ? x : x<<1; |
| } |
| else if (rsize <= MPD_MAXTRANSFORM_2N) { |
| x = ((mpd_size_t)1)<<log2rsize; |
| if (rsize == x) return x; |
| step = x>>1; |
| x += step; |
| return (rsize <= x) ? x : x + step; |
| } |
| else if (rsize <= MPD_MAXTRANSFORM_2N+MPD_MAXTRANSFORM_2N/2) { |
| return MPD_MAXTRANSFORM_2N+MPD_MAXTRANSFORM_2N/2; |
| } |
| else if (rsize <= 3*MPD_MAXTRANSFORM_2N) { |
| return 3*MPD_MAXTRANSFORM_2N; |
| } |
| else { |
| return MPD_SIZE_MAX; |
| } |
| } |
| |
| #ifdef PPRO |
| #ifndef _MSC_VER |
| static inline unsigned short |
| _mpd_get_control87(void) |
| { |
| unsigned short cw; |
| |
| __asm__ __volatile__ ("fnstcw %0" : "=m" (cw)); |
| return cw; |
| } |
| |
| static inline void |
| _mpd_set_control87(unsigned short cw) |
| { |
| __asm__ __volatile__ ("fldcw %0" : : "m" (cw)); |
| } |
| #endif |
| |
| static unsigned int |
| mpd_set_fenv(void) |
| { |
| unsigned int cw; |
| #ifdef _MSC_VER |
| unsigned int flags = |
| _EM_INVALID|_EM_DENORMAL|_EM_ZERODIVIDE|_EM_OVERFLOW| |
| _EM_UNDERFLOW|_EM_INEXACT|_RC_CHOP|_PC_64; |
| unsigned int mask = _MCW_EM|_MCW_RC|_MCW_PC; |
| unsigned int dummy; |
| |
| __control87_2(0, 0, &cw, NULL); |
| __control87_2(flags, mask, &dummy, NULL); |
| #else |
| cw = _mpd_get_control87(); |
| _mpd_set_control87(cw|0xF3F); |
| #endif |
| return cw; |
| } |
| |
| static void |
| mpd_restore_fenv(unsigned int cw) |
| { |
| #ifdef _MSC_VER |
| unsigned int mask = _MCW_EM|_MCW_RC|_MCW_PC; |
| unsigned int dummy; |
| |
| __control87_2(cw, mask, &dummy, NULL); |
| #else |
| _mpd_set_control87((unsigned short)cw); |
| #endif |
| } |
| #endif /* PPRO */ |
| |
| /* |
| * Multiply u and v, using the fast number theoretic transform. Returns |
| * a pointer to the result or NULL in case of failure (malloc error). |
| */ |
| static mpd_uint_t * |
| _mpd_fntmul(const mpd_uint_t *u, const mpd_uint_t *v, |
| mpd_size_t ulen, mpd_size_t vlen, |
| mpd_size_t *rsize) |
| { |
| mpd_uint_t *c1 = NULL, *c2 = NULL, *c3 = NULL, *vtmp = NULL; |
| mpd_size_t n; |
| |
| #ifdef PPRO |
| unsigned int cw; |
| cw = mpd_set_fenv(); |
| #endif |
| |
| *rsize = add_size_t(ulen, vlen); |
| if ((n = _mpd_get_transform_len(*rsize)) == MPD_SIZE_MAX) { |
| goto malloc_error; |
| } |
| |
| if ((c1 = mpd_calloc(n, sizeof *c1)) == NULL) { |
| goto malloc_error; |
| } |
| if ((c2 = mpd_calloc(n, sizeof *c2)) == NULL) { |
| goto malloc_error; |
| } |
| if ((c3 = mpd_calloc(n, sizeof *c3)) == NULL) { |
| goto malloc_error; |
| } |
| |
| memcpy(c1, u, ulen * (sizeof *c1)); |
| memcpy(c2, u, ulen * (sizeof *c2)); |
| memcpy(c3, u, ulen * (sizeof *c3)); |
| |
| if (u == v) { |
| if (!fnt_autoconvolute(c1, n, P1) || |
| !fnt_autoconvolute(c2, n, P2) || |
| !fnt_autoconvolute(c3, n, P3)) { |
| goto malloc_error; |
| } |
| } |
| else { |
| if ((vtmp = mpd_calloc(n, sizeof *vtmp)) == NULL) { |
| goto malloc_error; |
| } |
| |
| memcpy(vtmp, v, vlen * (sizeof *vtmp)); |
| if (!fnt_convolute(c1, vtmp, n, P1)) { |
| mpd_free(vtmp); |
| goto malloc_error; |
| } |
| |
| memcpy(vtmp, v, vlen * (sizeof *vtmp)); |
| mpd_uint_zero(vtmp+vlen, n-vlen); |
| if (!fnt_convolute(c2, vtmp, n, P2)) { |
| mpd_free(vtmp); |
| goto malloc_error; |
| } |
| |
| memcpy(vtmp, v, vlen * (sizeof *vtmp)); |
| mpd_uint_zero(vtmp+vlen, n-vlen); |
| if (!fnt_convolute(c3, vtmp, n, P3)) { |
| mpd_free(vtmp); |
| goto malloc_error; |
| } |
| |
| mpd_free(vtmp); |
| } |
| |
| crt3(c1, c2, c3, *rsize); |
| |
| out: |
| #ifdef PPRO |
| mpd_restore_fenv(cw); |
| #endif |
| if (c2) mpd_free(c2); |
| if (c3) mpd_free(c3); |
| return c1; |
| |
| malloc_error: |
| if (c1) mpd_free(c1); |
| c1 = NULL; |
| goto out; |
| } |
| |
| |
| /* |
| * Karatsuba multiplication with FNT/basemul as the base case. |
| */ |
| static int |
| _karatsuba_rec_fnt(mpd_uint_t *c, const mpd_uint_t *a, const mpd_uint_t *b, |
| mpd_uint_t *w, mpd_size_t la, mpd_size_t lb) |
| { |
| mpd_size_t m, lt; |
| |
| assert(la >= lb && lb > 0); |
| assert(la <= 3*(MPD_MAXTRANSFORM_2N/2) || w != NULL); |
| |
| if (la <= 3*(MPD_MAXTRANSFORM_2N/2)) { |
| |
| if (lb <= 192) { |
| _mpd_basemul(c, b, a, lb, la); |
| } |
| else { |
| mpd_uint_t *result; |
| mpd_size_t dummy; |
| |
| if ((result = _mpd_fntmul(a, b, la, lb, &dummy)) == NULL) { |
| return 0; |
| } |
| memcpy(c, result, (la+lb) * (sizeof *result)); |
| mpd_free(result); |
| } |
| return 1; |
| } |
| |
| m = (la+1)/2; /* ceil(la/2) */ |
| |
| /* lb <= m < la */ |
| if (lb <= m) { |
| |
| /* lb can now be larger than la-m */ |
| if (lb > la-m) { |
| lt = lb + lb + 1; /* space needed for result array */ |
| mpd_uint_zero(w, lt); /* clear result array */ |
| if (!_karatsuba_rec_fnt(w, b, a+m, w+lt, lb, la-m)) { /* b*ah */ |
| return 0; /* GCOV_UNLIKELY */ |
| } |
| } |
| else { |
| lt = (la-m) + (la-m) + 1; /* space needed for result array */ |
| mpd_uint_zero(w, lt); /* clear result array */ |
| if (!_karatsuba_rec_fnt(w, a+m, b, w+lt, la-m, lb)) { /* ah*b */ |
| return 0; /* GCOV_UNLIKELY */ |
| } |
| } |
| _mpd_baseaddto(c+m, w, (la-m)+lb); /* add ah*b*B**m */ |
| |
| lt = m + m + 1; /* space needed for the result array */ |
| mpd_uint_zero(w, lt); /* clear result array */ |
| if (!_karatsuba_rec_fnt(w, a, b, w+lt, m, lb)) { /* al*b */ |
| return 0; /* GCOV_UNLIKELY */ |
| } |
| _mpd_baseaddto(c, w, m+lb); /* add al*b */ |
| |
| return 1; |
| } |
| |
| /* la >= lb > m */ |
| memcpy(w, a, m * sizeof *w); |
| w[m] = 0; |
| _mpd_baseaddto(w, a+m, la-m); |
| |
| memcpy(w+(m+1), b, m * sizeof *w); |
| w[m+1+m] = 0; |
| _mpd_baseaddto(w+(m+1), b+m, lb-m); |
| |
| if (!_karatsuba_rec_fnt(c+m, w, w+(m+1), w+2*(m+1), m+1, m+1)) { |
| return 0; /* GCOV_UNLIKELY */ |
| } |
| |
| lt = (la-m) + (la-m) + 1; |
| mpd_uint_zero(w, lt); |
| |
| if (!_karatsuba_rec_fnt(w, a+m, b+m, w+lt, la-m, lb-m)) { |
| return 0; /* GCOV_UNLIKELY */ |
| } |
| |
| _mpd_baseaddto(c+2*m, w, (la-m) + (lb-m)); |
| _mpd_basesubfrom(c+m, w, (la-m) + (lb-m)); |
| |
| lt = m + m + 1; |
| mpd_uint_zero(w, lt); |
| |
| if (!_karatsuba_rec_fnt(w, a, b, w+lt, m, m)) { |
| return 0; /* GCOV_UNLIKELY */ |
| } |
| _mpd_baseaddto(c, w, m+m); |
| _mpd_basesubfrom(c+m, w, m+m); |
| |
| return 1; |
| } |
| |
| /* |
| * Multiply u and v, using Karatsuba multiplication with the FNT as the |
| * base case. Returns a pointer to the result or NULL in case of failure |
| * (malloc error). Conditions: ulen >= vlen, ulen >= 4. |
| */ |
| static mpd_uint_t * |
| _mpd_kmul_fnt(const mpd_uint_t *u, const mpd_uint_t *v, |
| mpd_size_t ulen, mpd_size_t vlen, |
| mpd_size_t *rsize) |
| { |
| mpd_uint_t *result = NULL, *w = NULL; |
| mpd_size_t m; |
| |
| assert(ulen >= 4); |
| assert(ulen >= vlen); |
| |
| *rsize = _kmul_resultsize(ulen, vlen); |
| if ((result = mpd_calloc(*rsize, sizeof *result)) == NULL) { |
| return NULL; |
| } |
| |
| m = _kmul_worksize(ulen, 3*(MPD_MAXTRANSFORM_2N/2)); |
| if (m && ((w = mpd_calloc(m, sizeof *w)) == NULL)) { |
| mpd_free(result); /* GCOV_UNLIKELY */ |
| return NULL; /* GCOV_UNLIKELY */ |
| } |
| |
| if (!_karatsuba_rec_fnt(result, u, v, w, ulen, vlen)) { |
| mpd_free(result); |
| result = NULL; |
| } |
| |
| |
| if (w) mpd_free(w); |
| return result; |
| } |
| |
| |
| /* Deal with the special cases of multiplying infinities. */ |
| static void |
| _mpd_qmul_inf(mpd_t *result, const mpd_t *a, const mpd_t *b, uint32_t *status) |
| { |
| if (mpd_isinfinite(a)) { |
| if (mpd_iszero(b)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| } |
| else { |
| mpd_setspecial(result, mpd_sign(a)^mpd_sign(b), MPD_INF); |
| } |
| return; |
| } |
| assert(mpd_isinfinite(b)); |
| if (mpd_iszero(a)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| } |
| else { |
| mpd_setspecial(result, mpd_sign(a)^mpd_sign(b), MPD_INF); |
| } |
| } |
| |
| /* |
| * Internal function: Multiply a and b. _mpd_qmul deals with specials but |
| * does NOT finalize the result. This is for use in mpd_fma(). |
| */ |
| static inline void |
| _mpd_qmul(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_t *big = (mpd_t *)a, *small = (mpd_t *)b; |
| mpd_uint_t *rdata = NULL; |
| mpd_uint_t rbuf[MPD_MINALLOC_MAX]; |
| mpd_size_t rsize, i; |
| |
| |
| if (mpd_isspecial(a) || mpd_isspecial(b)) { |
| if (mpd_qcheck_nans(result, a, b, ctx, status)) { |
| return; |
| } |
| _mpd_qmul_inf(result, a, b, status); |
| return; |
| } |
| |
| if (small->len > big->len) { |
| _mpd_ptrswap(&big, &small); |
| } |
| |
| rsize = big->len + small->len; |
| |
| if (big->len == 1) { |
| _mpd_singlemul(result->data, big->data[0], small->data[0]); |
| goto finish; |
| } |
| if (rsize <= (mpd_size_t)MPD_MINALLOC_MAX) { |
| if (big->len == 2) { |
| _mpd_mul_2_le2(rbuf, big->data, small->data, small->len); |
| } |
| else { |
| mpd_uint_zero(rbuf, rsize); |
| if (small->len == 1) { |
| _mpd_shortmul(rbuf, big->data, big->len, small->data[0]); |
| } |
| else { |
| _mpd_basemul(rbuf, small->data, big->data, small->len, big->len); |
| } |
| } |
| if (!mpd_qresize(result, rsize, status)) { |
| return; |
| } |
| for(i = 0; i < rsize; i++) { |
| result->data[i] = rbuf[i]; |
| } |
| goto finish; |
| } |
| |
| |
| if (small->len <= 256) { |
| rdata = mpd_calloc(rsize, sizeof *rdata); |
| if (rdata != NULL) { |
| if (small->len == 1) { |
| _mpd_shortmul(rdata, big->data, big->len, small->data[0]); |
| } |
| else { |
| _mpd_basemul(rdata, small->data, big->data, small->len, big->len); |
| } |
| } |
| } |
| else if (rsize <= 1024) { |
| rdata = _mpd_kmul(big->data, small->data, big->len, small->len, &rsize); |
| } |
| else if (rsize <= 3*MPD_MAXTRANSFORM_2N) { |
| rdata = _mpd_fntmul(big->data, small->data, big->len, small->len, &rsize); |
| } |
| else { |
| rdata = _mpd_kmul_fnt(big->data, small->data, big->len, small->len, &rsize); |
| } |
| |
| if (rdata == NULL) { |
| mpd_seterror(result, MPD_Malloc_error, status); |
| return; |
| } |
| |
| if (mpd_isdynamic_data(result)) { |
| mpd_free(result->data); |
| } |
| result->data = rdata; |
| result->alloc = rsize; |
| mpd_set_dynamic_data(result); |
| |
| |
| finish: |
| mpd_set_flags(result, mpd_sign(a)^mpd_sign(b)); |
| result->exp = big->exp + small->exp; |
| result->len = _mpd_real_size(result->data, rsize); |
| /* resize to smaller cannot fail */ |
| mpd_qresize(result, result->len, status); |
| mpd_setdigits(result); |
| } |
| |
| /* Multiply a and b. */ |
| void |
| mpd_qmul(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| _mpd_qmul(result, a, b, ctx, status); |
| mpd_qfinalize(result, ctx, status); |
| } |
| |
| /* Multiply a and b. Set NaN/Invalid_operation if the result is inexact. */ |
| static void |
| _mpd_qmul_exact(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| uint32_t workstatus = 0; |
| |
| mpd_qmul(result, a, b, ctx, &workstatus); |
| *status |= workstatus; |
| if (workstatus & (MPD_Inexact|MPD_Rounded|MPD_Clamped)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| } |
| } |
| |
| /* Multiply decimal and mpd_ssize_t. */ |
| void |
| mpd_qmul_ssize(mpd_t *result, const mpd_t *a, mpd_ssize_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_context_t maxcontext; |
| MPD_NEW_STATIC(bb,0,0,0,0); |
| |
| mpd_maxcontext(&maxcontext); |
| mpd_qsset_ssize(&bb, b, &maxcontext, status); |
| mpd_qmul(result, a, &bb, ctx, status); |
| mpd_del(&bb); |
| } |
| |
| /* Multiply decimal and mpd_uint_t. */ |
| void |
| mpd_qmul_uint(mpd_t *result, const mpd_t *a, mpd_uint_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_context_t maxcontext; |
| MPD_NEW_STATIC(bb,0,0,0,0); |
| |
| mpd_maxcontext(&maxcontext); |
| mpd_qsset_uint(&bb, b, &maxcontext, status); |
| mpd_qmul(result, a, &bb, ctx, status); |
| mpd_del(&bb); |
| } |
| |
| void |
| mpd_qmul_i32(mpd_t *result, const mpd_t *a, int32_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_qmul_ssize(result, a, b, ctx, status); |
| } |
| |
| void |
| mpd_qmul_u32(mpd_t *result, const mpd_t *a, uint32_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_qmul_uint(result, a, b, ctx, status); |
| } |
| |
| #ifdef CONFIG_64 |
| void |
| mpd_qmul_i64(mpd_t *result, const mpd_t *a, int64_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_qmul_ssize(result, a, b, ctx, status); |
| } |
| |
| void |
| mpd_qmul_u64(mpd_t *result, const mpd_t *a, uint64_t b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_qmul_uint(result, a, b, ctx, status); |
| } |
| #endif |
| |
| /* Like the minus operator. */ |
| void |
| mpd_qminus(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| if (mpd_isspecial(a)) { |
| if (mpd_qcheck_nan(result, a, ctx, status)) { |
| return; |
| } |
| } |
| |
| if (mpd_iszero(a) && ctx->round != MPD_ROUND_FLOOR) { |
| mpd_qcopy_abs(result, a, status); |
| } |
| else { |
| mpd_qcopy_negate(result, a, status); |
| } |
| |
| mpd_qfinalize(result, ctx, status); |
| } |
| |
| /* Like the plus operator. */ |
| void |
| mpd_qplus(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| if (mpd_isspecial(a)) { |
| if (mpd_qcheck_nan(result, a, ctx, status)) { |
| return; |
| } |
| } |
| |
| if (mpd_iszero(a) && ctx->round != MPD_ROUND_FLOOR) { |
| mpd_qcopy_abs(result, a, status); |
| } |
| else { |
| mpd_qcopy(result, a, status); |
| } |
| |
| mpd_qfinalize(result, ctx, status); |
| } |
| |
| /* The largest representable number that is smaller than the operand. */ |
| void |
| mpd_qnext_minus(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_context_t workctx; |
| MPD_NEW_CONST(tiny,MPD_POS,mpd_etiny(ctx)-1,1,1,1,1); |
| |
| if (mpd_isspecial(a)) { |
| if (mpd_qcheck_nan(result, a, ctx, status)) { |
| return; |
| } |
| |
| assert(mpd_isinfinite(a)); |
| if (mpd_isnegative(a)) { |
| mpd_qcopy(result, a, status); |
| return; |
| } |
| else { |
| mpd_clear_flags(result); |
| mpd_qmaxcoeff(result, ctx, status); |
| if (mpd_isnan(result)) { |
| return; |
| } |
| result->exp = mpd_etop(ctx); |
| return; |
| } |
| } |
| |
| mpd_workcontext(&workctx, ctx); |
| workctx.round = MPD_ROUND_FLOOR; |
| |
| if (!mpd_qcopy(result, a, status)) { |
| return; |
| } |
| |
| mpd_qfinalize(result, &workctx, &workctx.status); |
| if (workctx.status&(MPD_Inexact|MPD_Errors)) { |
| *status |= (workctx.status&MPD_Errors); |
| return; |
| } |
| |
| workctx.status = 0; |
| mpd_qsub(result, a, &tiny, &workctx, &workctx.status); |
| *status |= (workctx.status&MPD_Errors); |
| } |
| |
| /* The smallest representable number that is larger than the operand. */ |
| void |
| mpd_qnext_plus(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_context_t workctx; |
| MPD_NEW_CONST(tiny,MPD_POS,mpd_etiny(ctx)-1,1,1,1,1); |
| |
| if (mpd_isspecial(a)) { |
| if (mpd_qcheck_nan(result, a, ctx, status)) { |
| return; |
| } |
| |
| assert(mpd_isinfinite(a)); |
| if (mpd_ispositive(a)) { |
| mpd_qcopy(result, a, status); |
| } |
| else { |
| mpd_clear_flags(result); |
| mpd_qmaxcoeff(result, ctx, status); |
| if (mpd_isnan(result)) { |
| return; |
| } |
| mpd_set_flags(result, MPD_NEG); |
| result->exp = mpd_etop(ctx); |
| } |
| return; |
| } |
| |
| mpd_workcontext(&workctx, ctx); |
| workctx.round = MPD_ROUND_CEILING; |
| |
| if (!mpd_qcopy(result, a, status)) { |
| return; |
| } |
| |
| mpd_qfinalize(result, &workctx, &workctx.status); |
| if (workctx.status & (MPD_Inexact|MPD_Errors)) { |
| *status |= (workctx.status&MPD_Errors); |
| return; |
| } |
| |
| workctx.status = 0; |
| mpd_qadd(result, a, &tiny, &workctx, &workctx.status); |
| *status |= (workctx.status&MPD_Errors); |
| } |
| |
| /* |
| * The number closest to the first operand that is in the direction towards |
| * the second operand. |
| */ |
| void |
| mpd_qnext_toward(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| int c; |
| |
| if (mpd_qcheck_nans(result, a, b, ctx, status)) { |
| return; |
| } |
| |
| c = _mpd_cmp(a, b); |
| if (c == 0) { |
| mpd_qcopy_sign(result, a, b, status); |
| return; |
| } |
| |
| if (c < 0) { |
| mpd_qnext_plus(result, a, ctx, status); |
| } |
| else { |
| mpd_qnext_minus(result, a, ctx, status); |
| } |
| |
| if (mpd_isinfinite(result)) { |
| *status |= (MPD_Overflow|MPD_Rounded|MPD_Inexact); |
| } |
| else if (mpd_adjexp(result) < ctx->emin) { |
| *status |= (MPD_Underflow|MPD_Subnormal|MPD_Rounded|MPD_Inexact); |
| if (mpd_iszero(result)) { |
| *status |= MPD_Clamped; |
| } |
| } |
| } |
| |
| /* |
| * Internal function: Integer power with mpd_uint_t exponent. The function |
| * can fail with MPD_Malloc_error. |
| * |
| * The error is equal to the error incurred in k-1 multiplications. Assuming |
| * the upper bound for the relative error in each operation: |
| * |
| * abs(err) = 5 * 10**-prec |
| * result = x**k * (1 + err)**(k-1) |
| */ |
| static inline void |
| _mpd_qpow_uint(mpd_t *result, const mpd_t *base, mpd_uint_t exp, |
| uint8_t resultsign, const mpd_context_t *ctx, uint32_t *status) |
| { |
| uint32_t workstatus = 0; |
| mpd_uint_t n; |
| |
| if (exp == 0) { |
| _settriple(result, resultsign, 1, 0); /* GCOV_NOT_REACHED */ |
| return; /* GCOV_NOT_REACHED */ |
| } |
| |
| if (!mpd_qcopy(result, base, status)) { |
| return; |
| } |
| |
| n = mpd_bits[mpd_bsr(exp)]; |
| while (n >>= 1) { |
| mpd_qmul(result, result, result, ctx, &workstatus); |
| if (exp & n) { |
| mpd_qmul(result, result, base, ctx, &workstatus); |
| } |
| if (mpd_isspecial(result) || |
| (mpd_iszerocoeff(result) && (workstatus & MPD_Clamped))) { |
| break; |
| } |
| } |
| |
| *status |= workstatus; |
| mpd_set_sign(result, resultsign); |
| } |
| |
| /* |
| * Internal function: Integer power with mpd_t exponent, tbase and texp |
| * are modified!! Function can fail with MPD_Malloc_error. |
| * |
| * The error is equal to the error incurred in k multiplications. Assuming |
| * the upper bound for the relative error in each operation: |
| * |
| * abs(err) = 5 * 10**-prec |
| * result = x**k * (1 + err)**k |
| */ |
| static inline void |
| _mpd_qpow_mpd(mpd_t *result, mpd_t *tbase, mpd_t *texp, uint8_t resultsign, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| uint32_t workstatus = 0; |
| mpd_context_t maxctx; |
| MPD_NEW_CONST(two,0,0,1,1,1,2); |
| |
| |
| mpd_maxcontext(&maxctx); |
| |
| /* resize to smaller cannot fail */ |
| mpd_qcopy(result, &one, status); |
| |
| while (!mpd_iszero(texp)) { |
| if (mpd_isodd(texp)) { |
| mpd_qmul(result, result, tbase, ctx, &workstatus); |
| *status |= workstatus; |
| if (mpd_isspecial(result) || |
| (mpd_iszerocoeff(result) && (workstatus & MPD_Clamped))) { |
| break; |
| } |
| } |
| mpd_qmul(tbase, tbase, tbase, ctx, &workstatus); |
| mpd_qdivint(texp, texp, &two, &maxctx, &workstatus); |
| if (mpd_isnan(tbase) || mpd_isnan(texp)) { |
| mpd_seterror(result, workstatus&MPD_Errors, status); |
| return; |
| } |
| } |
| mpd_set_sign(result, resultsign); |
| } |
| |
| /* |
| * The power function for integer exponents. Relative error _before_ the |
| * final rounding to prec: |
| * abs(result - base**exp) < 0.1 * 10**-prec * abs(base**exp) |
| */ |
| static void |
| _mpd_qpow_int(mpd_t *result, const mpd_t *base, const mpd_t *exp, |
| uint8_t resultsign, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_context_t workctx; |
| MPD_NEW_STATIC(tbase,0,0,0,0); |
| MPD_NEW_STATIC(texp,0,0,0,0); |
| mpd_ssize_t n; |
| |
| |
| mpd_workcontext(&workctx, ctx); |
| workctx.prec += (exp->digits + exp->exp + 2); |
| workctx.round = MPD_ROUND_HALF_EVEN; |
| workctx.clamp = 0; |
| if (mpd_isnegative(exp)) { |
| workctx.prec += 1; |
| mpd_qdiv(&tbase, &one, base, &workctx, status); |
| if (*status&MPD_Errors) { |
| mpd_setspecial(result, MPD_POS, MPD_NAN); |
| goto finish; |
| } |
| } |
| else { |
| if (!mpd_qcopy(&tbase, base, status)) { |
| mpd_setspecial(result, MPD_POS, MPD_NAN); |
| goto finish; |
| } |
| } |
| |
| n = mpd_qabs_uint(exp, &workctx.status); |
| if (workctx.status&MPD_Invalid_operation) { |
| if (!mpd_qcopy(&texp, exp, status)) { |
| mpd_setspecial(result, MPD_POS, MPD_NAN); /* GCOV_UNLIKELY */ |
| goto finish; /* GCOV_UNLIKELY */ |
| } |
| _mpd_qpow_mpd(result, &tbase, &texp, resultsign, &workctx, status); |
| } |
| else { |
| _mpd_qpow_uint(result, &tbase, n, resultsign, &workctx, status); |
| } |
| |
| if (mpd_isinfinite(result)) { |
| /* for ROUND_DOWN, ROUND_FLOOR, etc. */ |
| _settriple(result, resultsign, 1, MPD_EXP_INF); |
| } |
| |
| finish: |
| mpd_del(&tbase); |
| mpd_del(&texp); |
| mpd_qfinalize(result, ctx, status); |
| } |
| |
| /* |
| * If the exponent is infinite and base equals one, the result is one |
| * with a coefficient of length prec. Otherwise, result is undefined. |
| * Return the value of the comparison against one. |
| */ |
| static int |
| _qcheck_pow_one_inf(mpd_t *result, const mpd_t *base, uint8_t resultsign, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_ssize_t shift; |
| int cmp; |
| |
| if ((cmp = _mpd_cmp(base, &one)) == 0) { |
| shift = ctx->prec-1; |
| mpd_qshiftl(result, &one, shift, status); |
| result->exp = -shift; |
| mpd_set_flags(result, resultsign); |
| *status |= (MPD_Inexact|MPD_Rounded); |
| } |
| |
| return cmp; |
| } |
| |
| /* |
| * If abs(base) equals one, calculate the correct power of one result. |
| * Otherwise, result is undefined. Return the value of the comparison |
| * against 1. |
| * |
| * This is an internal function that does not check for specials. |
| */ |
| static int |
| _qcheck_pow_one(mpd_t *result, const mpd_t *base, const mpd_t *exp, |
| uint8_t resultsign, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| uint32_t workstatus = 0; |
| mpd_ssize_t shift; |
| int cmp; |
| |
| if ((cmp = _mpd_cmp_abs(base, &one)) == 0) { |
| if (_mpd_isint(exp)) { |
| if (mpd_isnegative(exp)) { |
| _settriple(result, resultsign, 1, 0); |
| return 0; |
| } |
| /* 1.000**3 = 1.000000000 */ |
| mpd_qmul_ssize(result, exp, -base->exp, ctx, &workstatus); |
| if (workstatus&MPD_Errors) { |
| *status |= (workstatus&MPD_Errors); |
| return 0; |
| } |
| /* digits-1 after exponentiation */ |
| shift = mpd_qget_ssize(result, &workstatus); |
| /* shift is MPD_SSIZE_MAX if result is too large */ |
| if (shift > ctx->prec-1) { |
| shift = ctx->prec-1; |
| *status |= MPD_Rounded; |
| } |
| } |
| else if (mpd_ispositive(base)) { |
| shift = ctx->prec-1; |
| *status |= (MPD_Inexact|MPD_Rounded); |
| } |
| else { |
| return -2; /* GCOV_NOT_REACHED */ |
| } |
| if (!mpd_qshiftl(result, &one, shift, status)) { |
| return 0; |
| } |
| result->exp = -shift; |
| mpd_set_flags(result, resultsign); |
| } |
| |
| return cmp; |
| } |
| |
| /* |
| * Detect certain over/underflow of x**y. |
| * ACL2 proof: pow-bounds.lisp. |
| * |
| * Symbols: |
| * |
| * e: EXP_INF or EXP_CLAMP |
| * x: base |
| * y: exponent |
| * |
| * omega(e) = log10(abs(e)) |
| * zeta(x) = log10(abs(log10(x))) |
| * theta(y) = log10(abs(y)) |
| * |
| * Upper and lower bounds: |
| * |
| * ub_omega(e) = ceil(log10(abs(e))) |
| * lb_theta(y) = floor(log10(abs(y))) |
| * |
| * | floor(log10(floor(abs(log10(x))))) if x < 1/10 or x >= 10 |
| * lb_zeta(x) = | floor(log10(abs(x-1)/10)) if 1/10 <= x < 1 |
| * | floor(log10(abs((x-1)/100))) if 1 < x < 10 |
| * |
| * ub_omega(e) and lb_theta(y) are obviously upper and lower bounds |
| * for omega(e) and theta(y). |
| * |
| * lb_zeta is a lower bound for zeta(x): |
| * |
| * x < 1/10 or x >= 10: |
| * |
| * abs(log10(x)) >= 1, so the outer log10 is well defined. Since log10 |
| * is strictly increasing, the end result is a lower bound. |
| * |
| * 1/10 <= x < 1: |
| * |
| * We use: log10(x) <= (x-1)/log(10) |
| * abs(log10(x)) >= abs(x-1)/log(10) |
| * abs(log10(x)) >= abs(x-1)/10 |
| * |
| * 1 < x < 10: |
| * |
| * We use: (x-1)/(x*log(10)) < log10(x) |
| * abs((x-1)/100) < abs(log10(x)) |
| * |
| * XXX: abs((x-1)/10) would work, need ACL2 proof. |
| * |
| * |
| * Let (0 < x < 1 and y < 0) or (x > 1 and y > 0). (H1) |
| * Let ub_omega(exp_inf) < lb_zeta(x) + lb_theta(y) (H2) |
| * |
| * Then: |
| * log10(abs(exp_inf)) < log10(abs(log10(x))) + log10(abs(y)). (1) |
| * exp_inf < log10(x) * y (2) |
| * 10**exp_inf < x**y (3) |
| * |
| * Let (0 < x < 1 and y > 0) or (x > 1 and y < 0). (H3) |
| * Let ub_omega(exp_clamp) < lb_zeta(x) + lb_theta(y) (H4) |
| * |
| * Then: |
| * log10(abs(exp_clamp)) < log10(abs(log10(x))) + log10(abs(y)). (4) |
| * log10(x) * y < exp_clamp (5) |
| * x**y < 10**exp_clamp (6) |
| * |
| */ |
| static mpd_ssize_t |
| _lower_bound_zeta(const mpd_t *x, uint32_t *status) |
| { |
| mpd_context_t maxctx; |
| MPD_NEW_STATIC(scratch,0,0,0,0); |
| mpd_ssize_t t, u; |
| |
| t = mpd_adjexp(x); |
| if (t > 0) { |
| /* x >= 10 -> floor(log10(floor(abs(log10(x))))) */ |
| return mpd_exp_digits(t) - 1; |
| } |
| else if (t < -1) { |
| /* x < 1/10 -> floor(log10(floor(abs(log10(x))))) */ |
| return mpd_exp_digits(t+1) - 1; |
| } |
| else { |
| mpd_maxcontext(&maxctx); |
| mpd_qsub(&scratch, x, &one, &maxctx, status); |
| if (mpd_isspecial(&scratch)) { |
| mpd_del(&scratch); |
| return MPD_SSIZE_MAX; |
| } |
| u = mpd_adjexp(&scratch); |
| mpd_del(&scratch); |
| |
| /* t == -1, 1/10 <= x < 1 -> floor(log10(abs(x-1)/10)) |
| * t == 0, 1 < x < 10 -> floor(log10(abs(x-1)/100)) */ |
| return (t == 0) ? u-2 : u-1; |
| } |
| } |
| |
| /* |
| * Detect cases of certain overflow/underflow in the power function. |
| * Assumptions: x != 1, y != 0. The proof above is for positive x. |
| * If x is negative and y is an odd integer, x**y == -(abs(x)**y), |
| * so the analysis does not change. |
| */ |
| static int |
| _qcheck_pow_bounds(mpd_t *result, const mpd_t *x, const mpd_t *y, |
| uint8_t resultsign, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| MPD_NEW_SHARED(abs_x, x); |
| mpd_ssize_t ub_omega, lb_zeta, lb_theta; |
| uint8_t sign; |
| |
| mpd_set_positive(&abs_x); |
| |
| lb_theta = mpd_adjexp(y); |
| lb_zeta = _lower_bound_zeta(&abs_x, status); |
| if (lb_zeta == MPD_SSIZE_MAX) { |
| mpd_seterror(result, MPD_Malloc_error, status); |
| return 1; |
| } |
| |
| sign = (mpd_adjexp(&abs_x) < 0) ^ mpd_sign(y); |
| if (sign == 0) { |
| /* (0 < |x| < 1 and y < 0) or (|x| > 1 and y > 0) */ |
| ub_omega = mpd_exp_digits(ctx->emax); |
| if (ub_omega < lb_zeta + lb_theta) { |
| _settriple(result, resultsign, 1, MPD_EXP_INF); |
| mpd_qfinalize(result, ctx, status); |
| return 1; |
| } |
| } |
| else { |
| /* (0 < |x| < 1 and y > 0) or (|x| > 1 and y < 0). */ |
| ub_omega = mpd_exp_digits(mpd_etiny(ctx)); |
| if (ub_omega < lb_zeta + lb_theta) { |
| _settriple(result, resultsign, 1, mpd_etiny(ctx)-1); |
| mpd_qfinalize(result, ctx, status); |
| return 1; |
| } |
| } |
| |
| return 0; |
| } |
| |
| /* |
| * TODO: Implement algorithm for computing exact powers from decimal.py. |
| * In order to prevent infinite loops, this has to be called before |
| * using Ziv's strategy for correct rounding. |
| */ |
| /* |
| static int |
| _mpd_qpow_exact(mpd_t *result, const mpd_t *base, const mpd_t *exp, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| return 0; |
| } |
| */ |
| |
| /* |
| * The power function for real exponents. |
| * Relative error: abs(result - e**y) < e**y * 1/5 * 10**(-prec - 1) |
| */ |
| static void |
| _mpd_qpow_real(mpd_t *result, const mpd_t *base, const mpd_t *exp, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_context_t workctx; |
| MPD_NEW_STATIC(texp,0,0,0,0); |
| |
| if (!mpd_qcopy(&texp, exp, status)) { |
| mpd_seterror(result, MPD_Malloc_error, status); |
| return; |
| } |
| |
| mpd_maxcontext(&workctx); |
| workctx.prec = (base->digits > ctx->prec) ? base->digits : ctx->prec; |
| workctx.prec += (4 + MPD_EXPDIGITS); |
| workctx.round = MPD_ROUND_HALF_EVEN; |
| workctx.allcr = ctx->allcr; |
| |
| /* |
| * extra := MPD_EXPDIGITS = MPD_EXP_MAX_T |
| * wp := prec + 4 + extra |
| * abs(err) < 5 * 10**-wp |
| * y := log(base) * exp |
| * Calculate: |
| * 1) e**(y * (1 + err)**2) * (1 + err) |
| * = e**y * e**(y * (2*err + err**2)) * (1 + err) |
| * ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ |
| * Relative error of the underlined term: |
| * 2) abs(e**(y * (2*err + err**2)) - 1) |
| * Case abs(y) >= 10**extra: |
| * 3) adjexp(y)+1 > log10(abs(y)) >= extra |
| * This triggers the Overflow/Underflow shortcut in _mpd_qexp(), |
| * so no further analysis is necessary. |
| * Case abs(y) < 10**extra: |
| * 4) abs(y * (2*err + err**2)) < 1/5 * 10**(-prec - 2) |
| * Use (see _mpd_qexp): |
| * 5) abs(x) <= 9/10 * 10**-p ==> abs(e**x - 1) < 10**-p |
| * With 2), 4) and 5): |
| * 6) abs(e**(y * (2*err + err**2)) - 1) < 10**(-prec - 2) |
| * The complete relative error of 1) is: |
| * 7) abs(result - e**y) < e**y * 1/5 * 10**(-prec - 1) |
| */ |
| mpd_qln(result, base, &workctx, &workctx.status); |
| mpd_qmul(result, result, &texp, &workctx, &workctx.status); |
| mpd_qexp(result, result, &workctx, status); |
| |
| mpd_del(&texp); |
| *status |= (workctx.status&MPD_Errors); |
| *status |= (MPD_Inexact|MPD_Rounded); |
| } |
| |
| /* The power function: base**exp */ |
| void |
| mpd_qpow(mpd_t *result, const mpd_t *base, const mpd_t *exp, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| uint8_t resultsign = 0; |
| int intexp = 0; |
| int cmp; |
| |
| if (mpd_isspecial(base) || mpd_isspecial(exp)) { |
| if (mpd_qcheck_nans(result, base, exp, ctx, status)) { |
| return; |
| } |
| } |
| if (mpd_isinteger(exp)) { |
| intexp = 1; |
| resultsign = mpd_isnegative(base) && mpd_isodd(exp); |
| } |
| |
| if (mpd_iszero(base)) { |
| if (mpd_iszero(exp)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| } |
| else if (mpd_isnegative(exp)) { |
| mpd_setspecial(result, resultsign, MPD_INF); |
| } |
| else { |
| _settriple(result, resultsign, 0, 0); |
| } |
| return; |
| } |
| if (mpd_isnegative(base)) { |
| if (!intexp || mpd_isinfinite(exp)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| } |
| if (mpd_isinfinite(exp)) { |
| /* power of one */ |
| cmp = _qcheck_pow_one_inf(result, base, resultsign, ctx, status); |
| if (cmp == 0) { |
| return; |
| } |
| else { |
| cmp *= mpd_arith_sign(exp); |
| if (cmp < 0) { |
| _settriple(result, resultsign, 0, 0); |
| } |
| else { |
| mpd_setspecial(result, resultsign, MPD_INF); |
| } |
| } |
| return; |
| } |
| if (mpd_isinfinite(base)) { |
| if (mpd_iszero(exp)) { |
| _settriple(result, resultsign, 1, 0); |
| } |
| else if (mpd_isnegative(exp)) { |
| _settriple(result, resultsign, 0, 0); |
| } |
| else { |
| mpd_setspecial(result, resultsign, MPD_INF); |
| } |
| return; |
| } |
| if (mpd_iszero(exp)) { |
| _settriple(result, resultsign, 1, 0); |
| return; |
| } |
| if (_qcheck_pow_one(result, base, exp, resultsign, ctx, status) == 0) { |
| return; |
| } |
| if (_qcheck_pow_bounds(result, base, exp, resultsign, ctx, status)) { |
| return; |
| } |
| |
| if (intexp) { |
| _mpd_qpow_int(result, base, exp, resultsign, ctx, status); |
| } |
| else { |
| _mpd_qpow_real(result, base, exp, ctx, status); |
| if (!mpd_isspecial(result) && _mpd_cmp(result, &one) == 0) { |
| mpd_ssize_t shift = ctx->prec-1; |
| mpd_qshiftl(result, &one, shift, status); |
| result->exp = -shift; |
| } |
| if (mpd_isinfinite(result)) { |
| /* for ROUND_DOWN, ROUND_FLOOR, etc. */ |
| _settriple(result, MPD_POS, 1, MPD_EXP_INF); |
| } |
| mpd_qfinalize(result, ctx, status); |
| } |
| } |
| |
| /* |
| * Internal function: Integer powmod with mpd_uint_t exponent, base is modified! |
| * Function can fail with MPD_Malloc_error. |
| */ |
| static inline void |
| _mpd_qpowmod_uint(mpd_t *result, mpd_t *base, mpd_uint_t exp, |
| const mpd_t *mod, uint32_t *status) |
| { |
| mpd_context_t maxcontext; |
| |
| mpd_maxcontext(&maxcontext); |
| |
| /* resize to smaller cannot fail */ |
| mpd_qcopy(result, &one, status); |
| |
| while (exp > 0) { |
| if (exp & 1) { |
| _mpd_qmul_exact(result, result, base, &maxcontext, status); |
| mpd_qrem(result, result, mod, &maxcontext, status); |
| } |
| _mpd_qmul_exact(base, base, base, &maxcontext, status); |
| mpd_qrem(base, base, mod, &maxcontext, status); |
| exp >>= 1; |
| } |
| } |
| |
| /* The powmod function: (base**exp) % mod */ |
| void |
| mpd_qpowmod(mpd_t *result, const mpd_t *base, const mpd_t *exp, |
| const mpd_t *mod, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_context_t maxcontext; |
| MPD_NEW_STATIC(tbase,0,0,0,0); |
| MPD_NEW_STATIC(texp,0,0,0,0); |
| MPD_NEW_STATIC(tmod,0,0,0,0); |
| MPD_NEW_STATIC(tmp,0,0,0,0); |
| MPD_NEW_CONST(two,0,0,1,1,1,2); |
| mpd_ssize_t tbase_exp, texp_exp; |
| mpd_ssize_t i; |
| mpd_t t; |
| mpd_uint_t r; |
| uint8_t sign; |
| |
| |
| if (mpd_isspecial(base) || mpd_isspecial(exp) || mpd_isspecial(mod)) { |
| if (mpd_qcheck_3nans(result, base, exp, mod, ctx, status)) { |
| return; |
| } |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| |
| |
| if (!_mpd_isint(base) || !_mpd_isint(exp) || !_mpd_isint(mod)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| if (mpd_iszerocoeff(mod)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| if (mod->digits+mod->exp > ctx->prec) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| |
| sign = (mpd_isnegative(base)) && (mpd_isodd(exp)); |
| if (mpd_iszerocoeff(exp)) { |
| if (mpd_iszerocoeff(base)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| r = (_mpd_cmp_abs(mod, &one)==0) ? 0 : 1; |
| _settriple(result, sign, r, 0); |
| return; |
| } |
| if (mpd_isnegative(exp)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| if (mpd_iszerocoeff(base)) { |
| _settriple(result, sign, 0, 0); |
| return; |
| } |
| |
| mpd_maxcontext(&maxcontext); |
| |
| mpd_qrescale(&tmod, mod, 0, &maxcontext, &maxcontext.status); |
| if (maxcontext.status&MPD_Errors) { |
| mpd_seterror(result, maxcontext.status&MPD_Errors, status); |
| goto out; |
| } |
| maxcontext.status = 0; |
| mpd_set_positive(&tmod); |
| |
| mpd_qround_to_int(&tbase, base, &maxcontext, status); |
| mpd_set_positive(&tbase); |
| tbase_exp = tbase.exp; |
| tbase.exp = 0; |
| |
| mpd_qround_to_int(&texp, exp, &maxcontext, status); |
| texp_exp = texp.exp; |
| texp.exp = 0; |
| |
| /* base = (base.int % modulo * pow(10, base.exp, modulo)) % modulo */ |
| mpd_qrem(&tbase, &tbase, &tmod, &maxcontext, status); |
| mpd_qshiftl(result, &one, tbase_exp, status); |
| mpd_qrem(result, result, &tmod, &maxcontext, status); |
| _mpd_qmul_exact(&tbase, &tbase, result, &maxcontext, status); |
| mpd_qrem(&tbase, &tbase, &tmod, &maxcontext, status); |
| if (mpd_isspecial(&tbase) || |
| mpd_isspecial(&texp) || |
| mpd_isspecial(&tmod)) { |
| goto mpd_errors; |
| } |
| |
| for (i = 0; i < texp_exp; i++) { |
| _mpd_qpowmod_uint(&tmp, &tbase, 10, &tmod, status); |
| t = tmp; |
| tmp = tbase; |
| tbase = t; |
| } |
| if (mpd_isspecial(&tbase)) { |
| goto mpd_errors; /* GCOV_UNLIKELY */ |
| } |
| |
| /* resize to smaller cannot fail */ |
| mpd_qcopy(result, &one, status); |
| while (mpd_isfinite(&texp) && !mpd_iszero(&texp)) { |
| if (mpd_isodd(&texp)) { |
| _mpd_qmul_exact(result, result, &tbase, &maxcontext, status); |
| mpd_qrem(result, result, &tmod, &maxcontext, status); |
| } |
| _mpd_qmul_exact(&tbase, &tbase, &tbase, &maxcontext, status); |
| mpd_qrem(&tbase, &tbase, &tmod, &maxcontext, status); |
| mpd_qdivint(&texp, &texp, &two, &maxcontext, status); |
| } |
| if (mpd_isspecial(&texp) || mpd_isspecial(&tbase) || |
| mpd_isspecial(&tmod) || mpd_isspecial(result)) { |
| /* MPD_Malloc_error */ |
| goto mpd_errors; |
| } |
| else { |
| mpd_set_sign(result, sign); |
| } |
| |
| out: |
| mpd_del(&tbase); |
| mpd_del(&texp); |
| mpd_del(&tmod); |
| mpd_del(&tmp); |
| return; |
| |
| mpd_errors: |
| mpd_setspecial(result, MPD_POS, MPD_NAN); |
| goto out; |
| } |
| |
| void |
| mpd_qquantize(mpd_t *result, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| uint32_t workstatus = 0; |
| mpd_ssize_t b_exp = b->exp; |
| mpd_ssize_t expdiff, shift; |
| mpd_uint_t rnd; |
| |
| if (mpd_isspecial(a) || mpd_isspecial(b)) { |
| if (mpd_qcheck_nans(result, a, b, ctx, status)) { |
| return; |
| } |
| if (mpd_isinfinite(a) && mpd_isinfinite(b)) { |
| mpd_qcopy(result, a, status); |
| return; |
| } |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| |
| if (b->exp > ctx->emax || b->exp < mpd_etiny(ctx)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| |
| if (mpd_iszero(a)) { |
| _settriple(result, mpd_sign(a), 0, b->exp); |
| mpd_qfinalize(result, ctx, status); |
| return; |
| } |
| |
| |
| expdiff = a->exp - b->exp; |
| if (a->digits + expdiff > ctx->prec) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| |
| if (expdiff >= 0) { |
| shift = expdiff; |
| if (!mpd_qshiftl(result, a, shift, status)) { |
| return; |
| } |
| result->exp = b_exp; |
| } |
| else { |
| /* At this point expdiff < 0 and a->digits+expdiff <= prec, |
| * so the shift before an increment will fit in prec. */ |
| shift = -expdiff; |
| rnd = mpd_qshiftr(result, a, shift, status); |
| if (rnd == MPD_UINT_MAX) { |
| return; |
| } |
| result->exp = b_exp; |
| if (!_mpd_apply_round_fit(result, rnd, ctx, status)) { |
| return; |
| } |
| workstatus |= MPD_Rounded; |
| if (rnd) { |
| workstatus |= MPD_Inexact; |
| } |
| } |
| |
| if (mpd_adjexp(result) > ctx->emax || |
| mpd_adjexp(result) < mpd_etiny(ctx)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| |
| *status |= workstatus; |
| mpd_qfinalize(result, ctx, status); |
| } |
| |
| void |
| mpd_qreduce(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_ssize_t shift, maxexp, maxshift; |
| uint8_t sign_a = mpd_sign(a); |
| |
| if (mpd_isspecial(a)) { |
| if (mpd_qcheck_nan(result, a, ctx, status)) { |
| return; |
| } |
| mpd_qcopy(result, a, status); |
| return; |
| } |
| |
| if (!mpd_qcopy(result, a, status)) { |
| return; |
| } |
| mpd_qfinalize(result, ctx, status); |
| if (mpd_isspecial(result)) { |
| return; |
| } |
| if (mpd_iszero(result)) { |
| _settriple(result, sign_a, 0, 0); |
| return; |
| } |
| |
| shift = mpd_trail_zeros(result); |
| maxexp = (ctx->clamp) ? mpd_etop(ctx) : ctx->emax; |
| /* After the finalizing above result->exp <= maxexp. */ |
| maxshift = maxexp - result->exp; |
| shift = (shift > maxshift) ? maxshift : shift; |
| |
| mpd_qshiftr_inplace(result, shift); |
| result->exp += shift; |
| } |
| |
| void |
| mpd_qrem(mpd_t *r, const mpd_t *a, const mpd_t *b, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| MPD_NEW_STATIC(q,0,0,0,0); |
| |
| if (mpd_isspecial(a) || mpd_isspecial(b)) { |
| if (mpd_qcheck_nans(r, a, b, ctx, status)) { |
| return; |
| } |
| if (mpd_isinfinite(a)) { |
| mpd_seterror(r, MPD_Invalid_operation, status); |
| return; |
| } |
| if (mpd_isinfinite(b)) { |
| mpd_qcopy(r, a, status); |
| mpd_qfinalize(r, ctx, status); |
| return; |
| } |
| /* debug */ |
| abort(); /* GCOV_NOT_REACHED */ |
| } |
| if (mpd_iszerocoeff(b)) { |
| if (mpd_iszerocoeff(a)) { |
| mpd_seterror(r, MPD_Division_undefined, status); |
| } |
| else { |
| mpd_seterror(r, MPD_Invalid_operation, status); |
| } |
| return; |
| } |
| |
| _mpd_qdivmod(&q, r, a, b, ctx, status); |
| mpd_del(&q); |
| mpd_qfinalize(r, ctx, status); |
| } |
| |
| void |
| mpd_qrem_near(mpd_t *r, const mpd_t *a, const mpd_t *b, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_context_t workctx; |
| MPD_NEW_STATIC(btmp,0,0,0,0); |
| MPD_NEW_STATIC(q,0,0,0,0); |
| mpd_ssize_t expdiff, qdigits; |
| int cmp, isodd, allnine; |
| |
| if (mpd_isspecial(a) || mpd_isspecial(b)) { |
| if (mpd_qcheck_nans(r, a, b, ctx, status)) { |
| return; |
| } |
| if (mpd_isinfinite(a)) { |
| mpd_seterror(r, MPD_Invalid_operation, status); |
| return; |
| } |
| if (mpd_isinfinite(b)) { |
| mpd_qcopy(r, a, status); |
| mpd_qfinalize(r, ctx, status); |
| return; |
| } |
| /* debug */ |
| abort(); /* GCOV_NOT_REACHED */ |
| } |
| if (mpd_iszerocoeff(b)) { |
| if (mpd_iszerocoeff(a)) { |
| mpd_seterror(r, MPD_Division_undefined, status); |
| } |
| else { |
| mpd_seterror(r, MPD_Invalid_operation, status); |
| } |
| return; |
| } |
| |
| if (r == b) { |
| if (!mpd_qcopy(&btmp, b, status)) { |
| mpd_seterror(r, MPD_Malloc_error, status); |
| return; |
| } |
| b = &btmp; |
| } |
| |
| _mpd_qdivmod(&q, r, a, b, ctx, status); |
| if (mpd_isnan(&q) || mpd_isnan(r)) { |
| goto finish; |
| } |
| if (mpd_iszerocoeff(r)) { |
| goto finish; |
| } |
| |
| expdiff = mpd_adjexp(b) - mpd_adjexp(r); |
| if (-1 <= expdiff && expdiff <= 1) { |
| |
| allnine = mpd_coeff_isallnine(&q); |
| qdigits = q.digits; |
| isodd = mpd_isodd(&q); |
| |
| mpd_maxcontext(&workctx); |
| if (mpd_sign(a) == mpd_sign(b)) { |
| /* sign(r) == sign(b) */ |
| _mpd_qsub(&q, r, b, &workctx, &workctx.status); |
| } |
| else { |
| /* sign(r) != sign(b) */ |
| _mpd_qadd(&q, r, b, &workctx, &workctx.status); |
| } |
| |
| if (workctx.status&MPD_Errors) { |
| mpd_seterror(r, workctx.status&MPD_Errors, status); |
| goto finish; |
| } |
| |
| cmp = _mpd_cmp_abs(&q, r); |
| if (cmp < 0 || (cmp == 0 && isodd)) { |
| /* abs(r) > abs(b)/2 or abs(r) == abs(b)/2 and isodd(quotient) */ |
| if (allnine && qdigits == ctx->prec) { |
| /* abs(quotient) + 1 == 10**prec */ |
| mpd_seterror(r, MPD_Division_impossible, status); |
| goto finish; |
| } |
| mpd_qcopy(r, &q, status); |
| } |
| } |
| |
| |
| finish: |
| mpd_del(&btmp); |
| mpd_del(&q); |
| mpd_qfinalize(r, ctx, status); |
| } |
| |
| static void |
| _mpd_qrescale(mpd_t *result, const mpd_t *a, mpd_ssize_t exp, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_ssize_t expdiff, shift; |
| mpd_uint_t rnd; |
| |
| if (mpd_isspecial(a)) { |
| mpd_qcopy(result, a, status); |
| return; |
| } |
| |
| if (mpd_iszero(a)) { |
| _settriple(result, mpd_sign(a), 0, exp); |
| return; |
| } |
| |
| expdiff = a->exp - exp; |
| if (expdiff >= 0) { |
| shift = expdiff; |
| if (a->digits + shift > MPD_MAX_PREC+1) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| if (!mpd_qshiftl(result, a, shift, status)) { |
| return; |
| } |
| result->exp = exp; |
| } |
| else { |
| shift = -expdiff; |
| rnd = mpd_qshiftr(result, a, shift, status); |
| if (rnd == MPD_UINT_MAX) { |
| return; |
| } |
| result->exp = exp; |
| _mpd_apply_round_excess(result, rnd, ctx, status); |
| *status |= MPD_Rounded; |
| if (rnd) { |
| *status |= MPD_Inexact; |
| } |
| } |
| |
| if (mpd_issubnormal(result, ctx)) { |
| *status |= MPD_Subnormal; |
| } |
| } |
| |
| /* |
| * Rescale a number so that it has exponent 'exp'. Does not regard context |
| * precision, emax, emin, but uses the rounding mode. Special numbers are |
| * quietly copied. Restrictions: |
| * |
| * MPD_MIN_ETINY <= exp <= MPD_MAX_EMAX+1 |
| * result->digits <= MPD_MAX_PREC+1 |
| */ |
| void |
| mpd_qrescale(mpd_t *result, const mpd_t *a, mpd_ssize_t exp, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| if (exp > MPD_MAX_EMAX+1 || exp < MPD_MIN_ETINY) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| |
| _mpd_qrescale(result, a, exp, ctx, status); |
| } |
| |
| /* |
| * Same as mpd_qrescale, but with relaxed restrictions. The result of this |
| * function should only be used for formatting a number and never as input |
| * for other operations. |
| * |
| * MPD_MIN_ETINY-MPD_MAX_PREC <= exp <= MPD_MAX_EMAX+1 |
| * result->digits <= MPD_MAX_PREC+1 |
| */ |
| void |
| mpd_qrescale_fmt(mpd_t *result, const mpd_t *a, mpd_ssize_t exp, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| if (exp > MPD_MAX_EMAX+1 || exp < MPD_MIN_ETINY-MPD_MAX_PREC) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| |
| _mpd_qrescale(result, a, exp, ctx, status); |
| } |
| |
| /* Round to an integer according to 'action' and ctx->round. */ |
| enum {TO_INT_EXACT, TO_INT_SILENT, TO_INT_TRUNC}; |
| static void |
| _mpd_qround_to_integral(int action, mpd_t *result, const mpd_t *a, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_uint_t rnd; |
| |
| if (mpd_isspecial(a)) { |
| if (mpd_qcheck_nan(result, a, ctx, status)) { |
| return; |
| } |
| mpd_qcopy(result, a, status); |
| return; |
| } |
| if (a->exp >= 0) { |
| mpd_qcopy(result, a, status); |
| return; |
| } |
| if (mpd_iszerocoeff(a)) { |
| _settriple(result, mpd_sign(a), 0, 0); |
| return; |
| } |
| |
| rnd = mpd_qshiftr(result, a, -a->exp, status); |
| if (rnd == MPD_UINT_MAX) { |
| return; |
| } |
| result->exp = 0; |
| |
| if (action == TO_INT_EXACT || action == TO_INT_SILENT) { |
| _mpd_apply_round_excess(result, rnd, ctx, status); |
| if (action == TO_INT_EXACT) { |
| *status |= MPD_Rounded; |
| if (rnd) { |
| *status |= MPD_Inexact; |
| } |
| } |
| } |
| } |
| |
| void |
| mpd_qround_to_intx(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| (void)_mpd_qround_to_integral(TO_INT_EXACT, result, a, ctx, status); |
| } |
| |
| void |
| mpd_qround_to_int(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| (void)_mpd_qround_to_integral(TO_INT_SILENT, result, a, ctx, status); |
| } |
| |
| void |
| mpd_qtrunc(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| (void)_mpd_qround_to_integral(TO_INT_TRUNC, result, a, ctx, status); |
| } |
| |
| void |
| mpd_qfloor(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_context_t workctx = *ctx; |
| workctx.round = MPD_ROUND_FLOOR; |
| (void)_mpd_qround_to_integral(TO_INT_SILENT, result, a, |
| &workctx, status); |
| } |
| |
| void |
| mpd_qceil(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_context_t workctx = *ctx; |
| workctx.round = MPD_ROUND_CEILING; |
| (void)_mpd_qround_to_integral(TO_INT_SILENT, result, a, |
| &workctx, status); |
| } |
| |
| int |
| mpd_same_quantum(const mpd_t *a, const mpd_t *b) |
| { |
| if (mpd_isspecial(a) || mpd_isspecial(b)) { |
| return ((mpd_isnan(a) && mpd_isnan(b)) || |
| (mpd_isinfinite(a) && mpd_isinfinite(b))); |
| } |
| |
| return a->exp == b->exp; |
| } |
| |
| /* Schedule the increase in precision for the Newton iteration. */ |
| static inline int |
| recpr_schedule_prec(mpd_ssize_t klist[MPD_MAX_PREC_LOG2], |
| mpd_ssize_t maxprec, mpd_ssize_t initprec) |
| { |
| mpd_ssize_t k; |
| int i; |
| |
| assert(maxprec > 0 && initprec > 0); |
| if (maxprec <= initprec) return -1; |
| |
| i = 0; k = maxprec; |
| do { |
| k = (k+1) / 2; |
| klist[i++] = k; |
| } while (k > initprec); |
| |
| return i-1; |
| } |
| |
| /* |
| * Initial approximation for the reciprocal: |
| * k_0 := MPD_RDIGITS-2 |
| * z_0 := 10**(-k_0) * floor(10**(2*k_0 + 2) / floor(v * 10**(k_0 + 2))) |
| * Absolute error: |
| * |1/v - z_0| < 10**(-k_0) |
| * ACL2 proof: maxerror-inverse-approx |
| */ |
| static void |
| _mpd_qreciprocal_approx(mpd_t *z, const mpd_t *v, uint32_t *status) |
| { |
| mpd_uint_t p10data[2] = {0, mpd_pow10[MPD_RDIGITS-2]}; |
| mpd_uint_t dummy, word; |
| int n; |
| |
| assert(v->exp == -v->digits); |
| |
| _mpd_get_msdigits(&dummy, &word, v, MPD_RDIGITS); |
| n = mpd_word_digits(word); |
| word *= mpd_pow10[MPD_RDIGITS-n]; |
| |
| mpd_qresize(z, 2, status); |
| (void)_mpd_shortdiv(z->data, p10data, 2, word); |
| |
| mpd_clear_flags(z); |
| z->exp = -(MPD_RDIGITS-2); |
| z->len = (z->data[1] == 0) ? 1 : 2; |
| mpd_setdigits(z); |
| } |
| |
| /* |
| * Reciprocal, calculated with Newton's Method. Assumption: result != a. |
| * NOTE: The comments in the function show that certain operations are |
| * exact. The proof for the maximum error is too long to fit in here. |
| * ACL2 proof: maxerror-inverse-complete |
| */ |
| static void |
| _mpd_qreciprocal(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_context_t varcontext, maxcontext; |
| mpd_t *z = result; /* current approximation */ |
| mpd_t *v; /* a, normalized to a number between 0.1 and 1 */ |
| MPD_NEW_SHARED(vtmp, a); /* v shares data with a */ |
| MPD_NEW_STATIC(s,0,0,0,0); /* temporary variable */ |
| MPD_NEW_STATIC(t,0,0,0,0); /* temporary variable */ |
| MPD_NEW_CONST(two,0,0,1,1,1,2); /* const 2 */ |
| mpd_ssize_t klist[MPD_MAX_PREC_LOG2]; |
| mpd_ssize_t adj, maxprec, initprec; |
| uint8_t sign = mpd_sign(a); |
| int i; |
| |
| assert(result != a); |
| |
| v = &vtmp; |
| mpd_clear_flags(v); |
| adj = v->digits + v->exp; |
| v->exp = -v->digits; |
| |
| /* Initial approximation */ |
| _mpd_qreciprocal_approx(z, v, status); |
| |
| mpd_maxcontext(&varcontext); |
| mpd_maxcontext(&maxcontext); |
| varcontext.round = maxcontext.round = MPD_ROUND_TRUNC; |
| varcontext.emax = maxcontext.emax = MPD_MAX_EMAX + 100; |
| varcontext.emin = maxcontext.emin = MPD_MIN_EMIN - 100; |
| maxcontext.prec = MPD_MAX_PREC + 100; |
| |
| maxprec = ctx->prec; |
| maxprec += 2; |
| initprec = MPD_RDIGITS-3; |
| |
| i = recpr_schedule_prec(klist, maxprec, initprec); |
| for (; i >= 0; i--) { |
| /* Loop invariant: z->digits <= klist[i]+7 */ |
| /* Let s := z**2, exact result */ |
| _mpd_qmul_exact(&s, z, z, &maxcontext, status); |
| varcontext.prec = 2*klist[i] + 5; |
| if (v->digits > varcontext.prec) { |
| /* Let t := v, truncated to n >= 2*k+5 fraction digits */ |
| mpd_qshiftr(&t, v, v->digits-varcontext.prec, status); |
| t.exp = -varcontext.prec; |
| /* Let t := trunc(v)*s, truncated to n >= 2*k+1 fraction digits */ |
| mpd_qmul(&t, &t, &s, &varcontext, status); |
| } |
| else { /* v->digits <= 2*k+5 */ |
| /* Let t := v*s, truncated to n >= 2*k+1 fraction digits */ |
| mpd_qmul(&t, v, &s, &varcontext, status); |
| } |
| /* Let s := 2*z, exact result */ |
| _mpd_qmul_exact(&s, z, &two, &maxcontext, status); |
| /* s.digits < t.digits <= 2*k+5, |adjexp(s)-adjexp(t)| <= 1, |
| * so the subtraction generates at most 2*k+6 <= klist[i+1]+7 |
| * digits. The loop invariant is preserved. */ |
| _mpd_qsub_exact(z, &s, &t, &maxcontext, status); |
| } |
| |
| if (!mpd_isspecial(z)) { |
| z->exp -= adj; |
| mpd_set_flags(z, sign); |
| } |
| |
| mpd_del(&s); |
| mpd_del(&t); |
| mpd_qfinalize(z, ctx, status); |
| } |
| |
| /* |
| * Internal function for large numbers: |
| * |
| * q, r = divmod(coeff(a), coeff(b)) |
| * |
| * Strategy: Multiply the dividend by the reciprocal of the divisor. The |
| * inexact result is fixed by a small loop, using at most one iteration. |
| * |
| * ACL2 proofs: |
| * ------------ |
| * 1) q is a natural number. (ndivmod-quotient-natp) |
| * 2) r is a natural number. (ndivmod-remainder-natp) |
| * 3) a = q * b + r (ndivmod-q*b+r==a) |
| * 4) r < b (ndivmod-remainder-<-b) |
| */ |
| static void |
| _mpd_base_ndivmod(mpd_t *q, mpd_t *r, const mpd_t *a, const mpd_t *b, |
| uint32_t *status) |
| { |
| mpd_context_t workctx; |
| mpd_t *qq = q, *rr = r; |
| mpd_t aa, bb; |
| int k; |
| |
| _mpd_copy_shared(&aa, a); |
| _mpd_copy_shared(&bb, b); |
| |
| mpd_set_positive(&aa); |
| mpd_set_positive(&bb); |
| aa.exp = 0; |
| bb.exp = 0; |
| |
| if (q == a || q == b) { |
| if ((qq = mpd_qnew()) == NULL) { |
| *status |= MPD_Malloc_error; |
| goto nanresult; |
| } |
| } |
| if (r == a || r == b) { |
| if ((rr = mpd_qnew()) == NULL) { |
| *status |= MPD_Malloc_error; |
| goto nanresult; |
| } |
| } |
| |
| mpd_maxcontext(&workctx); |
| |
| /* Let prec := adigits - bdigits + 4 */ |
| workctx.prec = a->digits - b->digits + 1 + 3; |
| if (a->digits > MPD_MAX_PREC || workctx.prec > MPD_MAX_PREC) { |
| *status |= MPD_Division_impossible; |
| goto nanresult; |
| } |
| |
| /* Let x := _mpd_qreciprocal(b, prec) |
| * Then x is bounded by: |
| * 1) 1/b - 10**(-prec - bdigits) < x < 1/b + 10**(-prec - bdigits) |
| * 2) 1/b - 10**(-adigits - 4) < x < 1/b + 10**(-adigits - 4) |
| */ |
| _mpd_qreciprocal(rr, &bb, &workctx, &workctx.status); |
| |
| /* Get an estimate for the quotient. Let q := a * x |
| * Then q is bounded by: |
| * 3) a/b - 10**-4 < q < a/b + 10**-4 |
| */ |
| _mpd_qmul(qq, &aa, rr, &workctx, &workctx.status); |
| /* Truncate q to an integer: |
| * 4) a/b - 2 < trunc(q) < a/b + 1 |
| */ |
| mpd_qtrunc(qq, qq, &workctx, &workctx.status); |
| |
| workctx.prec = aa.digits + 3; |
| workctx.emax = MPD_MAX_EMAX + 3; |
| workctx.emin = MPD_MIN_EMIN - 3; |
| /* Multiply the estimate for q by b: |
| * 5) a - 2 * b < trunc(q) * b < a + b |
| */ |
| _mpd_qmul(rr, &bb, qq, &workctx, &workctx.status); |
| /* Get the estimate for r such that a = q * b + r. */ |
| _mpd_qsub_exact(rr, &aa, rr, &workctx, &workctx.status); |
| |
| /* Fix the result. At this point -b < r < 2*b, so the correction loop |
| takes at most one iteration. */ |
| for (k = 0;; k++) { |
| if (mpd_isspecial(qq) || mpd_isspecial(rr)) { |
| *status |= (workctx.status&MPD_Errors); |
| goto nanresult; |
| } |
| if (k > 2) { /* Allow two iterations despite the proof. */ |
| mpd_err_warn("libmpdec: internal error in " /* GCOV_NOT_REACHED */ |
| "_mpd_base_ndivmod: please report"); /* GCOV_NOT_REACHED */ |
| *status |= MPD_Invalid_operation; /* GCOV_NOT_REACHED */ |
| goto nanresult; /* GCOV_NOT_REACHED */ |
| } |
| /* r < 0 */ |
| else if (_mpd_cmp(&zero, rr) == 1) { |
| _mpd_qadd_exact(rr, rr, &bb, &workctx, &workctx.status); |
| _mpd_qadd_exact(qq, qq, &minus_one, &workctx, &workctx.status); |
| } |
| /* 0 <= r < b */ |
| else if (_mpd_cmp(rr, &bb) == -1) { |
| break; |
| } |
| /* r >= b */ |
| else { |
| _mpd_qsub_exact(rr, rr, &bb, &workctx, &workctx.status); |
| _mpd_qadd_exact(qq, qq, &one, &workctx, &workctx.status); |
| } |
| } |
| |
| if (qq != q) { |
| if (!mpd_qcopy(q, qq, status)) { |
| goto nanresult; /* GCOV_UNLIKELY */ |
| } |
| mpd_del(qq); |
| } |
| if (rr != r) { |
| if (!mpd_qcopy(r, rr, status)) { |
| goto nanresult; /* GCOV_UNLIKELY */ |
| } |
| mpd_del(rr); |
| } |
| |
| *status |= (workctx.status&MPD_Errors); |
| return; |
| |
| |
| nanresult: |
| if (qq && qq != q) mpd_del(qq); |
| if (rr && rr != r) mpd_del(rr); |
| mpd_setspecial(q, MPD_POS, MPD_NAN); |
| mpd_setspecial(r, MPD_POS, MPD_NAN); |
| } |
| |
| /* LIBMPDEC_ONLY */ |
| /* |
| * Schedule the optimal precision increase for the Newton iteration. |
| * v := input operand |
| * z_0 := initial approximation |
| * initprec := natural number such that abs(sqrt(v) - z_0) < 10**-initprec |
| * maxprec := target precision |
| * |
| * For convenience the output klist contains the elements in reverse order: |
| * klist := [k_n-1, ..., k_0], where |
| * 1) k_0 <= initprec and |
| * 2) abs(sqrt(v) - result) < 10**(-2*k_n-1 + 2) <= 10**-maxprec. |
| */ |
| static inline int |
| invroot_schedule_prec(mpd_ssize_t klist[MPD_MAX_PREC_LOG2], |
| mpd_ssize_t maxprec, mpd_ssize_t initprec) |
| { |
| mpd_ssize_t k; |
| int i; |
| |
| assert(maxprec >= 3 && initprec >= 3); |
| if (maxprec <= initprec) return -1; |
| |
| i = 0; k = maxprec; |
| do { |
| k = (k+3) / 2; |
| klist[i++] = k; |
| } while (k > initprec); |
| |
| return i-1; |
| } |
| |
| /* |
| * Initial approximation for the inverse square root function. |
| * Input: |
| * v := rational number, with 1 <= v < 100 |
| * vhat := floor(v * 10**6) |
| * Output: |
| * z := approximation to 1/sqrt(v), such that abs(z - 1/sqrt(v)) < 10**-3. |
| */ |
| static inline void |
| _invroot_init_approx(mpd_t *z, mpd_uint_t vhat) |
| { |
| mpd_uint_t lo = 1000; |
| mpd_uint_t hi = 10000; |
| mpd_uint_t a, sq; |
| |
| assert(lo*lo <= vhat && vhat < (hi+1)*(hi+1)); |
| |
| for(;;) { |
| a = (lo + hi) / 2; |
| sq = a * a; |
| if (vhat >= sq) { |
| if (vhat < sq + 2*a + 1) { |
| break; |
| } |
| lo = a + 1; |
| } |
| else { |
| hi = a - 1; |
| } |
| } |
| |
| /* |
| * After the binary search we have: |
| * 1) a**2 <= floor(v * 10**6) < (a + 1)**2 |
| * This implies: |
| * 2) a**2 <= v * 10**6 < (a + 1)**2 |
| * 3) a <= sqrt(v) * 10**3 < a + 1 |
| * Since 10**3 <= a: |
| * 4) 0 <= 10**prec/a - 1/sqrt(v) < 10**-prec |
| * We have: |
| * 5) 10**3/a - 10**-3 < floor(10**9/a) * 10**-6 <= 10**3/a |
| * Merging 4) and 5): |
| * 6) abs(floor(10**9/a) * 10**-6 - 1/sqrt(v)) < 10**-3 |
| */ |
| mpd_minalloc(z); |
| mpd_clear_flags(z); |
| z->data[0] = 1000000000UL / a; |
| z->len = 1; |
| z->exp = -6; |
| mpd_setdigits(z); |
| } |
| |
| /* |
| * Set 'result' to 1/sqrt(a). |
| * Relative error: abs(result - 1/sqrt(a)) < 10**-prec * 1/sqrt(a) |
| */ |
| static void |
| _mpd_qinvroot(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| uint32_t workstatus = 0; |
| mpd_context_t varcontext, maxcontext; |
| mpd_t *z = result; /* current approximation */ |
| mpd_t *v; /* a, normalized to a number between 1 and 100 */ |
| MPD_NEW_SHARED(vtmp, a); /* by default v will share data with a */ |
| MPD_NEW_STATIC(s,0,0,0,0); /* temporary variable */ |
| MPD_NEW_STATIC(t,0,0,0,0); /* temporary variable */ |
| MPD_NEW_CONST(one_half,0,-1,1,1,1,5); |
| MPD_NEW_CONST(three,0,0,1,1,1,3); |
| mpd_ssize_t klist[MPD_MAX_PREC_LOG2]; |
| mpd_ssize_t ideal_exp, shift; |
| mpd_ssize_t adj, tz; |
| mpd_ssize_t maxprec, fracdigits; |
| mpd_uint_t vhat, dummy; |
| int i, n; |
| |
| |
| ideal_exp = -(a->exp - (a->exp & 1)) / 2; |
| |
| v = &vtmp; |
| if (result == a) { |
| if ((v = mpd_qncopy(a)) == NULL) { |
| mpd_seterror(result, MPD_Malloc_error, status); |
| return; |
| } |
| } |
| |
| /* normalize a to 1 <= v < 100 */ |
| if ((v->digits+v->exp) & 1) { |
| fracdigits = v->digits - 1; |
| v->exp = -fracdigits; |
| n = (v->digits > 7) ? 7 : (int)v->digits; |
| /* Let vhat := floor(v * 10**(2*initprec)) */ |
| _mpd_get_msdigits(&dummy, &vhat, v, n); |
| if (n < 7) { |
| vhat *= mpd_pow10[7-n]; |
| } |
| } |
| else { |
| fracdigits = v->digits - 2; |
| v->exp = -fracdigits; |
| n = (v->digits > 8) ? 8 : (int)v->digits; |
| /* Let vhat := floor(v * 10**(2*initprec)) */ |
| _mpd_get_msdigits(&dummy, &vhat, v, n); |
| if (n < 8) { |
| vhat *= mpd_pow10[8-n]; |
| } |
| } |
| adj = (a->exp-v->exp) / 2; |
| |
| /* initial approximation */ |
| _invroot_init_approx(z, vhat); |
| |
| mpd_maxcontext(&maxcontext); |
| mpd_maxcontext(&varcontext); |
| varcontext.round = MPD_ROUND_TRUNC; |
| maxprec = ctx->prec + 1; |
| |
| /* initprec == 3 */ |
| i = invroot_schedule_prec(klist, maxprec, 3); |
| for (; i >= 0; i--) { |
| varcontext.prec = 2*klist[i]+2; |
| mpd_qmul(&s, z, z, &maxcontext, &workstatus); |
| if (v->digits > varcontext.prec) { |
| shift = v->digits - varcontext.prec; |
| mpd_qshiftr(&t, v, shift, &workstatus); |
| t.exp += shift; |
| mpd_qmul(&t, &t, &s, &varcontext, &workstatus); |
| } |
| else { |
| mpd_qmul(&t, v, &s, &varcontext, &workstatus); |
| } |
| mpd_qsub(&t, &three, &t, &maxcontext, &workstatus); |
| mpd_qmul(z, z, &t, &varcontext, &workstatus); |
| mpd_qmul(z, z, &one_half, &maxcontext, &workstatus); |
| } |
| |
| z->exp -= adj; |
| |
| tz = mpd_trail_zeros(result); |
| shift = ideal_exp - result->exp; |
| shift = (tz > shift) ? shift : tz; |
| if (shift > 0) { |
| mpd_qshiftr_inplace(result, shift); |
| result->exp += shift; |
| } |
| |
| |
| mpd_del(&s); |
| mpd_del(&t); |
| if (v != &vtmp) mpd_del(v); |
| *status |= (workstatus&MPD_Errors); |
| *status |= (MPD_Rounded|MPD_Inexact); |
| } |
| |
| void |
| mpd_qinvroot(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_context_t workctx; |
| |
| if (mpd_isspecial(a)) { |
| if (mpd_qcheck_nan(result, a, ctx, status)) { |
| return; |
| } |
| if (mpd_isnegative(a)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| /* positive infinity */ |
| _settriple(result, MPD_POS, 0, mpd_etiny(ctx)); |
| *status |= MPD_Clamped; |
| return; |
| } |
| if (mpd_iszero(a)) { |
| mpd_setspecial(result, mpd_sign(a), MPD_INF); |
| *status |= MPD_Division_by_zero; |
| return; |
| } |
| if (mpd_isnegative(a)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| |
| workctx = *ctx; |
| workctx.prec += 2; |
| workctx.round = MPD_ROUND_HALF_EVEN; |
| _mpd_qinvroot(result, a, &workctx, status); |
| mpd_qfinalize(result, ctx, status); |
| } |
| /* END LIBMPDEC_ONLY */ |
| |
| /* Algorithm from decimal.py */ |
| void |
| mpd_qsqrt(mpd_t *result, const mpd_t *a, const mpd_context_t *ctx, |
| uint32_t *status) |
| { |
| mpd_context_t maxcontext; |
| MPD_NEW_STATIC(c,0,0,0,0); |
| MPD_NEW_STATIC(q,0,0,0,0); |
| MPD_NEW_STATIC(r,0,0,0,0); |
| MPD_NEW_CONST(two,0,0,1,1,1,2); |
| mpd_ssize_t prec, ideal_exp; |
| mpd_ssize_t l, shift; |
| int exact = 0; |
| |
| |
| ideal_exp = (a->exp - (a->exp & 1)) / 2; |
| |
| if (mpd_isspecial(a)) { |
| if (mpd_qcheck_nan(result, a, ctx, status)) { |
| return; |
| } |
| if (mpd_isnegative(a)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| mpd_setspecial(result, MPD_POS, MPD_INF); |
| return; |
| } |
| if (mpd_iszero(a)) { |
| _settriple(result, mpd_sign(a), 0, ideal_exp); |
| mpd_qfinalize(result, ctx, status); |
| return; |
| } |
| if (mpd_isnegative(a)) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| |
| mpd_maxcontext(&maxcontext); |
| prec = ctx->prec + 1; |
| |
| if (!mpd_qcopy(&c, a, status)) { |
| goto malloc_error; |
| } |
| c.exp = 0; |
| |
| if (a->exp & 1) { |
| if (!mpd_qshiftl(&c, &c, 1, status)) { |
| goto malloc_error; |
| } |
| l = (a->digits >> 1) + 1; |
| } |
| else { |
| l = (a->digits + 1) >> 1; |
| } |
| |
| shift = prec - l; |
| if (shift >= 0) { |
| if (!mpd_qshiftl(&c, &c, 2*shift, status)) { |
| goto malloc_error; |
| } |
| exact = 1; |
| } |
| else { |
| exact = !mpd_qshiftr_inplace(&c, -2*shift); |
| } |
| |
| ideal_exp -= shift; |
| |
| /* find result = floor(sqrt(c)) using Newton's method */ |
| if (!mpd_qshiftl(result, &one, prec, status)) { |
| goto malloc_error; |
| } |
| |
| while (1) { |
| _mpd_qdivmod(&q, &r, &c, result, &maxcontext, &maxcontext.status); |
| if (mpd_isspecial(result) || mpd_isspecial(&q)) { |
| mpd_seterror(result, maxcontext.status&MPD_Errors, status); |
| goto out; |
| } |
| if (_mpd_cmp(result, &q) <= 0) { |
| break; |
| } |
| _mpd_qadd_exact(result, result, &q, &maxcontext, &maxcontext.status); |
| if (mpd_isspecial(result)) { |
| mpd_seterror(result, maxcontext.status&MPD_Errors, status); |
| goto out; |
| } |
| _mpd_qdivmod(result, &r, result, &two, &maxcontext, &maxcontext.status); |
| } |
| |
| if (exact) { |
| _mpd_qmul_exact(&r, result, result, &maxcontext, &maxcontext.status); |
| if (mpd_isspecial(&r)) { |
| mpd_seterror(result, maxcontext.status&MPD_Errors, status); |
| goto out; |
| } |
| exact = (_mpd_cmp(&r, &c) == 0); |
| } |
| |
| if (exact) { |
| if (shift >= 0) { |
| mpd_qshiftr_inplace(result, shift); |
| } |
| else { |
| if (!mpd_qshiftl(result, result, -shift, status)) { |
| goto malloc_error; |
| } |
| } |
| ideal_exp += shift; |
| } |
| else { |
| int lsd = (int)mpd_lsd(result->data[0]); |
| if (lsd == 0 || lsd == 5) { |
| result->data[0] += 1; |
| } |
| } |
| |
| result->exp = ideal_exp; |
| |
| |
| out: |
| mpd_del(&c); |
| mpd_del(&q); |
| mpd_del(&r); |
| maxcontext = *ctx; |
| maxcontext.round = MPD_ROUND_HALF_EVEN; |
| mpd_qfinalize(result, &maxcontext, status); |
| return; |
| |
| malloc_error: |
| mpd_seterror(result, MPD_Malloc_error, status); |
| goto out; |
| } |
| |
| |
| /******************************************************************************/ |
| /* Base conversions */ |
| /******************************************************************************/ |
| |
| /* Space needed to represent an integer mpd_t in base 'base'. */ |
| size_t |
| mpd_sizeinbase(const mpd_t *a, uint32_t base) |
| { |
| double x; |
| size_t digits; |
| |
| assert(mpd_isinteger(a)); |
| assert(base >= 2); |
| |
| if (mpd_iszero(a)) { |
| return 1; |
| } |
| |
| digits = a->digits+a->exp; |
| assert(digits > 0); |
| |
| #ifdef CONFIG_64 |
| /* ceil(2711437152599294 / log10(2)) + 4 == 2**53 */ |
| if (digits > 2711437152599294ULL) { |
| return SIZE_MAX; |
| } |
| #endif |
| |
| x = (double)digits / log10(base); |
| return (x > SIZE_MAX-1) ? SIZE_MAX : (size_t)x + 1; |
| } |
| |
| /* Space needed to import a base 'base' integer of length 'srclen'. */ |
| static mpd_ssize_t |
| _mpd_importsize(size_t srclen, uint32_t base) |
| { |
| double x; |
| |
| assert(srclen > 0); |
| assert(base >= 2); |
| |
| #if SIZE_MAX == UINT64_MAX |
| if (srclen > (1ULL<<53)) { |
| return MPD_SSIZE_MAX; |
| } |
| #endif |
| |
| x = (double)srclen * (log10(base)/MPD_RDIGITS); |
| return (x >= MPD_MAXIMPORT) ? MPD_SSIZE_MAX : (mpd_ssize_t)x + 1; |
| } |
| |
| static uint8_t |
| mpd_resize_u16(uint16_t **w, size_t nmemb) |
| { |
| uint8_t err = 0; |
| *w = mpd_realloc(*w, nmemb, sizeof **w, &err); |
| return !err; |
| } |
| |
| static uint8_t |
| mpd_resize_u32(uint32_t **w, size_t nmemb) |
| { |
| uint8_t err = 0; |
| *w = mpd_realloc(*w, nmemb, sizeof **w, &err); |
| return !err; |
| } |
| |
| static size_t |
| _baseconv_to_u16(uint16_t **w, size_t wlen, mpd_uint_t wbase, |
| mpd_uint_t *u, mpd_ssize_t ulen) |
| { |
| size_t n = 0; |
| |
| assert(wlen > 0 && ulen > 0); |
| assert(wbase <= (1U<<16)); |
| |
| do { |
| if (n >= wlen) { |
| if (!mpd_resize_u16(w, n+1)) { |
| return SIZE_MAX; |
| } |
| wlen = n+1; |
| } |
| (*w)[n++] = (uint16_t)_mpd_shortdiv(u, u, ulen, wbase); |
| /* ulen is at least 1. u[ulen-1] can only be zero if ulen == 1. */ |
| ulen = _mpd_real_size(u, ulen); |
| |
| } while (u[ulen-1] != 0); |
| |
| return n; |
| } |
| |
| static size_t |
| _coeff_from_u16(mpd_t *w, mpd_ssize_t wlen, |
| const mpd_uint_t *u, size_t ulen, uint32_t ubase, |
| uint32_t *status) |
| { |
| mpd_ssize_t n = 0; |
| mpd_uint_t carry; |
| |
| assert(wlen > 0 && ulen > 0); |
| assert(ubase <= (1U<<16)); |
| |
| w->data[n++] = u[--ulen]; |
| while (--ulen != SIZE_MAX) { |
| carry = _mpd_shortmul_c(w->data, w->data, n, ubase); |
| if (carry) { |
| if (n >= wlen) { |
| if (!mpd_qresize(w, n+1, status)) { |
| return SIZE_MAX; |
| } |
| wlen = n+1; |
| } |
| w->data[n++] = carry; |
| } |
| carry = _mpd_shortadd(w->data, n, u[ulen]); |
| if (carry) { |
| if (n >= wlen) { |
| if (!mpd_qresize(w, n+1, status)) { |
| return SIZE_MAX; |
| } |
| wlen = n+1; |
| } |
| w->data[n++] = carry; |
| } |
| } |
| |
| return n; |
| } |
| |
| /* target base wbase < source base ubase */ |
| static size_t |
| _baseconv_to_smaller(uint32_t **w, size_t wlen, uint32_t wbase, |
| mpd_uint_t *u, mpd_ssize_t ulen, mpd_uint_t ubase) |
| { |
| size_t n = 0; |
| |
| assert(wlen > 0 && ulen > 0); |
| assert(wbase < ubase); |
| |
| do { |
| if (n >= wlen) { |
| if (!mpd_resize_u32(w, n+1)) { |
| return SIZE_MAX; |
| } |
| wlen = n+1; |
| } |
| (*w)[n++] = (uint32_t)_mpd_shortdiv_b(u, u, ulen, wbase, ubase); |
| /* ulen is at least 1. u[ulen-1] can only be zero if ulen == 1. */ |
| ulen = _mpd_real_size(u, ulen); |
| |
| } while (u[ulen-1] != 0); |
| |
| return n; |
| } |
| |
| #ifdef CONFIG_32 |
| /* target base 'wbase' == source base 'ubase' */ |
| static size_t |
| _copy_equal_base(uint32_t **w, size_t wlen, |
| const uint32_t *u, size_t ulen) |
| { |
| if (wlen < ulen) { |
| if (!mpd_resize_u32(w, ulen)) { |
| return SIZE_MAX; |
| } |
| } |
| |
| memcpy(*w, u, ulen * (sizeof **w)); |
| return ulen; |
| } |
| |
| /* target base 'wbase' > source base 'ubase' */ |
| static size_t |
| _baseconv_to_larger(uint32_t **w, size_t wlen, mpd_uint_t wbase, |
| const mpd_uint_t *u, size_t ulen, mpd_uint_t ubase) |
| { |
| size_t n = 0; |
| mpd_uint_t carry; |
| |
| assert(wlen > 0 && ulen > 0); |
| assert(ubase < wbase); |
| |
| (*w)[n++] = u[--ulen]; |
| while (--ulen != SIZE_MAX) { |
| carry = _mpd_shortmul_b(*w, *w, n, ubase, wbase); |
| if (carry) { |
| if (n >= wlen) { |
| if (!mpd_resize_u32(w, n+1)) { |
| return SIZE_MAX; |
| } |
| wlen = n+1; |
| } |
| (*w)[n++] = carry; |
| } |
| carry = _mpd_shortadd_b(*w, n, u[ulen], wbase); |
| if (carry) { |
| if (n >= wlen) { |
| if (!mpd_resize_u32(w, n+1)) { |
| return SIZE_MAX; |
| } |
| wlen = n+1; |
| } |
| (*w)[n++] = carry; |
| } |
| } |
| |
| return n; |
| } |
| |
| /* target base wbase < source base ubase */ |
| static size_t |
| _coeff_from_larger_base(mpd_t *w, size_t wlen, mpd_uint_t wbase, |
| mpd_uint_t *u, mpd_ssize_t ulen, mpd_uint_t ubase, |
| uint32_t *status) |
| { |
| size_t n = 0; |
| |
| assert(wlen > 0 && ulen > 0); |
| assert(wbase < ubase); |
| |
| do { |
| if (n >= wlen) { |
| if (!mpd_qresize(w, n+1, status)) { |
| return SIZE_MAX; |
| } |
| wlen = n+1; |
| } |
| w->data[n++] = (uint32_t)_mpd_shortdiv_b(u, u, ulen, wbase, ubase); |
| /* ulen is at least 1. u[ulen-1] can only be zero if ulen == 1. */ |
| ulen = _mpd_real_size(u, ulen); |
| |
| } while (u[ulen-1] != 0); |
| |
| return n; |
| } |
| #endif |
| |
| /* target base 'wbase' > source base 'ubase' */ |
| static size_t |
| _coeff_from_smaller_base(mpd_t *w, mpd_ssize_t wlen, mpd_uint_t wbase, |
| const uint32_t *u, size_t ulen, mpd_uint_t ubase, |
| uint32_t *status) |
| { |
| mpd_ssize_t n = 0; |
| mpd_uint_t carry; |
| |
| assert(wlen > 0 && ulen > 0); |
| assert(wbase > ubase); |
| |
| w->data[n++] = u[--ulen]; |
| while (--ulen != SIZE_MAX) { |
| carry = _mpd_shortmul_b(w->data, w->data, n, ubase, wbase); |
| if (carry) { |
| if (n >= wlen) { |
| if (!mpd_qresize(w, n+1, status)) { |
| return SIZE_MAX; |
| } |
| wlen = n+1; |
| } |
| w->data[n++] = carry; |
| } |
| carry = _mpd_shortadd_b(w->data, n, u[ulen], wbase); |
| if (carry) { |
| if (n >= wlen) { |
| if (!mpd_qresize(w, n+1, status)) { |
| return SIZE_MAX; |
| } |
| wlen = n+1; |
| } |
| w->data[n++] = carry; |
| } |
| } |
| |
| return n; |
| } |
| |
| /* |
| * Convert an integer mpd_t to a multiprecision integer with base <= 2**16. |
| * The least significant word of the result is (*rdata)[0]. |
| * |
| * If rdata is NULL, space is allocated by the function and rlen is irrelevant. |
| * In case of an error any allocated storage is freed and rdata is set back to |
| * NULL. |
| * |
| * If rdata is non-NULL, it MUST be allocated by one of libmpdec's allocation |
| * functions and rlen MUST be correct. If necessary, the function will resize |
| * rdata. In case of an error the caller must free rdata. |
| * |
| * Return value: In case of success, the exact length of rdata, SIZE_MAX |
| * otherwise. |
| */ |
| size_t |
| mpd_qexport_u16(uint16_t **rdata, size_t rlen, uint32_t rbase, |
| const mpd_t *src, uint32_t *status) |
| { |
| MPD_NEW_STATIC(tsrc,0,0,0,0); |
| int alloc = 0; /* rdata == NULL */ |
| size_t n; |
| |
| assert(rbase <= (1U<<16)); |
| |
| if (mpd_isspecial(src) || !_mpd_isint(src)) { |
| *status |= MPD_Invalid_operation; |
| return SIZE_MAX; |
| } |
| |
| if (*rdata == NULL) { |
| rlen = mpd_sizeinbase(src, rbase); |
| if (rlen == SIZE_MAX) { |
| *status |= MPD_Invalid_operation; |
| return SIZE_MAX; |
| } |
| *rdata = mpd_alloc(rlen, sizeof **rdata); |
| if (*rdata == NULL) { |
| goto malloc_error; |
| } |
| alloc = 1; |
| } |
| |
| if (mpd_iszero(src)) { |
| **rdata = 0; |
| return 1; |
| } |
| |
| if (src->exp >= 0) { |
| if (!mpd_qshiftl(&tsrc, src, src->exp, status)) { |
| goto malloc_error; |
| } |
| } |
| else { |
| if (mpd_qshiftr(&tsrc, src, -src->exp, status) == MPD_UINT_MAX) { |
| goto malloc_error; |
| } |
| } |
| |
| n = _baseconv_to_u16(rdata, rlen, rbase, tsrc.data, tsrc.len); |
| if (n == SIZE_MAX) { |
| goto malloc_error; |
| } |
| |
| |
| out: |
| mpd_del(&tsrc); |
| return n; |
| |
| malloc_error: |
| if (alloc) { |
| mpd_free(*rdata); |
| *rdata = NULL; |
| } |
| n = SIZE_MAX; |
| *status |= MPD_Malloc_error; |
| goto out; |
| } |
| |
| /* |
| * Convert an integer mpd_t to a multiprecision integer with base<=UINT32_MAX. |
| * The least significant word of the result is (*rdata)[0]. |
| * |
| * If rdata is NULL, space is allocated by the function and rlen is irrelevant. |
| * In case of an error any allocated storage is freed and rdata is set back to |
| * NULL. |
| * |
| * If rdata is non-NULL, it MUST be allocated by one of libmpdec's allocation |
| * functions and rlen MUST be correct. If necessary, the function will resize |
| * rdata. In case of an error the caller must free rdata. |
| * |
| * Return value: In case of success, the exact length of rdata, SIZE_MAX |
| * otherwise. |
| */ |
| size_t |
| mpd_qexport_u32(uint32_t **rdata, size_t rlen, uint32_t rbase, |
| const mpd_t *src, uint32_t *status) |
| { |
| MPD_NEW_STATIC(tsrc,0,0,0,0); |
| int alloc = 0; /* rdata == NULL */ |
| size_t n; |
| |
| if (mpd_isspecial(src) || !_mpd_isint(src)) { |
| *status |= MPD_Invalid_operation; |
| return SIZE_MAX; |
| } |
| |
| if (*rdata == NULL) { |
| rlen = mpd_sizeinbase(src, rbase); |
| if (rlen == SIZE_MAX) { |
| *status |= MPD_Invalid_operation; |
| return SIZE_MAX; |
| } |
| *rdata = mpd_alloc(rlen, sizeof **rdata); |
| if (*rdata == NULL) { |
| goto malloc_error; |
| } |
| alloc = 1; |
| } |
| |
| if (mpd_iszero(src)) { |
| **rdata = 0; |
| return 1; |
| } |
| |
| if (src->exp >= 0) { |
| if (!mpd_qshiftl(&tsrc, src, src->exp, status)) { |
| goto malloc_error; |
| } |
| } |
| else { |
| if (mpd_qshiftr(&tsrc, src, -src->exp, status) == MPD_UINT_MAX) { |
| goto malloc_error; |
| } |
| } |
| |
| #ifdef CONFIG_64 |
| n = _baseconv_to_smaller(rdata, rlen, rbase, |
| tsrc.data, tsrc.len, MPD_RADIX); |
| #else |
| if (rbase == MPD_RADIX) { |
| n = _copy_equal_base(rdata, rlen, tsrc.data, tsrc.len); |
| } |
| else if (rbase < MPD_RADIX) { |
| n = _baseconv_to_smaller(rdata, rlen, rbase, |
| tsrc.data, tsrc.len, MPD_RADIX); |
| } |
| else { |
| n = _baseconv_to_larger(rdata, rlen, rbase, |
| tsrc.data, tsrc.len, MPD_RADIX); |
| } |
| #endif |
| |
| if (n == SIZE_MAX) { |
| goto malloc_error; |
| } |
| |
| |
| out: |
| mpd_del(&tsrc); |
| return n; |
| |
| malloc_error: |
| if (alloc) { |
| mpd_free(*rdata); |
| *rdata = NULL; |
| } |
| n = SIZE_MAX; |
| *status |= MPD_Malloc_error; |
| goto out; |
| } |
| |
| |
| /* |
| * Converts a multiprecision integer with base <= UINT16_MAX+1 to an mpd_t. |
| * The least significant word of the source is srcdata[0]. |
| */ |
| void |
| mpd_qimport_u16(mpd_t *result, |
| const uint16_t *srcdata, size_t srclen, |
| uint8_t srcsign, uint32_t srcbase, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_uint_t *usrc; /* uint16_t src copied to an mpd_uint_t array */ |
| mpd_ssize_t rlen; /* length of the result */ |
| size_t n; |
| |
| assert(srclen > 0); |
| assert(srcbase <= (1U<<16)); |
| |
| rlen = _mpd_importsize(srclen, srcbase); |
| if (rlen == MPD_SSIZE_MAX) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| |
| usrc = mpd_alloc((mpd_size_t)srclen, sizeof *usrc); |
| if (usrc == NULL) { |
| mpd_seterror(result, MPD_Malloc_error, status); |
| return; |
| } |
| for (n = 0; n < srclen; n++) { |
| usrc[n] = srcdata[n]; |
| } |
| |
| if (!mpd_qresize(result, rlen, status)) { |
| goto finish; |
| } |
| |
| n = _coeff_from_u16(result, rlen, usrc, srclen, srcbase, status); |
| if (n == SIZE_MAX) { |
| goto finish; |
| } |
| |
| mpd_set_flags(result, srcsign); |
| result->exp = 0; |
| result->len = n; |
| mpd_setdigits(result); |
| |
| mpd_qresize(result, result->len, status); |
| mpd_qfinalize(result, ctx, status); |
| |
| |
| finish: |
| mpd_free(usrc); |
| } |
| |
| /* |
| * Converts a multiprecision integer with base <= UINT32_MAX to an mpd_t. |
| * The least significant word of the source is srcdata[0]. |
| */ |
| void |
| mpd_qimport_u32(mpd_t *result, |
| const uint32_t *srcdata, size_t srclen, |
| uint8_t srcsign, uint32_t srcbase, |
| const mpd_context_t *ctx, uint32_t *status) |
| { |
| mpd_ssize_t rlen; /* length of the result */ |
| size_t n; |
| |
| assert(srclen > 0); |
| |
| rlen = _mpd_importsize(srclen, srcbase); |
| if (rlen == MPD_SSIZE_MAX) { |
| mpd_seterror(result, MPD_Invalid_operation, status); |
| return; |
| } |
| |
| if (!mpd_qresize(result, rlen, status)) { |
| return; |
| } |
| |
| #ifdef CONFIG_64 |
| n = _coeff_from_smaller_base(result, rlen, MPD_RADIX, |
| srcdata, srclen, srcbase, |
| status); |
| #else |
| if (srcbase == MPD_RADIX) { |
| if (!mpd_qresize(result, srclen, status)) { |
| return; |
| } |
| memcpy(result->data, srcdata, srclen * (sizeof *srcdata)); |
| n = srclen; |
| } |
| else if (srcbase < MPD_RADIX) { |
| n = _coeff_from_smaller_base(result, rlen, MPD_RADIX, |
| srcdata, srclen, srcbase, |
| status); |
| } |
| else { |
| mpd_uint_t *usrc = mpd_alloc((mpd_size_t)srclen, sizeof *usrc); |
| if (usrc == NULL) { |
| mpd_seterror(result, MPD_Malloc_error, status); |
| return; |
| } |
| for (n = 0; n < srclen; n++) { |
| usrc[n] = srcdata[n]; |
| } |
| |
| n = _coeff_from_larger_base(result, rlen, MPD_RADIX, |
| usrc, (mpd_ssize_t)srclen, srcbase, |
| status); |
| mpd_free(usrc); |
| } |
| #endif |
| |
| if (n == SIZE_MAX) { |
| return; |
| } |
| |
| mpd_set_flags(result, srcsign); |
| result->exp = 0; |
| result->len = n; |
| mpd_setdigits(result); |
| |
| mpd_qresize(result, result->len, status); |
| mpd_qfinalize(result, ctx, status); |
| } |
| |
| |
| |