math/pow.c

/*
 * Double-precision x^y function.
 *
 * Copyright (c) 2018, Arm Limited.
 * SPDX-License-Identifier: Apache-2.0
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *     http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */

#include <math.h>
#include <stdint.h>
#include "math_config.h"

/*
Worst-case error: 0.67 ULP (~= ulperr_exp + 1024*Ln2*relerr_log*2^53)
relerr_log: 1.8 * 2^-66 (Relative error of log)
ulperr_exp: 0.509 ULP (ULP error of exp)
*/

#define T __pow_log_data.tab
#define B __pow_log_data.poly1
#define A __pow_log_data.poly
#define Ln2hi __pow_log_data.ln2hi
#define Ln2lo __pow_log_data.ln2lo
#define N (1 << POW_LOG_TABLE_BITS)
#define OFF 0x3fe6000000000000

static inline uint32_t
top12 (double x)
{
  return asuint64 (x) >> 52;
}

static inline double_t
log_inline (uint64_t ix, double_t *tail)
{
  /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
  double_t w, z, zhi, zlo, r, r2, r3, y, invc, logc, kd, hi, lo, khi, rhi, rlo, p, q;
  uint64_t iz, tmp;
  int k, i;

#if POW_LOG_POLY1_ORDER == 9
# define LO asuint64 (1 - 0x1.1p-7)
# define HI asuint64 (1 + 0x1.98p-7)
#endif
  if (unlikely (ix - LO < HI - LO))
    {
      r = asdouble (ix) - 1.0;
      /* Split r into top and bottom half.  */
      w = r * 0x1p27;
      rhi = r + w - w;
      rlo = r - rhi;
      /* Compute r - r*r/2 precisely into hi+lo.  */
      w = rhi*rhi*B[0]; /* B[0] == -0.5.  */
      hi = r + w;
      lo = r - hi + w;
      lo += B[0]*rlo*(rhi + r);
      r2 = r*r;
      r3 = r*r2;
#if POW_LOG_POLY1_ORDER == 9
      p = B[1] + r*(B[2] + r*B[3] + r2*B[4] + r3*(B[5] + r*B[6] + r2*B[7]));
#endif
      q = lo + r3*p;
      y = hi + q;
      *tail = (hi - y) + q;
      return y;
    }

  /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
     The range is split into N subintervals.
     The ith subinterval contains z and c is near its center.  */
  tmp = ix - OFF;
  i = (tmp >> (52 - POW_LOG_TABLE_BITS)) % N;
  k = (int64_t) tmp >> 52; /* arithmetic shift */
  iz = ix - (tmp & 0xfffULL << 52);
  invc = T[i].invc;
  logc = T[i].logc;
  z = asdouble (iz);
  zhi = asdouble ((iz + (1ULL<<31)) & (-1ULL << 32));
  zlo = z - zhi;

  /* log(x) = log1p(z/c-1) + log(c) + k*Ln2 */
  rhi = zhi * invc - 1.0;
  rlo = zlo * invc;
  kd = (double_t) k;

  /* hi + lo = r + log(c) + k*Ln2.  */
  khi = kd * Ln2hi;
  w = khi + logc;
  lo = khi - w + logc;
  hi = w + rhi;
  lo = w - hi + rhi + (lo + kd*Ln2lo) + rlo;

  /* log(x) = lo + (log1p(r) - r) + hi.  */
  /* Evaluation is optimized assuming superscalar pipelined execution.  */
#if HAVE_FAST_FMA
  r = fma (z, invc, -1.0);
#else
  r = rhi + rlo;
#endif
  r2 = r * r;

#if POW_LOG_POLY_ORDER == 7
  p = lo + r*r2*(A[1] + r*A[2] + r2*(A[3] + r*A[4] + r2*A[5]));
#endif
  q = A[0]*r2; /* A[0] == -0.5.  */
  w = q + hi;
  p += hi - w + q;
  y = p + w;
  *tail = w - y + p;
  return y;
}

#undef N
#undef T
#define N (1 << EXP_TABLE_BITS)
#define InvLn2N __exp_data.invln2N
#define NegLn2hiN __exp_data.negln2hiN
#define NegLn2loN __exp_data.negln2loN
#define Shift __exp_data.shift
#define T __exp_data.tab
#define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
#define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
#define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
#define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
#define C6 __exp_data.poly[9 - EXP_POLY_ORDER]

static inline double
specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
{
  double_t scale, y;

  if ((ki & 0x80000000) == 0)
    {
      /* k > 0, the exponent of scale might have overflowed by <= 460.  */
      sbits -= 1009ull << 52;
      scale = asdouble (sbits);
      y = 0x1p1009 * (scale + scale * tmp);
      return check_oflow (y);
    }
  /* k < 0, need special care in the subnormal range.  */
  sbits += 1022ull << 52;
  /* Note: sbits is signed scale.  */
  scale = asdouble (sbits);
  y = scale + scale * tmp;
  if (fabs (y) < 1.0)
    {
      /* Round y to the right precision before scaling it into the subnormal
	 range to avoid double rounding that can cause 0.5+E/2 ulp error where
	 E is the worst-case ulp error outside the subnormal range.  So this
	 is only useful if the goal is better than 1 ulp worst-case error.  */
      double_t hi, lo, one = 1.0;
      if (y < 0.0)
	one = -1.0;
      lo = scale - y + scale * tmp;
      hi = one + y;
      lo = one - hi + y + lo;
      y = eval_as_double (hi + lo) - one;
      /* Avoid -0.0 with downward rounding.  */
      if (WANT_ROUNDING && y == 0.0)
	y = asdouble (sbits & 0x8000000000000000);
      /* The underflow exception needs to be signaled explicitly.  */
      force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
    }
  y = 0x1p-1022 * y;
  return check_uflow (y);
}

#define SIGN_BIAS (0x800 << EXP_TABLE_BITS)

static inline double
exp_inline (double x, double xtail, uint32_t sign_bias)
{
  uint32_t abstop;
  uint64_t ki, idx, top, sbits;
  /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
  double_t kd, z, r, r2, scale, tail, tmp;

  abstop = top12 (x) & 0x7ff;
  if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
    {
      if (abstop - top12 (0x1p-54) >= 0x80000000)
	{
	  /* Avoid spurious underflow for tiny x.  */
	  /* Note: 0 is common input.  */
	  double_t one = WANT_ROUNDING ? 1.0 + x : 1.0;
	  return sign_bias ? -one : one;
	}
      if (abstop >= top12 (1024.0))
	{
	  /* Note: inf and nan are already handled.  */
	  if (asuint64 (x) >> 63)
	    return __math_uflow (sign_bias);
	  else
	    return __math_oflow (sign_bias);
	}
      /* Large x is special cased below.  */
      abstop = 0;
    }

  /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)].  */
  /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N].  */
  z = InvLn2N * x;
#if TOINT_INTRINSICS
  kd = roundtoint (z);
  ki = converttoint (z);
#elif EXP_USE_TOINT_NARROW
  /* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes.  */
  kd = eval_as_double (z + Shift);
  ki = asuint64 (kd) >> 16;
  kd = (double_t) (int32_t) ki;
#else
  /* z - kd is in [-1, 1] in non-nearest rounding modes.  */
  kd = eval_as_double (z + Shift);
  ki = asuint64 (kd);
  kd -= Shift;
#endif
  r = x + kd*NegLn2hiN + kd*NegLn2loN;
  /* The code assumes 2^-200 < |xtail| < 2^-8/N.  */
  r += xtail;
  /* 2^(k/N) ~= scale * (1 + tail).  */
  idx = 2*(ki % N);
  top = (ki + sign_bias) << (52 - EXP_TABLE_BITS);
  tail = asdouble (T[idx]);
  /* This is only a valid scale when -1023*N < k < 1024*N.  */
  sbits = T[idx + 1] + top;
  /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1).  */
  /* Evaluation is optimized assuming superscalar pipelined execution.  */
  r2 = r*r;
  /* Without fma the worst case error is 0.25/N ulp larger.  */
  /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp.  */
#if EXP_POLY_ORDER == 4
  tmp = tail + r + r2*C2 + r*r2*(C3 + r*C4);
#elif EXP_POLY_ORDER == 5
  tmp = tail + r + r2*(C2 + r*C3) + r2*r2*(C4 + r*C5);
#elif EXP_POLY_ORDER == 6
  tmp = tail + r + r2*(0.5 + r*C3) + r2*r2*(C4 + r*C5 + r2*C6);
#endif
  if (unlikely (abstop == 0))
    return specialcase (tmp, sbits, ki);
  scale = asdouble (sbits);
  /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
     is no spurious underflow here even without fma.  */
  return scale + scale * tmp;
}

/* Returns 0 if not int, 1 if odd int, 2 if even int.  */
static inline int
checkint (uint64_t iy)
{
  int e = iy >> 52 & 0x7ff;
  if (e < 0x3ff)
    return 0;
  if (e > 0x3ff + 52)
    return 2;
  if (iy & ((1ULL << (0x3ff + 52 - e)) - 1))
    return 0;
  if (iy & (1ULL << (0x3ff + 52 - e)))
    return 1;
  return 2;
}

static inline int
zeroinfnan (uint64_t i)
{
  return 2 * i - 1 >= 2 * asuint64 (INFINITY) - 1;
}

double
pow (double x, double y)
{
  uint64_t sign_bias = 0;
  uint64_t ix, iy;
  uint32_t topx, topy;

  ix = asuint64 (x);
  iy = asuint64 (y);
  topx = top12 (x);
  topy = top12 (y);
  if (unlikely (topx - 0x001 >= 0x7ff - 0x001 || (topy & 0x7ff) - 0x3be >= 0x43e - 0x3be))
    {
      /* Note: if |y| > 1075 * ln2 * 2^53 ~= 0x1.749p62 then pow(x,y) = inf/0
	 and if |y| < 2^-54 / 1075 ~= 0x1.e7b6p-65 then pow(x,y) = +-1.  */
      /* Special cases: (x < 0x1p-126 or inf or nan) or
	 (|y| < 0x1p-65 or |y| >= 0x1p63 or nan).  */
      if (unlikely (zeroinfnan (iy)))
	{
	  if (2 * iy == 0)
	    return issignaling_inline (x) ? x + y : 1.0;
	  if (ix == asuint64 (1.0))
	    return issignaling_inline (y) ? x + y : 1.0;
	  if (2 * ix > 2 * asuint64 (INFINITY) || 2 * iy > 2 * asuint64 (INFINITY))
	    return x + y;
	  if (2 * ix == 2 * asuint64 (1.0))
	    return 1.0;
	  if ((2 * ix < 2 * asuint64 (1.0)) == !(iy >> 63))
	    return 0.0; /* |x|<1 && y==inf or |x|>1 && y==-inf.  */
	  return y * y;
	}
      if (unlikely (zeroinfnan (ix)))
	{
	  double_t x2 = x * x;
	  if (ix >> 63 && checkint (iy) == 1)
	    {
	      x2 = -x2;
	      sign_bias = 1;
	    }
	  if (WANT_ERRNO && 2 * ix == 0 && iy >> 63)
	    return __math_divzero (sign_bias);
	  /* Without the barrier some versions of clang hoist the 1/x2 and
	     thus division by zero exception can be signaled spuriously.  */
	  return iy >> 63 ? opt_barrier_double (1 / x2) : x2;
	}
      /* Here x and y are non-zero finite.  */
      if (ix >> 63)
	{
	  /* Finite x < 0.  */
	  int yint = checkint (iy);
	  if (yint == 0)
	    return __math_invalid (x);
	  if (yint == 1)
	    sign_bias = SIGN_BIAS;
	  ix &= 0x7fffffffffffffff;
	  topx &= 0x7ff;
	}
      if ((topy & 0x7ff) - 0x3be >= 0x43e - 0x3be)
	{
	  /* Note: sign_bias == 0 here because y is not odd.  */
	  if (ix == asuint64 (1.0))
	    return 1.0;
	  if ((topy & 0x7ff) < 0x3be)
	    {
	      /* |y| < 2^-65, x^y ~= 1 + y*log(x).  */
	      if (WANT_ROUNDING)
		return ix > asuint64 (1.0) ? 1.0 + y : 1.0 - y;
	      else
		return 1.0;
	    }
	  return (ix > asuint64 (1.0)) == (topy < 0x800)
		 ? __math_oflow (0) : __math_uflow (0);
	}
      if (topx == 0)
	{
	  /* Normalize subnormal x so exponent becomes negative.  */
	  ix = asuint64 (x * 0x1p52);
	  ix &= 0x7fffffffffffffff;
	  ix -= 52ULL << 52;
	}
    }

  double_t lo;
  double_t hi = log_inline (ix, &lo);
  double_t ehi, elo;
#if HAVE_FAST_FMA
  ehi = y*hi;
  elo = y*lo + fma (y, hi, -ehi);
#else
  double_t yhi = asdouble (iy & -1ULL << 27);
  double_t ylo = y - yhi;
  double_t lhi = asdouble (asuint64 (hi) & -1ULL << 27);
  double_t llo = hi - lhi + lo;
  ehi = yhi*lhi;
  elo = ylo*lhi + y*llo; /* |elo| < |ehi| * 2^-25.  */
#endif
  return exp_inline (ehi, elo, sign_bias);
}
#if USE_GLIBC_ABI
strong_alias (pow, __pow_finite)
hidden_alias (pow, __ieee754_pow)
#endif