Ben Murdoch | 61f157c | 2016-09-16 13:49:30 +0100 | [diff] [blame] | 1 | // The following is adapted from fdlibm (http://www.netlib.org/fdlibm). |
| 2 | // |
| 3 | // ==================================================== |
| 4 | // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
| 5 | // |
| 6 | // Developed at SunSoft, a Sun Microsystems, Inc. business. |
| 7 | // Permission to use, copy, modify, and distribute this |
| 8 | // software is freely granted, provided that this notice |
| 9 | // is preserved. |
| 10 | // ==================================================== |
| 11 | // |
| 12 | // The original source code covered by the above license above has been |
| 13 | // modified significantly by Google Inc. |
| 14 | // Copyright 2016 the V8 project authors. All rights reserved. |
| 15 | |
| 16 | #include "src/base/ieee754.h" |
| 17 | |
| 18 | #include <cmath> |
| 19 | #include <limits> |
| 20 | |
| 21 | #include "src/base/build_config.h" |
| 22 | #include "src/base/macros.h" |
| 23 | |
| 24 | namespace v8 { |
| 25 | namespace base { |
| 26 | namespace ieee754 { |
| 27 | |
| 28 | namespace { |
| 29 | |
| 30 | /* Disable "potential divide by 0" warning in Visual Studio compiler. */ |
| 31 | |
| 32 | #if V8_CC_MSVC |
| 33 | |
| 34 | #pragma warning(disable : 4723) |
| 35 | |
| 36 | #endif |
| 37 | |
| 38 | /* |
| 39 | * The original fdlibm code used statements like: |
| 40 | * n0 = ((*(int*)&one)>>29)^1; * index of high word * |
| 41 | * ix0 = *(n0+(int*)&x); * high word of x * |
| 42 | * ix1 = *((1-n0)+(int*)&x); * low word of x * |
| 43 | * to dig two 32 bit words out of the 64 bit IEEE floating point |
| 44 | * value. That is non-ANSI, and, moreover, the gcc instruction |
| 45 | * scheduler gets it wrong. We instead use the following macros. |
| 46 | * Unlike the original code, we determine the endianness at compile |
| 47 | * time, not at run time; I don't see much benefit to selecting |
| 48 | * endianness at run time. |
| 49 | */ |
| 50 | |
| 51 | /* |
| 52 | * A union which permits us to convert between a double and two 32 bit |
| 53 | * ints. |
| 54 | */ |
| 55 | |
| 56 | #if V8_TARGET_LITTLE_ENDIAN |
| 57 | |
| 58 | typedef union { |
| 59 | double value; |
| 60 | struct { |
| 61 | uint32_t lsw; |
| 62 | uint32_t msw; |
| 63 | } parts; |
| 64 | struct { |
| 65 | uint64_t w; |
| 66 | } xparts; |
| 67 | } ieee_double_shape_type; |
| 68 | |
| 69 | #else |
| 70 | |
| 71 | typedef union { |
| 72 | double value; |
| 73 | struct { |
| 74 | uint32_t msw; |
| 75 | uint32_t lsw; |
| 76 | } parts; |
| 77 | struct { |
| 78 | uint64_t w; |
| 79 | } xparts; |
| 80 | } ieee_double_shape_type; |
| 81 | |
| 82 | #endif |
| 83 | |
| 84 | /* Get two 32 bit ints from a double. */ |
| 85 | |
| 86 | #define EXTRACT_WORDS(ix0, ix1, d) \ |
| 87 | do { \ |
| 88 | ieee_double_shape_type ew_u; \ |
| 89 | ew_u.value = (d); \ |
| 90 | (ix0) = ew_u.parts.msw; \ |
| 91 | (ix1) = ew_u.parts.lsw; \ |
| 92 | } while (0) |
| 93 | |
| 94 | /* Get a 64-bit int from a double. */ |
| 95 | #define EXTRACT_WORD64(ix, d) \ |
| 96 | do { \ |
| 97 | ieee_double_shape_type ew_u; \ |
| 98 | ew_u.value = (d); \ |
| 99 | (ix) = ew_u.xparts.w; \ |
| 100 | } while (0) |
| 101 | |
| 102 | /* Get the more significant 32 bit int from a double. */ |
| 103 | |
| 104 | #define GET_HIGH_WORD(i, d) \ |
| 105 | do { \ |
| 106 | ieee_double_shape_type gh_u; \ |
| 107 | gh_u.value = (d); \ |
| 108 | (i) = gh_u.parts.msw; \ |
| 109 | } while (0) |
| 110 | |
| 111 | /* Get the less significant 32 bit int from a double. */ |
| 112 | |
| 113 | #define GET_LOW_WORD(i, d) \ |
| 114 | do { \ |
| 115 | ieee_double_shape_type gl_u; \ |
| 116 | gl_u.value = (d); \ |
| 117 | (i) = gl_u.parts.lsw; \ |
| 118 | } while (0) |
| 119 | |
| 120 | /* Set a double from two 32 bit ints. */ |
| 121 | |
| 122 | #define INSERT_WORDS(d, ix0, ix1) \ |
| 123 | do { \ |
| 124 | ieee_double_shape_type iw_u; \ |
| 125 | iw_u.parts.msw = (ix0); \ |
| 126 | iw_u.parts.lsw = (ix1); \ |
| 127 | (d) = iw_u.value; \ |
| 128 | } while (0) |
| 129 | |
| 130 | /* Set a double from a 64-bit int. */ |
| 131 | #define INSERT_WORD64(d, ix) \ |
| 132 | do { \ |
| 133 | ieee_double_shape_type iw_u; \ |
| 134 | iw_u.xparts.w = (ix); \ |
| 135 | (d) = iw_u.value; \ |
| 136 | } while (0) |
| 137 | |
| 138 | /* Set the more significant 32 bits of a double from an int. */ |
| 139 | |
| 140 | #define SET_HIGH_WORD(d, v) \ |
| 141 | do { \ |
| 142 | ieee_double_shape_type sh_u; \ |
| 143 | sh_u.value = (d); \ |
| 144 | sh_u.parts.msw = (v); \ |
| 145 | (d) = sh_u.value; \ |
| 146 | } while (0) |
| 147 | |
| 148 | /* Set the less significant 32 bits of a double from an int. */ |
| 149 | |
| 150 | #define SET_LOW_WORD(d, v) \ |
| 151 | do { \ |
| 152 | ieee_double_shape_type sl_u; \ |
| 153 | sl_u.value = (d); \ |
| 154 | sl_u.parts.lsw = (v); \ |
| 155 | (d) = sl_u.value; \ |
| 156 | } while (0) |
| 157 | |
| 158 | /* Support macro. */ |
| 159 | |
| 160 | #define STRICT_ASSIGN(type, lval, rval) ((lval) = (rval)) |
| 161 | |
| 162 | int32_t __ieee754_rem_pio2(double x, double *y) WARN_UNUSED_RESULT; |
| 163 | double __kernel_cos(double x, double y) WARN_UNUSED_RESULT; |
| 164 | int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, |
| 165 | const int32_t *ipio2) WARN_UNUSED_RESULT; |
| 166 | double __kernel_sin(double x, double y, int iy) WARN_UNUSED_RESULT; |
| 167 | |
| 168 | /* __ieee754_rem_pio2(x,y) |
| 169 | * |
| 170 | * return the remainder of x rem pi/2 in y[0]+y[1] |
| 171 | * use __kernel_rem_pio2() |
| 172 | */ |
| 173 | int32_t __ieee754_rem_pio2(double x, double *y) { |
| 174 | /* |
| 175 | * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi |
| 176 | */ |
| 177 | static const int32_t two_over_pi[] = { |
| 178 | 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, 0x95993C, |
| 179 | 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, 0x424DD2, 0xE00649, |
| 180 | 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, 0xA73EE8, 0x8235F5, 0x2EBB44, |
| 181 | 0x84E99C, 0x7026B4, 0x5F7E41, 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, |
| 182 | 0xBDF928, 0x3B1FF8, 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, |
| 183 | 0x367ECF, 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, |
| 184 | 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, 0x560330, |
| 185 | 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, 0x91615E, 0xE61B08, |
| 186 | 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, 0x4D7327, 0x310606, 0x1556CA, |
| 187 | 0x73A8C9, 0x60E27B, 0xC08C6B, |
| 188 | }; |
| 189 | |
| 190 | static const int32_t npio2_hw[] = { |
| 191 | 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C, |
| 192 | 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C, |
| 193 | 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A, |
| 194 | 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C, |
| 195 | 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB, |
| 196 | 0x404858EB, 0x404921FB, |
| 197 | }; |
| 198 | |
| 199 | /* |
| 200 | * invpio2: 53 bits of 2/pi |
| 201 | * pio2_1: first 33 bit of pi/2 |
| 202 | * pio2_1t: pi/2 - pio2_1 |
| 203 | * pio2_2: second 33 bit of pi/2 |
| 204 | * pio2_2t: pi/2 - (pio2_1+pio2_2) |
| 205 | * pio2_3: third 33 bit of pi/2 |
| 206 | * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) |
| 207 | */ |
| 208 | |
| 209 | static const double |
| 210 | zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ |
| 211 | half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ |
| 212 | two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ |
| 213 | invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ |
| 214 | pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */ |
| 215 | pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */ |
| 216 | pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */ |
| 217 | pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */ |
| 218 | pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */ |
| 219 | pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */ |
| 220 | |
| 221 | double z, w, t, r, fn; |
| 222 | double tx[3]; |
| 223 | int32_t e0, i, j, nx, n, ix, hx; |
| 224 | uint32_t low; |
| 225 | |
| 226 | z = 0; |
| 227 | GET_HIGH_WORD(hx, x); /* high word of x */ |
| 228 | ix = hx & 0x7fffffff; |
| 229 | if (ix <= 0x3fe921fb) { /* |x| ~<= pi/4 , no need for reduction */ |
| 230 | y[0] = x; |
| 231 | y[1] = 0; |
| 232 | return 0; |
| 233 | } |
| 234 | if (ix < 0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */ |
| 235 | if (hx > 0) { |
| 236 | z = x - pio2_1; |
| 237 | if (ix != 0x3ff921fb) { /* 33+53 bit pi is good enough */ |
| 238 | y[0] = z - pio2_1t; |
| 239 | y[1] = (z - y[0]) - pio2_1t; |
| 240 | } else { /* near pi/2, use 33+33+53 bit pi */ |
| 241 | z -= pio2_2; |
| 242 | y[0] = z - pio2_2t; |
| 243 | y[1] = (z - y[0]) - pio2_2t; |
| 244 | } |
| 245 | return 1; |
| 246 | } else { /* negative x */ |
| 247 | z = x + pio2_1; |
| 248 | if (ix != 0x3ff921fb) { /* 33+53 bit pi is good enough */ |
| 249 | y[0] = z + pio2_1t; |
| 250 | y[1] = (z - y[0]) + pio2_1t; |
| 251 | } else { /* near pi/2, use 33+33+53 bit pi */ |
| 252 | z += pio2_2; |
| 253 | y[0] = z + pio2_2t; |
| 254 | y[1] = (z - y[0]) + pio2_2t; |
| 255 | } |
| 256 | return -1; |
| 257 | } |
| 258 | } |
| 259 | if (ix <= 0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */ |
| 260 | t = fabs(x); |
| 261 | n = static_cast<int32_t>(t * invpio2 + half); |
| 262 | fn = static_cast<double>(n); |
| 263 | r = t - fn * pio2_1; |
| 264 | w = fn * pio2_1t; /* 1st round good to 85 bit */ |
| 265 | if (n < 32 && ix != npio2_hw[n - 1]) { |
| 266 | y[0] = r - w; /* quick check no cancellation */ |
| 267 | } else { |
| 268 | uint32_t high; |
| 269 | j = ix >> 20; |
| 270 | y[0] = r - w; |
| 271 | GET_HIGH_WORD(high, y[0]); |
| 272 | i = j - ((high >> 20) & 0x7ff); |
| 273 | if (i > 16) { /* 2nd iteration needed, good to 118 */ |
| 274 | t = r; |
| 275 | w = fn * pio2_2; |
| 276 | r = t - w; |
| 277 | w = fn * pio2_2t - ((t - r) - w); |
| 278 | y[0] = r - w; |
| 279 | GET_HIGH_WORD(high, y[0]); |
| 280 | i = j - ((high >> 20) & 0x7ff); |
| 281 | if (i > 49) { /* 3rd iteration need, 151 bits acc */ |
| 282 | t = r; /* will cover all possible cases */ |
| 283 | w = fn * pio2_3; |
| 284 | r = t - w; |
| 285 | w = fn * pio2_3t - ((t - r) - w); |
| 286 | y[0] = r - w; |
| 287 | } |
| 288 | } |
| 289 | } |
| 290 | y[1] = (r - y[0]) - w; |
| 291 | if (hx < 0) { |
| 292 | y[0] = -y[0]; |
| 293 | y[1] = -y[1]; |
| 294 | return -n; |
| 295 | } else { |
| 296 | return n; |
| 297 | } |
| 298 | } |
| 299 | /* |
| 300 | * all other (large) arguments |
| 301 | */ |
| 302 | if (ix >= 0x7ff00000) { /* x is inf or NaN */ |
| 303 | y[0] = y[1] = x - x; |
| 304 | return 0; |
| 305 | } |
| 306 | /* set z = scalbn(|x|,ilogb(x)-23) */ |
| 307 | GET_LOW_WORD(low, x); |
| 308 | SET_LOW_WORD(z, low); |
| 309 | e0 = (ix >> 20) - 1046; /* e0 = ilogb(z)-23; */ |
| 310 | SET_HIGH_WORD(z, ix - static_cast<int32_t>(e0 << 20)); |
| 311 | for (i = 0; i < 2; i++) { |
| 312 | tx[i] = static_cast<double>(static_cast<int32_t>(z)); |
| 313 | z = (z - tx[i]) * two24; |
| 314 | } |
| 315 | tx[2] = z; |
| 316 | nx = 3; |
| 317 | while (tx[nx - 1] == zero) nx--; /* skip zero term */ |
| 318 | n = __kernel_rem_pio2(tx, y, e0, nx, 2, two_over_pi); |
| 319 | if (hx < 0) { |
| 320 | y[0] = -y[0]; |
| 321 | y[1] = -y[1]; |
| 322 | return -n; |
| 323 | } |
| 324 | return n; |
| 325 | } |
| 326 | |
| 327 | /* __kernel_cos( x, y ) |
| 328 | * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 |
| 329 | * Input x is assumed to be bounded by ~pi/4 in magnitude. |
| 330 | * Input y is the tail of x. |
| 331 | * |
| 332 | * Algorithm |
| 333 | * 1. Since cos(-x) = cos(x), we need only to consider positive x. |
| 334 | * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. |
| 335 | * 3. cos(x) is approximated by a polynomial of degree 14 on |
| 336 | * [0,pi/4] |
| 337 | * 4 14 |
| 338 | * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x |
| 339 | * where the remez error is |
| 340 | * |
| 341 | * | 2 4 6 8 10 12 14 | -58 |
| 342 | * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 |
| 343 | * | | |
| 344 | * |
| 345 | * 4 6 8 10 12 14 |
| 346 | * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then |
| 347 | * cos(x) = 1 - x*x/2 + r |
| 348 | * since cos(x+y) ~ cos(x) - sin(x)*y |
| 349 | * ~ cos(x) - x*y, |
| 350 | * a correction term is necessary in cos(x) and hence |
| 351 | * cos(x+y) = 1 - (x*x/2 - (r - x*y)) |
| 352 | * For better accuracy when x > 0.3, let qx = |x|/4 with |
| 353 | * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. |
| 354 | * Then |
| 355 | * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). |
| 356 | * Note that 1-qx and (x*x/2-qx) is EXACT here, and the |
| 357 | * magnitude of the latter is at least a quarter of x*x/2, |
| 358 | * thus, reducing the rounding error in the subtraction. |
| 359 | */ |
| 360 | V8_INLINE double __kernel_cos(double x, double y) { |
| 361 | static const double |
| 362 | one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ |
| 363 | C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ |
| 364 | C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ |
| 365 | C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ |
| 366 | C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ |
| 367 | C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ |
| 368 | C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ |
| 369 | |
| 370 | double a, iz, z, r, qx; |
| 371 | int32_t ix; |
| 372 | GET_HIGH_WORD(ix, x); |
| 373 | ix &= 0x7fffffff; /* ix = |x|'s high word*/ |
| 374 | if (ix < 0x3e400000) { /* if x < 2**27 */ |
| 375 | if (static_cast<int>(x) == 0) return one; /* generate inexact */ |
| 376 | } |
| 377 | z = x * x; |
| 378 | r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6))))); |
| 379 | if (ix < 0x3FD33333) { /* if |x| < 0.3 */ |
| 380 | return one - (0.5 * z - (z * r - x * y)); |
| 381 | } else { |
| 382 | if (ix > 0x3fe90000) { /* x > 0.78125 */ |
| 383 | qx = 0.28125; |
| 384 | } else { |
| 385 | INSERT_WORDS(qx, ix - 0x00200000, 0); /* x/4 */ |
| 386 | } |
| 387 | iz = 0.5 * z - qx; |
| 388 | a = one - qx; |
| 389 | return a - (iz - (z * r - x * y)); |
| 390 | } |
| 391 | } |
| 392 | |
| 393 | /* __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) |
| 394 | * double x[],y[]; int e0,nx,prec; int ipio2[]; |
| 395 | * |
| 396 | * __kernel_rem_pio2 return the last three digits of N with |
| 397 | * y = x - N*pi/2 |
| 398 | * so that |y| < pi/2. |
| 399 | * |
| 400 | * The method is to compute the integer (mod 8) and fraction parts of |
| 401 | * (2/pi)*x without doing the full multiplication. In general we |
| 402 | * skip the part of the product that are known to be a huge integer ( |
| 403 | * more accurately, = 0 mod 8 ). Thus the number of operations are |
| 404 | * independent of the exponent of the input. |
| 405 | * |
| 406 | * (2/pi) is represented by an array of 24-bit integers in ipio2[]. |
| 407 | * |
| 408 | * Input parameters: |
| 409 | * x[] The input value (must be positive) is broken into nx |
| 410 | * pieces of 24-bit integers in double precision format. |
| 411 | * x[i] will be the i-th 24 bit of x. The scaled exponent |
| 412 | * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 |
| 413 | * match x's up to 24 bits. |
| 414 | * |
| 415 | * Example of breaking a double positive z into x[0]+x[1]+x[2]: |
| 416 | * e0 = ilogb(z)-23 |
| 417 | * z = scalbn(z,-e0) |
| 418 | * for i = 0,1,2 |
| 419 | * x[i] = floor(z) |
| 420 | * z = (z-x[i])*2**24 |
| 421 | * |
| 422 | * |
| 423 | * y[] output result in an array of double precision numbers. |
| 424 | * The dimension of y[] is: |
| 425 | * 24-bit precision 1 |
| 426 | * 53-bit precision 2 |
| 427 | * 64-bit precision 2 |
| 428 | * 113-bit precision 3 |
| 429 | * The actual value is the sum of them. Thus for 113-bit |
| 430 | * precison, one may have to do something like: |
| 431 | * |
| 432 | * long double t,w,r_head, r_tail; |
| 433 | * t = (long double)y[2] + (long double)y[1]; |
| 434 | * w = (long double)y[0]; |
| 435 | * r_head = t+w; |
| 436 | * r_tail = w - (r_head - t); |
| 437 | * |
| 438 | * e0 The exponent of x[0] |
| 439 | * |
| 440 | * nx dimension of x[] |
| 441 | * |
| 442 | * prec an integer indicating the precision: |
| 443 | * 0 24 bits (single) |
| 444 | * 1 53 bits (double) |
| 445 | * 2 64 bits (extended) |
| 446 | * 3 113 bits (quad) |
| 447 | * |
| 448 | * ipio2[] |
| 449 | * integer array, contains the (24*i)-th to (24*i+23)-th |
| 450 | * bit of 2/pi after binary point. The corresponding |
| 451 | * floating value is |
| 452 | * |
| 453 | * ipio2[i] * 2^(-24(i+1)). |
| 454 | * |
| 455 | * External function: |
| 456 | * double scalbn(), floor(); |
| 457 | * |
| 458 | * |
| 459 | * Here is the description of some local variables: |
| 460 | * |
| 461 | * jk jk+1 is the initial number of terms of ipio2[] needed |
| 462 | * in the computation. The recommended value is 2,3,4, |
| 463 | * 6 for single, double, extended,and quad. |
| 464 | * |
| 465 | * jz local integer variable indicating the number of |
| 466 | * terms of ipio2[] used. |
| 467 | * |
| 468 | * jx nx - 1 |
| 469 | * |
| 470 | * jv index for pointing to the suitable ipio2[] for the |
| 471 | * computation. In general, we want |
| 472 | * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 |
| 473 | * is an integer. Thus |
| 474 | * e0-3-24*jv >= 0 or (e0-3)/24 >= jv |
| 475 | * Hence jv = max(0,(e0-3)/24). |
| 476 | * |
| 477 | * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. |
| 478 | * |
| 479 | * q[] double array with integral value, representing the |
| 480 | * 24-bits chunk of the product of x and 2/pi. |
| 481 | * |
| 482 | * q0 the corresponding exponent of q[0]. Note that the |
| 483 | * exponent for q[i] would be q0-24*i. |
| 484 | * |
| 485 | * PIo2[] double precision array, obtained by cutting pi/2 |
| 486 | * into 24 bits chunks. |
| 487 | * |
| 488 | * f[] ipio2[] in floating point |
| 489 | * |
| 490 | * iq[] integer array by breaking up q[] in 24-bits chunk. |
| 491 | * |
| 492 | * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] |
| 493 | * |
| 494 | * ih integer. If >0 it indicates q[] is >= 0.5, hence |
| 495 | * it also indicates the *sign* of the result. |
| 496 | * |
| 497 | */ |
| 498 | int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, |
| 499 | const int32_t *ipio2) { |
| 500 | /* Constants: |
| 501 | * The hexadecimal values are the intended ones for the following |
| 502 | * constants. The decimal values may be used, provided that the |
| 503 | * compiler will convert from decimal to binary accurately enough |
| 504 | * to produce the hexadecimal values shown. |
| 505 | */ |
| 506 | static const int init_jk[] = {2, 3, 4, 6}; /* initial value for jk */ |
| 507 | |
| 508 | static const double PIo2[] = { |
| 509 | 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ |
| 510 | 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ |
| 511 | 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ |
| 512 | 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ |
| 513 | 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ |
| 514 | 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ |
| 515 | 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ |
| 516 | 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ |
| 517 | }; |
| 518 | |
| 519 | static const double |
| 520 | zero = 0.0, |
| 521 | one = 1.0, |
| 522 | two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ |
| 523 | twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ |
| 524 | |
| 525 | int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih; |
| 526 | double z, fw, f[20], fq[20], q[20]; |
| 527 | |
| 528 | /* initialize jk*/ |
| 529 | jk = init_jk[prec]; |
| 530 | jp = jk; |
| 531 | |
| 532 | /* determine jx,jv,q0, note that 3>q0 */ |
| 533 | jx = nx - 1; |
| 534 | jv = (e0 - 3) / 24; |
| 535 | if (jv < 0) jv = 0; |
| 536 | q0 = e0 - 24 * (jv + 1); |
| 537 | |
| 538 | /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ |
| 539 | j = jv - jx; |
| 540 | m = jx + jk; |
| 541 | for (i = 0; i <= m; i++, j++) { |
| 542 | f[i] = (j < 0) ? zero : static_cast<double>(ipio2[j]); |
| 543 | } |
| 544 | |
| 545 | /* compute q[0],q[1],...q[jk] */ |
| 546 | for (i = 0; i <= jk; i++) { |
| 547 | for (j = 0, fw = 0.0; j <= jx; j++) fw += x[j] * f[jx + i - j]; |
| 548 | q[i] = fw; |
| 549 | } |
| 550 | |
| 551 | jz = jk; |
| 552 | recompute: |
| 553 | /* distill q[] into iq[] reversingly */ |
| 554 | for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) { |
| 555 | fw = static_cast<double>(static_cast<int32_t>(twon24 * z)); |
| 556 | iq[i] = static_cast<int32_t>(z - two24 * fw); |
| 557 | z = q[j - 1] + fw; |
| 558 | } |
| 559 | |
| 560 | /* compute n */ |
| 561 | z = scalbn(z, q0); /* actual value of z */ |
| 562 | z -= 8.0 * floor(z * 0.125); /* trim off integer >= 8 */ |
| 563 | n = static_cast<int32_t>(z); |
| 564 | z -= static_cast<double>(n); |
| 565 | ih = 0; |
| 566 | if (q0 > 0) { /* need iq[jz-1] to determine n */ |
| 567 | i = (iq[jz - 1] >> (24 - q0)); |
| 568 | n += i; |
| 569 | iq[jz - 1] -= i << (24 - q0); |
| 570 | ih = iq[jz - 1] >> (23 - q0); |
| 571 | } else if (q0 == 0) { |
| 572 | ih = iq[jz - 1] >> 23; |
| 573 | } else if (z >= 0.5) { |
| 574 | ih = 2; |
| 575 | } |
| 576 | |
| 577 | if (ih > 0) { /* q > 0.5 */ |
| 578 | n += 1; |
| 579 | carry = 0; |
| 580 | for (i = 0; i < jz; i++) { /* compute 1-q */ |
| 581 | j = iq[i]; |
| 582 | if (carry == 0) { |
| 583 | if (j != 0) { |
| 584 | carry = 1; |
| 585 | iq[i] = 0x1000000 - j; |
| 586 | } |
| 587 | } else { |
| 588 | iq[i] = 0xffffff - j; |
| 589 | } |
| 590 | } |
| 591 | if (q0 > 0) { /* rare case: chance is 1 in 12 */ |
| 592 | switch (q0) { |
| 593 | case 1: |
| 594 | iq[jz - 1] &= 0x7fffff; |
| 595 | break; |
| 596 | case 2: |
| 597 | iq[jz - 1] &= 0x3fffff; |
| 598 | break; |
| 599 | } |
| 600 | } |
| 601 | if (ih == 2) { |
| 602 | z = one - z; |
| 603 | if (carry != 0) z -= scalbn(one, q0); |
| 604 | } |
| 605 | } |
| 606 | |
| 607 | /* check if recomputation is needed */ |
| 608 | if (z == zero) { |
| 609 | j = 0; |
| 610 | for (i = jz - 1; i >= jk; i--) j |= iq[i]; |
| 611 | if (j == 0) { /* need recomputation */ |
| 612 | for (k = 1; jk >= k && iq[jk - k] == 0; k++) { |
| 613 | /* k = no. of terms needed */ |
| 614 | } |
| 615 | |
| 616 | for (i = jz + 1; i <= jz + k; i++) { /* add q[jz+1] to q[jz+k] */ |
| 617 | f[jx + i] = ipio2[jv + i]; |
| 618 | for (j = 0, fw = 0.0; j <= jx; j++) fw += x[j] * f[jx + i - j]; |
| 619 | q[i] = fw; |
| 620 | } |
| 621 | jz += k; |
| 622 | goto recompute; |
| 623 | } |
| 624 | } |
| 625 | |
| 626 | /* chop off zero terms */ |
| 627 | if (z == 0.0) { |
| 628 | jz -= 1; |
| 629 | q0 -= 24; |
| 630 | while (iq[jz] == 0) { |
| 631 | jz--; |
| 632 | q0 -= 24; |
| 633 | } |
| 634 | } else { /* break z into 24-bit if necessary */ |
| 635 | z = scalbn(z, -q0); |
| 636 | if (z >= two24) { |
| 637 | fw = static_cast<double>(static_cast<int32_t>(twon24 * z)); |
| 638 | iq[jz] = z - two24 * fw; |
| 639 | jz += 1; |
| 640 | q0 += 24; |
| 641 | iq[jz] = fw; |
| 642 | } else { |
| 643 | iq[jz] = z; |
| 644 | } |
| 645 | } |
| 646 | |
| 647 | /* convert integer "bit" chunk to floating-point value */ |
| 648 | fw = scalbn(one, q0); |
| 649 | for (i = jz; i >= 0; i--) { |
| 650 | q[i] = fw * iq[i]; |
| 651 | fw *= twon24; |
| 652 | } |
| 653 | |
| 654 | /* compute PIo2[0,...,jp]*q[jz,...,0] */ |
| 655 | for (i = jz; i >= 0; i--) { |
| 656 | for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++) fw += PIo2[k] * q[i + k]; |
| 657 | fq[jz - i] = fw; |
| 658 | } |
| 659 | |
| 660 | /* compress fq[] into y[] */ |
| 661 | switch (prec) { |
| 662 | case 0: |
| 663 | fw = 0.0; |
| 664 | for (i = jz; i >= 0; i--) fw += fq[i]; |
| 665 | y[0] = (ih == 0) ? fw : -fw; |
| 666 | break; |
| 667 | case 1: |
| 668 | case 2: |
| 669 | fw = 0.0; |
| 670 | for (i = jz; i >= 0; i--) fw += fq[i]; |
| 671 | y[0] = (ih == 0) ? fw : -fw; |
| 672 | fw = fq[0] - fw; |
| 673 | for (i = 1; i <= jz; i++) fw += fq[i]; |
| 674 | y[1] = (ih == 0) ? fw : -fw; |
| 675 | break; |
| 676 | case 3: /* painful */ |
| 677 | for (i = jz; i > 0; i--) { |
| 678 | fw = fq[i - 1] + fq[i]; |
| 679 | fq[i] += fq[i - 1] - fw; |
| 680 | fq[i - 1] = fw; |
| 681 | } |
| 682 | for (i = jz; i > 1; i--) { |
| 683 | fw = fq[i - 1] + fq[i]; |
| 684 | fq[i] += fq[i - 1] - fw; |
| 685 | fq[i - 1] = fw; |
| 686 | } |
| 687 | for (fw = 0.0, i = jz; i >= 2; i--) fw += fq[i]; |
| 688 | if (ih == 0) { |
| 689 | y[0] = fq[0]; |
| 690 | y[1] = fq[1]; |
| 691 | y[2] = fw; |
| 692 | } else { |
| 693 | y[0] = -fq[0]; |
| 694 | y[1] = -fq[1]; |
| 695 | y[2] = -fw; |
| 696 | } |
| 697 | } |
| 698 | return n & 7; |
| 699 | } |
| 700 | |
| 701 | /* __kernel_sin( x, y, iy) |
| 702 | * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 |
| 703 | * Input x is assumed to be bounded by ~pi/4 in magnitude. |
| 704 | * Input y is the tail of x. |
| 705 | * Input iy indicates whether y is 0. (if iy=0, y assume to be 0). |
| 706 | * |
| 707 | * Algorithm |
| 708 | * 1. Since sin(-x) = -sin(x), we need only to consider positive x. |
| 709 | * 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0. |
| 710 | * 3. sin(x) is approximated by a polynomial of degree 13 on |
| 711 | * [0,pi/4] |
| 712 | * 3 13 |
| 713 | * sin(x) ~ x + S1*x + ... + S6*x |
| 714 | * where |
| 715 | * |
| 716 | * |sin(x) 2 4 6 8 10 12 | -58 |
| 717 | * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 |
| 718 | * | x | |
| 719 | * |
| 720 | * 4. sin(x+y) = sin(x) + sin'(x')*y |
| 721 | * ~ sin(x) + (1-x*x/2)*y |
| 722 | * For better accuracy, let |
| 723 | * 3 2 2 2 2 |
| 724 | * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6)))) |
| 725 | * then 3 2 |
| 726 | * sin(x) = x + (S1*x + (x *(r-y/2)+y)) |
| 727 | */ |
| 728 | V8_INLINE double __kernel_sin(double x, double y, int iy) { |
| 729 | static const double |
| 730 | half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ |
| 731 | S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */ |
| 732 | S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */ |
| 733 | S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */ |
| 734 | S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */ |
| 735 | S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */ |
| 736 | S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */ |
| 737 | |
| 738 | double z, r, v; |
| 739 | int32_t ix; |
| 740 | GET_HIGH_WORD(ix, x); |
| 741 | ix &= 0x7fffffff; /* high word of x */ |
| 742 | if (ix < 0x3e400000) { /* |x| < 2**-27 */ |
| 743 | if (static_cast<int>(x) == 0) return x; |
| 744 | } /* generate inexact */ |
| 745 | z = x * x; |
| 746 | v = z * x; |
| 747 | r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6))); |
| 748 | if (iy == 0) { |
| 749 | return x + v * (S1 + z * r); |
| 750 | } else { |
| 751 | return x - ((z * (half * y - v * r) - y) - v * S1); |
| 752 | } |
| 753 | } |
| 754 | |
| 755 | /* __kernel_tan( x, y, k ) |
| 756 | * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 |
| 757 | * Input x is assumed to be bounded by ~pi/4 in magnitude. |
| 758 | * Input y is the tail of x. |
| 759 | * Input k indicates whether tan (if k=1) or |
| 760 | * -1/tan (if k= -1) is returned. |
| 761 | * |
| 762 | * Algorithm |
| 763 | * 1. Since tan(-x) = -tan(x), we need only to consider positive x. |
| 764 | * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. |
| 765 | * 3. tan(x) is approximated by a odd polynomial of degree 27 on |
| 766 | * [0,0.67434] |
| 767 | * 3 27 |
| 768 | * tan(x) ~ x + T1*x + ... + T13*x |
| 769 | * where |
| 770 | * |
| 771 | * |tan(x) 2 4 26 | -59.2 |
| 772 | * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 |
| 773 | * | x | |
| 774 | * |
| 775 | * Note: tan(x+y) = tan(x) + tan'(x)*y |
| 776 | * ~ tan(x) + (1+x*x)*y |
| 777 | * Therefore, for better accuracy in computing tan(x+y), let |
| 778 | * 3 2 2 2 2 |
| 779 | * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) |
| 780 | * then |
| 781 | * 3 2 |
| 782 | * tan(x+y) = x + (T1*x + (x *(r+y)+y)) |
| 783 | * |
| 784 | * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then |
| 785 | * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) |
| 786 | * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) |
| 787 | */ |
| 788 | double __kernel_tan(double x, double y, int iy) { |
| 789 | static const double xxx[] = { |
| 790 | 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ |
| 791 | 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ |
| 792 | 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ |
| 793 | 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ |
| 794 | 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ |
| 795 | 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ |
| 796 | 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ |
| 797 | 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ |
| 798 | 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ |
| 799 | 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ |
| 800 | 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ |
| 801 | -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ |
| 802 | 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ |
| 803 | /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ |
| 804 | /* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ |
| 805 | /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ |
| 806 | }; |
| 807 | #define one xxx[13] |
| 808 | #define pio4 xxx[14] |
| 809 | #define pio4lo xxx[15] |
| 810 | #define T xxx |
| 811 | |
| 812 | double z, r, v, w, s; |
| 813 | int32_t ix, hx; |
| 814 | |
| 815 | GET_HIGH_WORD(hx, x); /* high word of x */ |
| 816 | ix = hx & 0x7fffffff; /* high word of |x| */ |
| 817 | if (ix < 0x3e300000) { /* x < 2**-28 */ |
| 818 | if (static_cast<int>(x) == 0) { /* generate inexact */ |
| 819 | uint32_t low; |
| 820 | GET_LOW_WORD(low, x); |
| 821 | if (((ix | low) | (iy + 1)) == 0) { |
| 822 | return one / fabs(x); |
| 823 | } else { |
| 824 | if (iy == 1) { |
| 825 | return x; |
| 826 | } else { /* compute -1 / (x+y) carefully */ |
| 827 | double a, t; |
| 828 | |
| 829 | z = w = x + y; |
| 830 | SET_LOW_WORD(z, 0); |
| 831 | v = y - (z - x); |
| 832 | t = a = -one / w; |
| 833 | SET_LOW_WORD(t, 0); |
| 834 | s = one + t * z; |
| 835 | return t + a * (s + t * v); |
| 836 | } |
| 837 | } |
| 838 | } |
| 839 | } |
| 840 | if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ |
| 841 | if (hx < 0) { |
| 842 | x = -x; |
| 843 | y = -y; |
| 844 | } |
| 845 | z = pio4 - x; |
| 846 | w = pio4lo - y; |
| 847 | x = z + w; |
| 848 | y = 0.0; |
| 849 | } |
| 850 | z = x * x; |
| 851 | w = z * z; |
| 852 | /* |
| 853 | * Break x^5*(T[1]+x^2*T[2]+...) into |
| 854 | * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + |
| 855 | * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) |
| 856 | */ |
| 857 | r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + w * T[11])))); |
| 858 | v = z * |
| 859 | (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + w * T[12]))))); |
| 860 | s = z * x; |
| 861 | r = y + z * (s * (r + v) + y); |
| 862 | r += T[0] * s; |
| 863 | w = x + r; |
| 864 | if (ix >= 0x3FE59428) { |
| 865 | v = iy; |
| 866 | return (1 - ((hx >> 30) & 2)) * (v - 2.0 * (x - (w * w / (w + v) - r))); |
| 867 | } |
| 868 | if (iy == 1) { |
| 869 | return w; |
| 870 | } else { |
| 871 | /* |
| 872 | * if allow error up to 2 ulp, simply return |
| 873 | * -1.0 / (x+r) here |
| 874 | */ |
| 875 | /* compute -1.0 / (x+r) accurately */ |
| 876 | double a, t; |
| 877 | z = w; |
| 878 | SET_LOW_WORD(z, 0); |
| 879 | v = r - (z - x); /* z+v = r+x */ |
| 880 | t = a = -1.0 / w; /* a = -1.0/w */ |
| 881 | SET_LOW_WORD(t, 0); |
| 882 | s = 1.0 + t * z; |
| 883 | return t + a * (s + t * v); |
| 884 | } |
| 885 | |
| 886 | #undef one |
| 887 | #undef pio4 |
| 888 | #undef pio4lo |
| 889 | #undef T |
| 890 | } |
| 891 | |
| 892 | } // namespace |
| 893 | |
| 894 | /* atan(x) |
| 895 | * Method |
| 896 | * 1. Reduce x to positive by atan(x) = -atan(-x). |
| 897 | * 2. According to the integer k=4t+0.25 chopped, t=x, the argument |
| 898 | * is further reduced to one of the following intervals and the |
| 899 | * arctangent of t is evaluated by the corresponding formula: |
| 900 | * |
| 901 | * [0,7/16] atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...) |
| 902 | * [7/16,11/16] atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) ) |
| 903 | * [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) ) |
| 904 | * [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) ) |
| 905 | * [39/16,INF] atan(x) = atan(INF) + atan( -1/t ) |
| 906 | * |
| 907 | * Constants: |
| 908 | * The hexadecimal values are the intended ones for the following |
| 909 | * constants. The decimal values may be used, provided that the |
| 910 | * compiler will convert from decimal to binary accurately enough |
| 911 | * to produce the hexadecimal values shown. |
| 912 | */ |
| 913 | double atan(double x) { |
| 914 | static const double atanhi[] = { |
| 915 | 4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */ |
| 916 | 7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */ |
| 917 | 9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */ |
| 918 | 1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */ |
| 919 | }; |
| 920 | |
| 921 | static const double atanlo[] = { |
| 922 | 2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */ |
| 923 | 3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */ |
| 924 | 1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */ |
| 925 | 6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */ |
| 926 | }; |
| 927 | |
| 928 | static const double aT[] = { |
| 929 | 3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */ |
| 930 | -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */ |
| 931 | 1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */ |
| 932 | -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */ |
| 933 | 9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */ |
| 934 | -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */ |
| 935 | 6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */ |
| 936 | -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */ |
| 937 | 4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */ |
| 938 | -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */ |
| 939 | 1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */ |
| 940 | }; |
| 941 | |
| 942 | static const double one = 1.0, huge = 1.0e300; |
| 943 | |
| 944 | double w, s1, s2, z; |
| 945 | int32_t ix, hx, id; |
| 946 | |
| 947 | GET_HIGH_WORD(hx, x); |
| 948 | ix = hx & 0x7fffffff; |
| 949 | if (ix >= 0x44100000) { /* if |x| >= 2^66 */ |
| 950 | uint32_t low; |
| 951 | GET_LOW_WORD(low, x); |
| 952 | if (ix > 0x7ff00000 || (ix == 0x7ff00000 && (low != 0))) |
| 953 | return x + x; /* NaN */ |
| 954 | if (hx > 0) |
| 955 | return atanhi[3] + *(volatile double *)&atanlo[3]; |
| 956 | else |
| 957 | return -atanhi[3] - *(volatile double *)&atanlo[3]; |
| 958 | } |
| 959 | if (ix < 0x3fdc0000) { /* |x| < 0.4375 */ |
| 960 | if (ix < 0x3e400000) { /* |x| < 2^-27 */ |
| 961 | if (huge + x > one) return x; /* raise inexact */ |
| 962 | } |
| 963 | id = -1; |
| 964 | } else { |
| 965 | x = fabs(x); |
| 966 | if (ix < 0x3ff30000) { /* |x| < 1.1875 */ |
| 967 | if (ix < 0x3fe60000) { /* 7/16 <=|x|<11/16 */ |
| 968 | id = 0; |
| 969 | x = (2.0 * x - one) / (2.0 + x); |
| 970 | } else { /* 11/16<=|x|< 19/16 */ |
| 971 | id = 1; |
| 972 | x = (x - one) / (x + one); |
| 973 | } |
| 974 | } else { |
| 975 | if (ix < 0x40038000) { /* |x| < 2.4375 */ |
| 976 | id = 2; |
| 977 | x = (x - 1.5) / (one + 1.5 * x); |
| 978 | } else { /* 2.4375 <= |x| < 2^66 */ |
| 979 | id = 3; |
| 980 | x = -1.0 / x; |
| 981 | } |
| 982 | } |
| 983 | } |
| 984 | /* end of argument reduction */ |
| 985 | z = x * x; |
| 986 | w = z * z; |
| 987 | /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */ |
| 988 | s1 = z * (aT[0] + |
| 989 | w * (aT[2] + w * (aT[4] + w * (aT[6] + w * (aT[8] + w * aT[10]))))); |
| 990 | s2 = w * (aT[1] + w * (aT[3] + w * (aT[5] + w * (aT[7] + w * aT[9])))); |
| 991 | if (id < 0) { |
| 992 | return x - x * (s1 + s2); |
| 993 | } else { |
| 994 | z = atanhi[id] - ((x * (s1 + s2) - atanlo[id]) - x); |
| 995 | return (hx < 0) ? -z : z; |
| 996 | } |
| 997 | } |
| 998 | |
| 999 | /* atan2(y,x) |
| 1000 | * Method : |
| 1001 | * 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x). |
| 1002 | * 2. Reduce x to positive by (if x and y are unexceptional): |
| 1003 | * ARG (x+iy) = arctan(y/x) ... if x > 0, |
| 1004 | * ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0, |
| 1005 | * |
| 1006 | * Special cases: |
| 1007 | * |
| 1008 | * ATAN2((anything), NaN ) is NaN; |
| 1009 | * ATAN2(NAN , (anything) ) is NaN; |
| 1010 | * ATAN2(+-0, +(anything but NaN)) is +-0 ; |
| 1011 | * ATAN2(+-0, -(anything but NaN)) is +-pi ; |
| 1012 | * ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2; |
| 1013 | * ATAN2(+-(anything but INF and NaN), +INF) is +-0 ; |
| 1014 | * ATAN2(+-(anything but INF and NaN), -INF) is +-pi; |
| 1015 | * ATAN2(+-INF,+INF ) is +-pi/4 ; |
| 1016 | * ATAN2(+-INF,-INF ) is +-3pi/4; |
| 1017 | * ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2; |
| 1018 | * |
| 1019 | * Constants: |
| 1020 | * The hexadecimal values are the intended ones for the following |
| 1021 | * constants. The decimal values may be used, provided that the |
| 1022 | * compiler will convert from decimal to binary accurately enough |
| 1023 | * to produce the hexadecimal values shown. |
| 1024 | */ |
| 1025 | double atan2(double y, double x) { |
| 1026 | static volatile double tiny = 1.0e-300; |
| 1027 | static const double |
| 1028 | zero = 0.0, |
| 1029 | pi_o_4 = 7.8539816339744827900E-01, /* 0x3FE921FB, 0x54442D18 */ |
| 1030 | pi_o_2 = 1.5707963267948965580E+00, /* 0x3FF921FB, 0x54442D18 */ |
| 1031 | pi = 3.1415926535897931160E+00; /* 0x400921FB, 0x54442D18 */ |
| 1032 | static volatile double pi_lo = |
| 1033 | 1.2246467991473531772E-16; /* 0x3CA1A626, 0x33145C07 */ |
| 1034 | |
| 1035 | double z; |
| 1036 | int32_t k, m, hx, hy, ix, iy; |
| 1037 | uint32_t lx, ly; |
| 1038 | |
| 1039 | EXTRACT_WORDS(hx, lx, x); |
| 1040 | ix = hx & 0x7fffffff; |
| 1041 | EXTRACT_WORDS(hy, ly, y); |
| 1042 | iy = hy & 0x7fffffff; |
| 1043 | if (((ix | ((lx | -static_cast<int32_t>(lx)) >> 31)) > 0x7ff00000) || |
| 1044 | ((iy | ((ly | -static_cast<int32_t>(ly)) >> 31)) > 0x7ff00000)) { |
| 1045 | return x + y; /* x or y is NaN */ |
| 1046 | } |
| 1047 | if (((hx - 0x3ff00000) | lx) == 0) return atan(y); /* x=1.0 */ |
| 1048 | m = ((hy >> 31) & 1) | ((hx >> 30) & 2); /* 2*sign(x)+sign(y) */ |
| 1049 | |
| 1050 | /* when y = 0 */ |
| 1051 | if ((iy | ly) == 0) { |
| 1052 | switch (m) { |
| 1053 | case 0: |
| 1054 | case 1: |
| 1055 | return y; /* atan(+-0,+anything)=+-0 */ |
| 1056 | case 2: |
| 1057 | return pi + tiny; /* atan(+0,-anything) = pi */ |
| 1058 | case 3: |
| 1059 | return -pi - tiny; /* atan(-0,-anything) =-pi */ |
| 1060 | } |
| 1061 | } |
| 1062 | /* when x = 0 */ |
| 1063 | if ((ix | lx) == 0) return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny; |
| 1064 | |
| 1065 | /* when x is INF */ |
| 1066 | if (ix == 0x7ff00000) { |
| 1067 | if (iy == 0x7ff00000) { |
| 1068 | switch (m) { |
| 1069 | case 0: |
| 1070 | return pi_o_4 + tiny; /* atan(+INF,+INF) */ |
| 1071 | case 1: |
| 1072 | return -pi_o_4 - tiny; /* atan(-INF,+INF) */ |
| 1073 | case 2: |
| 1074 | return 3.0 * pi_o_4 + tiny; /*atan(+INF,-INF)*/ |
| 1075 | case 3: |
| 1076 | return -3.0 * pi_o_4 - tiny; /*atan(-INF,-INF)*/ |
| 1077 | } |
| 1078 | } else { |
| 1079 | switch (m) { |
| 1080 | case 0: |
| 1081 | return zero; /* atan(+...,+INF) */ |
| 1082 | case 1: |
| 1083 | return -zero; /* atan(-...,+INF) */ |
| 1084 | case 2: |
| 1085 | return pi + tiny; /* atan(+...,-INF) */ |
| 1086 | case 3: |
| 1087 | return -pi - tiny; /* atan(-...,-INF) */ |
| 1088 | } |
| 1089 | } |
| 1090 | } |
| 1091 | /* when y is INF */ |
| 1092 | if (iy == 0x7ff00000) return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny; |
| 1093 | |
| 1094 | /* compute y/x */ |
| 1095 | k = (iy - ix) >> 20; |
| 1096 | if (k > 60) { /* |y/x| > 2**60 */ |
| 1097 | z = pi_o_2 + 0.5 * pi_lo; |
| 1098 | m &= 1; |
| 1099 | } else if (hx < 0 && k < -60) { |
| 1100 | z = 0.0; /* 0 > |y|/x > -2**-60 */ |
| 1101 | } else { |
| 1102 | z = atan(fabs(y / x)); /* safe to do y/x */ |
| 1103 | } |
| 1104 | switch (m) { |
| 1105 | case 0: |
| 1106 | return z; /* atan(+,+) */ |
| 1107 | case 1: |
| 1108 | return -z; /* atan(-,+) */ |
| 1109 | case 2: |
| 1110 | return pi - (z - pi_lo); /* atan(+,-) */ |
| 1111 | default: /* case 3 */ |
| 1112 | return (z - pi_lo) - pi; /* atan(-,-) */ |
| 1113 | } |
| 1114 | } |
| 1115 | |
| 1116 | /* cos(x) |
| 1117 | * Return cosine function of x. |
| 1118 | * |
| 1119 | * kernel function: |
| 1120 | * __kernel_sin ... sine function on [-pi/4,pi/4] |
| 1121 | * __kernel_cos ... cosine function on [-pi/4,pi/4] |
| 1122 | * __ieee754_rem_pio2 ... argument reduction routine |
| 1123 | * |
| 1124 | * Method. |
| 1125 | * Let S,C and T denote the sin, cos and tan respectively on |
| 1126 | * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 |
| 1127 | * in [-pi/4 , +pi/4], and let n = k mod 4. |
| 1128 | * We have |
| 1129 | * |
| 1130 | * n sin(x) cos(x) tan(x) |
| 1131 | * ---------------------------------------------------------- |
| 1132 | * 0 S C T |
| 1133 | * 1 C -S -1/T |
| 1134 | * 2 -S -C T |
| 1135 | * 3 -C S -1/T |
| 1136 | * ---------------------------------------------------------- |
| 1137 | * |
| 1138 | * Special cases: |
| 1139 | * Let trig be any of sin, cos, or tan. |
| 1140 | * trig(+-INF) is NaN, with signals; |
| 1141 | * trig(NaN) is that NaN; |
| 1142 | * |
| 1143 | * Accuracy: |
| 1144 | * TRIG(x) returns trig(x) nearly rounded |
| 1145 | */ |
| 1146 | double cos(double x) { |
| 1147 | double y[2], z = 0.0; |
| 1148 | int32_t n, ix; |
| 1149 | |
| 1150 | /* High word of x. */ |
| 1151 | GET_HIGH_WORD(ix, x); |
| 1152 | |
| 1153 | /* |x| ~< pi/4 */ |
| 1154 | ix &= 0x7fffffff; |
| 1155 | if (ix <= 0x3fe921fb) { |
| 1156 | return __kernel_cos(x, z); |
| 1157 | } else if (ix >= 0x7ff00000) { |
| 1158 | /* cos(Inf or NaN) is NaN */ |
| 1159 | return x - x; |
| 1160 | } else { |
| 1161 | /* argument reduction needed */ |
| 1162 | n = __ieee754_rem_pio2(x, y); |
| 1163 | switch (n & 3) { |
| 1164 | case 0: |
| 1165 | return __kernel_cos(y[0], y[1]); |
| 1166 | case 1: |
| 1167 | return -__kernel_sin(y[0], y[1], 1); |
| 1168 | case 2: |
| 1169 | return -__kernel_cos(y[0], y[1]); |
| 1170 | default: |
| 1171 | return __kernel_sin(y[0], y[1], 1); |
| 1172 | } |
| 1173 | } |
| 1174 | } |
| 1175 | |
| 1176 | /* exp(x) |
| 1177 | * Returns the exponential of x. |
| 1178 | * |
| 1179 | * Method |
| 1180 | * 1. Argument reduction: |
| 1181 | * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. |
| 1182 | * Given x, find r and integer k such that |
| 1183 | * |
| 1184 | * x = k*ln2 + r, |r| <= 0.5*ln2. |
| 1185 | * |
| 1186 | * Here r will be represented as r = hi-lo for better |
| 1187 | * accuracy. |
| 1188 | * |
| 1189 | * 2. Approximation of exp(r) by a special rational function on |
| 1190 | * the interval [0,0.34658]: |
| 1191 | * Write |
| 1192 | * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... |
| 1193 | * We use a special Remes algorithm on [0,0.34658] to generate |
| 1194 | * a polynomial of degree 5 to approximate R. The maximum error |
| 1195 | * of this polynomial approximation is bounded by 2**-59. In |
| 1196 | * other words, |
| 1197 | * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 |
| 1198 | * (where z=r*r, and the values of P1 to P5 are listed below) |
| 1199 | * and |
| 1200 | * | 5 | -59 |
| 1201 | * | 2.0+P1*z+...+P5*z - R(z) | <= 2 |
| 1202 | * | | |
| 1203 | * The computation of exp(r) thus becomes |
| 1204 | * 2*r |
| 1205 | * exp(r) = 1 + ------- |
| 1206 | * R - r |
| 1207 | * r*R1(r) |
| 1208 | * = 1 + r + ----------- (for better accuracy) |
| 1209 | * 2 - R1(r) |
| 1210 | * where |
| 1211 | * 2 4 10 |
| 1212 | * R1(r) = r - (P1*r + P2*r + ... + P5*r ). |
| 1213 | * |
| 1214 | * 3. Scale back to obtain exp(x): |
| 1215 | * From step 1, we have |
| 1216 | * exp(x) = 2^k * exp(r) |
| 1217 | * |
| 1218 | * Special cases: |
| 1219 | * exp(INF) is INF, exp(NaN) is NaN; |
| 1220 | * exp(-INF) is 0, and |
| 1221 | * for finite argument, only exp(0)=1 is exact. |
| 1222 | * |
| 1223 | * Accuracy: |
| 1224 | * according to an error analysis, the error is always less than |
| 1225 | * 1 ulp (unit in the last place). |
| 1226 | * |
| 1227 | * Misc. info. |
| 1228 | * For IEEE double |
| 1229 | * if x > 7.09782712893383973096e+02 then exp(x) overflow |
| 1230 | * if x < -7.45133219101941108420e+02 then exp(x) underflow |
| 1231 | * |
| 1232 | * Constants: |
| 1233 | * The hexadecimal values are the intended ones for the following |
| 1234 | * constants. The decimal values may be used, provided that the |
| 1235 | * compiler will convert from decimal to binary accurately enough |
| 1236 | * to produce the hexadecimal values shown. |
| 1237 | */ |
| 1238 | double exp(double x) { |
| 1239 | static const double |
| 1240 | one = 1.0, |
| 1241 | halF[2] = {0.5, -0.5}, |
| 1242 | o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ |
| 1243 | u_threshold = -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */ |
| 1244 | ln2HI[2] = {6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ |
| 1245 | -6.93147180369123816490e-01}, /* 0xbfe62e42, 0xfee00000 */ |
| 1246 | ln2LO[2] = {1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ |
| 1247 | -1.90821492927058770002e-10}, /* 0xbdea39ef, 0x35793c76 */ |
| 1248 | invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ |
| 1249 | P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ |
| 1250 | P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ |
| 1251 | P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ |
| 1252 | P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ |
| 1253 | P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ |
| 1254 | E = 2.718281828459045; /* 0x4005bf0a, 0x8b145769 */ |
| 1255 | |
| 1256 | static volatile double |
| 1257 | huge = 1.0e+300, |
| 1258 | twom1000 = 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/ |
| 1259 | two1023 = 8.988465674311579539e307; /* 0x1p1023 */ |
| 1260 | |
| 1261 | double y, hi = 0.0, lo = 0.0, c, t, twopk; |
| 1262 | int32_t k = 0, xsb; |
| 1263 | uint32_t hx; |
| 1264 | |
| 1265 | GET_HIGH_WORD(hx, x); |
| 1266 | xsb = (hx >> 31) & 1; /* sign bit of x */ |
| 1267 | hx &= 0x7fffffff; /* high word of |x| */ |
| 1268 | |
| 1269 | /* filter out non-finite argument */ |
| 1270 | if (hx >= 0x40862E42) { /* if |x|>=709.78... */ |
| 1271 | if (hx >= 0x7ff00000) { |
| 1272 | uint32_t lx; |
| 1273 | GET_LOW_WORD(lx, x); |
| 1274 | if (((hx & 0xfffff) | lx) != 0) |
| 1275 | return x + x; /* NaN */ |
| 1276 | else |
| 1277 | return (xsb == 0) ? x : 0.0; /* exp(+-inf)={inf,0} */ |
| 1278 | } |
| 1279 | if (x > o_threshold) return huge * huge; /* overflow */ |
| 1280 | if (x < u_threshold) return twom1000 * twom1000; /* underflow */ |
| 1281 | } |
| 1282 | |
| 1283 | /* argument reduction */ |
| 1284 | if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ |
| 1285 | if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ |
| 1286 | /* TODO(rtoy): We special case exp(1) here to return the correct |
| 1287 | * value of E, as the computation below would get the last bit |
| 1288 | * wrong. We should probably fix the algorithm instead. |
| 1289 | */ |
| 1290 | if (x == 1.0) return E; |
| 1291 | hi = x - ln2HI[xsb]; |
| 1292 | lo = ln2LO[xsb]; |
| 1293 | k = 1 - xsb - xsb; |
| 1294 | } else { |
| 1295 | k = static_cast<int>(invln2 * x + halF[xsb]); |
| 1296 | t = k; |
| 1297 | hi = x - t * ln2HI[0]; /* t*ln2HI is exact here */ |
| 1298 | lo = t * ln2LO[0]; |
| 1299 | } |
| 1300 | STRICT_ASSIGN(double, x, hi - lo); |
| 1301 | } else if (hx < 0x3e300000) { /* when |x|<2**-28 */ |
| 1302 | if (huge + x > one) return one + x; /* trigger inexact */ |
| 1303 | } else { |
| 1304 | k = 0; |
| 1305 | } |
| 1306 | |
| 1307 | /* x is now in primary range */ |
| 1308 | t = x * x; |
| 1309 | if (k >= -1021) { |
| 1310 | INSERT_WORDS(twopk, 0x3ff00000 + (k << 20), 0); |
| 1311 | } else { |
| 1312 | INSERT_WORDS(twopk, 0x3ff00000 + ((k + 1000) << 20), 0); |
| 1313 | } |
| 1314 | c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); |
| 1315 | if (k == 0) { |
| 1316 | return one - ((x * c) / (c - 2.0) - x); |
| 1317 | } else { |
| 1318 | y = one - ((lo - (x * c) / (2.0 - c)) - hi); |
| 1319 | } |
| 1320 | if (k >= -1021) { |
| 1321 | if (k == 1024) return y * 2.0 * two1023; |
| 1322 | return y * twopk; |
| 1323 | } else { |
| 1324 | return y * twopk * twom1000; |
| 1325 | } |
| 1326 | } |
| 1327 | |
| 1328 | /* |
| 1329 | * Method : |
| 1330 | * 1.Reduced x to positive by atanh(-x) = -atanh(x) |
| 1331 | * 2.For x>=0.5 |
| 1332 | * 1 2x x |
| 1333 | * atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------) |
| 1334 | * 2 1 - x 1 - x |
| 1335 | * |
| 1336 | * For x<0.5 |
| 1337 | * atanh(x) = 0.5*log1p(2x+2x*x/(1-x)) |
| 1338 | * |
| 1339 | * Special cases: |
| 1340 | * atanh(x) is NaN if |x| > 1 with signal; |
| 1341 | * atanh(NaN) is that NaN with no signal; |
| 1342 | * atanh(+-1) is +-INF with signal. |
| 1343 | * |
| 1344 | */ |
| 1345 | double atanh(double x) { |
| 1346 | static const double one = 1.0, huge = 1e300; |
| 1347 | static const double zero = 0.0; |
| 1348 | |
| 1349 | double t; |
| 1350 | int32_t hx, ix; |
| 1351 | uint32_t lx; |
| 1352 | EXTRACT_WORDS(hx, lx, x); |
| 1353 | ix = hx & 0x7fffffff; |
| 1354 | if ((ix | ((lx | -static_cast<int32_t>(lx)) >> 31)) > 0x3ff00000) /* |x|>1 */ |
| 1355 | return (x - x) / (x - x); |
| 1356 | if (ix == 0x3ff00000) return x / zero; |
| 1357 | if (ix < 0x3e300000 && (huge + x) > zero) return x; /* x<2**-28 */ |
| 1358 | SET_HIGH_WORD(x, ix); |
| 1359 | if (ix < 0x3fe00000) { /* x < 0.5 */ |
| 1360 | t = x + x; |
| 1361 | t = 0.5 * log1p(t + t * x / (one - x)); |
| 1362 | } else { |
| 1363 | t = 0.5 * log1p((x + x) / (one - x)); |
| 1364 | } |
| 1365 | if (hx >= 0) |
| 1366 | return t; |
| 1367 | else |
| 1368 | return -t; |
| 1369 | } |
| 1370 | |
| 1371 | /* log(x) |
| 1372 | * Return the logrithm of x |
| 1373 | * |
| 1374 | * Method : |
| 1375 | * 1. Argument Reduction: find k and f such that |
| 1376 | * x = 2^k * (1+f), |
| 1377 | * where sqrt(2)/2 < 1+f < sqrt(2) . |
| 1378 | * |
| 1379 | * 2. Approximation of log(1+f). |
| 1380 | * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) |
| 1381 | * = 2s + 2/3 s**3 + 2/5 s**5 + ....., |
| 1382 | * = 2s + s*R |
| 1383 | * We use a special Reme algorithm on [0,0.1716] to generate |
| 1384 | * a polynomial of degree 14 to approximate R The maximum error |
| 1385 | * of this polynomial approximation is bounded by 2**-58.45. In |
| 1386 | * other words, |
| 1387 | * 2 4 6 8 10 12 14 |
| 1388 | * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s |
| 1389 | * (the values of Lg1 to Lg7 are listed in the program) |
| 1390 | * and |
| 1391 | * | 2 14 | -58.45 |
| 1392 | * | Lg1*s +...+Lg7*s - R(z) | <= 2 |
| 1393 | * | | |
| 1394 | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. |
| 1395 | * In order to guarantee error in log below 1ulp, we compute log |
| 1396 | * by |
| 1397 | * log(1+f) = f - s*(f - R) (if f is not too large) |
| 1398 | * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) |
| 1399 | * |
| 1400 | * 3. Finally, log(x) = k*ln2 + log(1+f). |
| 1401 | * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) |
| 1402 | * Here ln2 is split into two floating point number: |
| 1403 | * ln2_hi + ln2_lo, |
| 1404 | * where n*ln2_hi is always exact for |n| < 2000. |
| 1405 | * |
| 1406 | * Special cases: |
| 1407 | * log(x) is NaN with signal if x < 0 (including -INF) ; |
| 1408 | * log(+INF) is +INF; log(0) is -INF with signal; |
| 1409 | * log(NaN) is that NaN with no signal. |
| 1410 | * |
| 1411 | * Accuracy: |
| 1412 | * according to an error analysis, the error is always less than |
| 1413 | * 1 ulp (unit in the last place). |
| 1414 | * |
| 1415 | * Constants: |
| 1416 | * The hexadecimal values are the intended ones for the following |
| 1417 | * constants. The decimal values may be used, provided that the |
| 1418 | * compiler will convert from decimal to binary accurately enough |
| 1419 | * to produce the hexadecimal values shown. |
| 1420 | */ |
| 1421 | double log(double x) { |
| 1422 | static const double /* -- */ |
| 1423 | ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ |
| 1424 | ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ |
| 1425 | two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ |
| 1426 | Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ |
| 1427 | Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ |
| 1428 | Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ |
| 1429 | Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ |
| 1430 | Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ |
| 1431 | Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ |
| 1432 | Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ |
| 1433 | |
| 1434 | static const double zero = 0.0; |
| 1435 | static volatile double vzero = 0.0; |
| 1436 | |
| 1437 | double hfsq, f, s, z, R, w, t1, t2, dk; |
| 1438 | int32_t k, hx, i, j; |
| 1439 | uint32_t lx; |
| 1440 | |
| 1441 | EXTRACT_WORDS(hx, lx, x); |
| 1442 | |
| 1443 | k = 0; |
| 1444 | if (hx < 0x00100000) { /* x < 2**-1022 */ |
| 1445 | if (((hx & 0x7fffffff) | lx) == 0) |
| 1446 | return -two54 / vzero; /* log(+-0)=-inf */ |
| 1447 | if (hx < 0) return (x - x) / zero; /* log(-#) = NaN */ |
| 1448 | k -= 54; |
| 1449 | x *= two54; /* subnormal number, scale up x */ |
| 1450 | GET_HIGH_WORD(hx, x); |
| 1451 | } |
| 1452 | if (hx >= 0x7ff00000) return x + x; |
| 1453 | k += (hx >> 20) - 1023; |
| 1454 | hx &= 0x000fffff; |
| 1455 | i = (hx + 0x95f64) & 0x100000; |
| 1456 | SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */ |
| 1457 | k += (i >> 20); |
| 1458 | f = x - 1.0; |
| 1459 | if ((0x000fffff & (2 + hx)) < 3) { /* -2**-20 <= f < 2**-20 */ |
| 1460 | if (f == zero) { |
| 1461 | if (k == 0) { |
| 1462 | return zero; |
| 1463 | } else { |
| 1464 | dk = static_cast<double>(k); |
| 1465 | return dk * ln2_hi + dk * ln2_lo; |
| 1466 | } |
| 1467 | } |
| 1468 | R = f * f * (0.5 - 0.33333333333333333 * f); |
| 1469 | if (k == 0) { |
| 1470 | return f - R; |
| 1471 | } else { |
| 1472 | dk = static_cast<double>(k); |
| 1473 | return dk * ln2_hi - ((R - dk * ln2_lo) - f); |
| 1474 | } |
| 1475 | } |
| 1476 | s = f / (2.0 + f); |
| 1477 | dk = static_cast<double>(k); |
| 1478 | z = s * s; |
| 1479 | i = hx - 0x6147a; |
| 1480 | w = z * z; |
| 1481 | j = 0x6b851 - hx; |
| 1482 | t1 = w * (Lg2 + w * (Lg4 + w * Lg6)); |
| 1483 | t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7))); |
| 1484 | i |= j; |
| 1485 | R = t2 + t1; |
| 1486 | if (i > 0) { |
| 1487 | hfsq = 0.5 * f * f; |
| 1488 | if (k == 0) |
| 1489 | return f - (hfsq - s * (hfsq + R)); |
| 1490 | else |
| 1491 | return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f); |
| 1492 | } else { |
| 1493 | if (k == 0) |
| 1494 | return f - s * (f - R); |
| 1495 | else |
| 1496 | return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f); |
| 1497 | } |
| 1498 | } |
| 1499 | |
| 1500 | /* double log1p(double x) |
| 1501 | * |
| 1502 | * Method : |
| 1503 | * 1. Argument Reduction: find k and f such that |
| 1504 | * 1+x = 2^k * (1+f), |
| 1505 | * where sqrt(2)/2 < 1+f < sqrt(2) . |
| 1506 | * |
| 1507 | * Note. If k=0, then f=x is exact. However, if k!=0, then f |
| 1508 | * may not be representable exactly. In that case, a correction |
| 1509 | * term is need. Let u=1+x rounded. Let c = (1+x)-u, then |
| 1510 | * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), |
| 1511 | * and add back the correction term c/u. |
| 1512 | * (Note: when x > 2**53, one can simply return log(x)) |
| 1513 | * |
| 1514 | * 2. Approximation of log1p(f). |
| 1515 | * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) |
| 1516 | * = 2s + 2/3 s**3 + 2/5 s**5 + ....., |
| 1517 | * = 2s + s*R |
| 1518 | * We use a special Reme algorithm on [0,0.1716] to generate |
| 1519 | * a polynomial of degree 14 to approximate R The maximum error |
| 1520 | * of this polynomial approximation is bounded by 2**-58.45. In |
| 1521 | * other words, |
| 1522 | * 2 4 6 8 10 12 14 |
| 1523 | * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s |
| 1524 | * (the values of Lp1 to Lp7 are listed in the program) |
| 1525 | * and |
| 1526 | * | 2 14 | -58.45 |
| 1527 | * | Lp1*s +...+Lp7*s - R(z) | <= 2 |
| 1528 | * | | |
| 1529 | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. |
| 1530 | * In order to guarantee error in log below 1ulp, we compute log |
| 1531 | * by |
| 1532 | * log1p(f) = f - (hfsq - s*(hfsq+R)). |
| 1533 | * |
| 1534 | * 3. Finally, log1p(x) = k*ln2 + log1p(f). |
| 1535 | * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) |
| 1536 | * Here ln2 is split into two floating point number: |
| 1537 | * ln2_hi + ln2_lo, |
| 1538 | * where n*ln2_hi is always exact for |n| < 2000. |
| 1539 | * |
| 1540 | * Special cases: |
| 1541 | * log1p(x) is NaN with signal if x < -1 (including -INF) ; |
| 1542 | * log1p(+INF) is +INF; log1p(-1) is -INF with signal; |
| 1543 | * log1p(NaN) is that NaN with no signal. |
| 1544 | * |
| 1545 | * Accuracy: |
| 1546 | * according to an error analysis, the error is always less than |
| 1547 | * 1 ulp (unit in the last place). |
| 1548 | * |
| 1549 | * Constants: |
| 1550 | * The hexadecimal values are the intended ones for the following |
| 1551 | * constants. The decimal values may be used, provided that the |
| 1552 | * compiler will convert from decimal to binary accurately enough |
| 1553 | * to produce the hexadecimal values shown. |
| 1554 | * |
| 1555 | * Note: Assuming log() return accurate answer, the following |
| 1556 | * algorithm can be used to compute log1p(x) to within a few ULP: |
| 1557 | * |
| 1558 | * u = 1+x; |
| 1559 | * if(u==1.0) return x ; else |
| 1560 | * return log(u)*(x/(u-1.0)); |
| 1561 | * |
| 1562 | * See HP-15C Advanced Functions Handbook, p.193. |
| 1563 | */ |
| 1564 | double log1p(double x) { |
| 1565 | static const double /* -- */ |
| 1566 | ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ |
| 1567 | ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ |
| 1568 | two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ |
| 1569 | Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ |
| 1570 | Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ |
| 1571 | Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ |
| 1572 | Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ |
| 1573 | Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ |
| 1574 | Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ |
| 1575 | Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ |
| 1576 | |
| 1577 | static const double zero = 0.0; |
| 1578 | static volatile double vzero = 0.0; |
| 1579 | |
| 1580 | double hfsq, f, c, s, z, R, u; |
| 1581 | int32_t k, hx, hu, ax; |
| 1582 | |
| 1583 | GET_HIGH_WORD(hx, x); |
| 1584 | ax = hx & 0x7fffffff; |
| 1585 | |
| 1586 | k = 1; |
| 1587 | if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */ |
| 1588 | if (ax >= 0x3ff00000) { /* x <= -1.0 */ |
| 1589 | if (x == -1.0) |
| 1590 | return -two54 / vzero; /* log1p(-1)=+inf */ |
| 1591 | else |
| 1592 | return (x - x) / (x - x); /* log1p(x<-1)=NaN */ |
| 1593 | } |
| 1594 | if (ax < 0x3e200000) { /* |x| < 2**-29 */ |
| 1595 | if (two54 + x > zero /* raise inexact */ |
| 1596 | && ax < 0x3c900000) /* |x| < 2**-54 */ |
| 1597 | return x; |
| 1598 | else |
| 1599 | return x - x * x * 0.5; |
| 1600 | } |
| 1601 | if (hx > 0 || hx <= static_cast<int32_t>(0xbfd2bec4)) { |
| 1602 | k = 0; |
| 1603 | f = x; |
| 1604 | hu = 1; |
| 1605 | } /* sqrt(2)/2- <= 1+x < sqrt(2)+ */ |
| 1606 | } |
| 1607 | if (hx >= 0x7ff00000) return x + x; |
| 1608 | if (k != 0) { |
| 1609 | if (hx < 0x43400000) { |
| 1610 | STRICT_ASSIGN(double, u, 1.0 + x); |
| 1611 | GET_HIGH_WORD(hu, u); |
| 1612 | k = (hu >> 20) - 1023; |
| 1613 | c = (k > 0) ? 1.0 - (u - x) : x - (u - 1.0); /* correction term */ |
| 1614 | c /= u; |
| 1615 | } else { |
| 1616 | u = x; |
| 1617 | GET_HIGH_WORD(hu, u); |
| 1618 | k = (hu >> 20) - 1023; |
| 1619 | c = 0; |
| 1620 | } |
| 1621 | hu &= 0x000fffff; |
| 1622 | /* |
| 1623 | * The approximation to sqrt(2) used in thresholds is not |
| 1624 | * critical. However, the ones used above must give less |
| 1625 | * strict bounds than the one here so that the k==0 case is |
| 1626 | * never reached from here, since here we have committed to |
| 1627 | * using the correction term but don't use it if k==0. |
| 1628 | */ |
| 1629 | if (hu < 0x6a09e) { /* u ~< sqrt(2) */ |
| 1630 | SET_HIGH_WORD(u, hu | 0x3ff00000); /* normalize u */ |
| 1631 | } else { |
| 1632 | k += 1; |
| 1633 | SET_HIGH_WORD(u, hu | 0x3fe00000); /* normalize u/2 */ |
| 1634 | hu = (0x00100000 - hu) >> 2; |
| 1635 | } |
| 1636 | f = u - 1.0; |
| 1637 | } |
| 1638 | hfsq = 0.5 * f * f; |
| 1639 | if (hu == 0) { /* |f| < 2**-20 */ |
| 1640 | if (f == zero) { |
| 1641 | if (k == 0) { |
| 1642 | return zero; |
| 1643 | } else { |
| 1644 | c += k * ln2_lo; |
| 1645 | return k * ln2_hi + c; |
| 1646 | } |
| 1647 | } |
| 1648 | R = hfsq * (1.0 - 0.66666666666666666 * f); |
| 1649 | if (k == 0) |
| 1650 | return f - R; |
| 1651 | else |
| 1652 | return k * ln2_hi - ((R - (k * ln2_lo + c)) - f); |
| 1653 | } |
| 1654 | s = f / (2.0 + f); |
| 1655 | z = s * s; |
| 1656 | R = z * (Lp1 + |
| 1657 | z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 + z * Lp7)))))); |
| 1658 | if (k == 0) |
| 1659 | return f - (hfsq - s * (hfsq + R)); |
| 1660 | else |
| 1661 | return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f); |
| 1662 | } |
| 1663 | |
| 1664 | /* |
| 1665 | * k_log1p(f): |
| 1666 | * Return log(1+f) - f for 1+f in ~[sqrt(2)/2, sqrt(2)]. |
| 1667 | * |
| 1668 | * The following describes the overall strategy for computing |
| 1669 | * logarithms in base e. The argument reduction and adding the final |
| 1670 | * term of the polynomial are done by the caller for increased accuracy |
| 1671 | * when different bases are used. |
| 1672 | * |
| 1673 | * Method : |
| 1674 | * 1. Argument Reduction: find k and f such that |
| 1675 | * x = 2^k * (1+f), |
| 1676 | * where sqrt(2)/2 < 1+f < sqrt(2) . |
| 1677 | * |
| 1678 | * 2. Approximation of log(1+f). |
| 1679 | * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) |
| 1680 | * = 2s + 2/3 s**3 + 2/5 s**5 + ....., |
| 1681 | * = 2s + s*R |
| 1682 | * We use a special Reme algorithm on [0,0.1716] to generate |
| 1683 | * a polynomial of degree 14 to approximate R The maximum error |
| 1684 | * of this polynomial approximation is bounded by 2**-58.45. In |
| 1685 | * other words, |
| 1686 | * 2 4 6 8 10 12 14 |
| 1687 | * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s |
| 1688 | * (the values of Lg1 to Lg7 are listed in the program) |
| 1689 | * and |
| 1690 | * | 2 14 | -58.45 |
| 1691 | * | Lg1*s +...+Lg7*s - R(z) | <= 2 |
| 1692 | * | | |
| 1693 | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. |
| 1694 | * In order to guarantee error in log below 1ulp, we compute log |
| 1695 | * by |
| 1696 | * log(1+f) = f - s*(f - R) (if f is not too large) |
| 1697 | * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) |
| 1698 | * |
| 1699 | * 3. Finally, log(x) = k*ln2 + log(1+f). |
| 1700 | * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) |
| 1701 | * Here ln2 is split into two floating point number: |
| 1702 | * ln2_hi + ln2_lo, |
| 1703 | * where n*ln2_hi is always exact for |n| < 2000. |
| 1704 | * |
| 1705 | * Special cases: |
| 1706 | * log(x) is NaN with signal if x < 0 (including -INF) ; |
| 1707 | * log(+INF) is +INF; log(0) is -INF with signal; |
| 1708 | * log(NaN) is that NaN with no signal. |
| 1709 | * |
| 1710 | * Accuracy: |
| 1711 | * according to an error analysis, the error is always less than |
| 1712 | * 1 ulp (unit in the last place). |
| 1713 | * |
| 1714 | * Constants: |
| 1715 | * The hexadecimal values are the intended ones for the following |
| 1716 | * constants. The decimal values may be used, provided that the |
| 1717 | * compiler will convert from decimal to binary accurately enough |
| 1718 | * to produce the hexadecimal values shown. |
| 1719 | */ |
| 1720 | |
| 1721 | static const double Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ |
| 1722 | Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ |
| 1723 | Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ |
| 1724 | Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ |
| 1725 | Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ |
| 1726 | Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ |
| 1727 | Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ |
| 1728 | |
| 1729 | /* |
| 1730 | * We always inline k_log1p(), since doing so produces a |
| 1731 | * substantial performance improvement (~40% on amd64). |
| 1732 | */ |
| 1733 | static inline double k_log1p(double f) { |
| 1734 | double hfsq, s, z, R, w, t1, t2; |
| 1735 | |
| 1736 | s = f / (2.0 + f); |
| 1737 | z = s * s; |
| 1738 | w = z * z; |
| 1739 | t1 = w * (Lg2 + w * (Lg4 + w * Lg6)); |
| 1740 | t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7))); |
| 1741 | R = t2 + t1; |
| 1742 | hfsq = 0.5 * f * f; |
| 1743 | return s * (hfsq + R); |
| 1744 | } |
| 1745 | |
| 1746 | /* |
| 1747 | * Return the base 2 logarithm of x. See e_log.c and k_log.h for most |
| 1748 | * comments. |
| 1749 | * |
| 1750 | * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel, |
| 1751 | * then does the combining and scaling steps |
| 1752 | * log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k |
| 1753 | * in not-quite-routine extra precision. |
| 1754 | */ |
| 1755 | double log2(double x) { |
| 1756 | static const double |
| 1757 | two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ |
| 1758 | ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */ |
| 1759 | ivln2lo = 1.67517131648865118353e-10; /* 0x3de705fc, 0x2eefa200 */ |
| 1760 | |
| 1761 | static const double zero = 0.0; |
| 1762 | static volatile double vzero = 0.0; |
| 1763 | |
| 1764 | double f, hfsq, hi, lo, r, val_hi, val_lo, w, y; |
| 1765 | int32_t i, k, hx; |
| 1766 | uint32_t lx; |
| 1767 | |
| 1768 | EXTRACT_WORDS(hx, lx, x); |
| 1769 | |
| 1770 | k = 0; |
| 1771 | if (hx < 0x00100000) { /* x < 2**-1022 */ |
| 1772 | if (((hx & 0x7fffffff) | lx) == 0) |
| 1773 | return -two54 / vzero; /* log(+-0)=-inf */ |
| 1774 | if (hx < 0) return (x - x) / zero; /* log(-#) = NaN */ |
| 1775 | k -= 54; |
| 1776 | x *= two54; /* subnormal number, scale up x */ |
| 1777 | GET_HIGH_WORD(hx, x); |
| 1778 | } |
| 1779 | if (hx >= 0x7ff00000) return x + x; |
| 1780 | if (hx == 0x3ff00000 && lx == 0) return zero; /* log(1) = +0 */ |
| 1781 | k += (hx >> 20) - 1023; |
| 1782 | hx &= 0x000fffff; |
| 1783 | i = (hx + 0x95f64) & 0x100000; |
| 1784 | SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */ |
| 1785 | k += (i >> 20); |
| 1786 | y = static_cast<double>(k); |
| 1787 | f = x - 1.0; |
| 1788 | hfsq = 0.5 * f * f; |
| 1789 | r = k_log1p(f); |
| 1790 | |
| 1791 | /* |
| 1792 | * f-hfsq must (for args near 1) be evaluated in extra precision |
| 1793 | * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2). |
| 1794 | * This is fairly efficient since f-hfsq only depends on f, so can |
| 1795 | * be evaluated in parallel with R. Not combining hfsq with R also |
| 1796 | * keeps R small (though not as small as a true `lo' term would be), |
| 1797 | * so that extra precision is not needed for terms involving R. |
| 1798 | * |
| 1799 | * Compiler bugs involving extra precision used to break Dekker's |
| 1800 | * theorem for spitting f-hfsq as hi+lo, unless double_t was used |
| 1801 | * or the multi-precision calculations were avoided when double_t |
| 1802 | * has extra precision. These problems are now automatically |
| 1803 | * avoided as a side effect of the optimization of combining the |
| 1804 | * Dekker splitting step with the clear-low-bits step. |
| 1805 | * |
| 1806 | * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra |
| 1807 | * precision to avoid a very large cancellation when x is very near |
| 1808 | * these values. Unlike the above cancellations, this problem is |
| 1809 | * specific to base 2. It is strange that adding +-1 is so much |
| 1810 | * harder than adding +-ln2 or +-log10_2. |
| 1811 | * |
| 1812 | * This uses Dekker's theorem to normalize y+val_hi, so the |
| 1813 | * compiler bugs are back in some configurations, sigh. And I |
| 1814 | * don't want to used double_t to avoid them, since that gives a |
| 1815 | * pessimization and the support for avoiding the pessimization |
| 1816 | * is not yet available. |
| 1817 | * |
| 1818 | * The multi-precision calculations for the multiplications are |
| 1819 | * routine. |
| 1820 | */ |
| 1821 | hi = f - hfsq; |
| 1822 | SET_LOW_WORD(hi, 0); |
| 1823 | lo = (f - hi) - hfsq + r; |
| 1824 | val_hi = hi * ivln2hi; |
| 1825 | val_lo = (lo + hi) * ivln2lo + lo * ivln2hi; |
| 1826 | |
| 1827 | /* spadd(val_hi, val_lo, y), except for not using double_t: */ |
| 1828 | w = y + val_hi; |
| 1829 | val_lo += (y - w) + val_hi; |
| 1830 | val_hi = w; |
| 1831 | |
| 1832 | return val_lo + val_hi; |
| 1833 | } |
| 1834 | |
| 1835 | /* |
| 1836 | * Return the base 10 logarithm of x |
| 1837 | * |
| 1838 | * Method : |
| 1839 | * Let log10_2hi = leading 40 bits of log10(2) and |
| 1840 | * log10_2lo = log10(2) - log10_2hi, |
| 1841 | * ivln10 = 1/log(10) rounded. |
| 1842 | * Then |
| 1843 | * n = ilogb(x), |
| 1844 | * if(n<0) n = n+1; |
| 1845 | * x = scalbn(x,-n); |
| 1846 | * log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x)) |
| 1847 | * |
| 1848 | * Note 1: |
| 1849 | * To guarantee log10(10**n)=n, where 10**n is normal, the rounding |
| 1850 | * mode must set to Round-to-Nearest. |
| 1851 | * Note 2: |
| 1852 | * [1/log(10)] rounded to 53 bits has error .198 ulps; |
| 1853 | * log10 is monotonic at all binary break points. |
| 1854 | * |
| 1855 | * Special cases: |
| 1856 | * log10(x) is NaN if x < 0; |
| 1857 | * log10(+INF) is +INF; log10(0) is -INF; |
| 1858 | * log10(NaN) is that NaN; |
| 1859 | * log10(10**N) = N for N=0,1,...,22. |
| 1860 | */ |
| 1861 | double log10(double x) { |
| 1862 | static const double |
| 1863 | two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ |
| 1864 | ivln10 = 4.34294481903251816668e-01, |
| 1865 | log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */ |
| 1866 | log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */ |
| 1867 | |
| 1868 | static const double zero = 0.0; |
| 1869 | static volatile double vzero = 0.0; |
| 1870 | |
| 1871 | double y; |
| 1872 | int32_t i, k, hx; |
| 1873 | uint32_t lx; |
| 1874 | |
| 1875 | EXTRACT_WORDS(hx, lx, x); |
| 1876 | |
| 1877 | k = 0; |
| 1878 | if (hx < 0x00100000) { /* x < 2**-1022 */ |
| 1879 | if (((hx & 0x7fffffff) | lx) == 0) |
| 1880 | return -two54 / vzero; /* log(+-0)=-inf */ |
| 1881 | if (hx < 0) return (x - x) / zero; /* log(-#) = NaN */ |
| 1882 | k -= 54; |
| 1883 | x *= two54; /* subnormal number, scale up x */ |
| 1884 | GET_HIGH_WORD(hx, x); |
| 1885 | GET_LOW_WORD(lx, x); |
| 1886 | } |
| 1887 | if (hx >= 0x7ff00000) return x + x; |
| 1888 | if (hx == 0x3ff00000 && lx == 0) return zero; /* log(1) = +0 */ |
| 1889 | k += (hx >> 20) - 1023; |
| 1890 | |
| 1891 | i = (k & 0x80000000) >> 31; |
| 1892 | hx = (hx & 0x000fffff) | ((0x3ff - i) << 20); |
| 1893 | y = k + i; |
| 1894 | SET_HIGH_WORD(x, hx); |
| 1895 | SET_LOW_WORD(x, lx); |
| 1896 | |
| 1897 | double z = y * log10_2lo + ivln10 * log(x); |
| 1898 | return z + y * log10_2hi; |
| 1899 | } |
| 1900 | |
| 1901 | /* expm1(x) |
| 1902 | * Returns exp(x)-1, the exponential of x minus 1. |
| 1903 | * |
| 1904 | * Method |
| 1905 | * 1. Argument reduction: |
| 1906 | * Given x, find r and integer k such that |
| 1907 | * |
| 1908 | * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 |
| 1909 | * |
| 1910 | * Here a correction term c will be computed to compensate |
| 1911 | * the error in r when rounded to a floating-point number. |
| 1912 | * |
| 1913 | * 2. Approximating expm1(r) by a special rational function on |
| 1914 | * the interval [0,0.34658]: |
| 1915 | * Since |
| 1916 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... |
| 1917 | * we define R1(r*r) by |
| 1918 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) |
| 1919 | * That is, |
| 1920 | * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) |
| 1921 | * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) |
| 1922 | * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... |
| 1923 | * We use a special Reme algorithm on [0,0.347] to generate |
| 1924 | * a polynomial of degree 5 in r*r to approximate R1. The |
| 1925 | * maximum error of this polynomial approximation is bounded |
| 1926 | * by 2**-61. In other words, |
| 1927 | * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 |
| 1928 | * where Q1 = -1.6666666666666567384E-2, |
| 1929 | * Q2 = 3.9682539681370365873E-4, |
| 1930 | * Q3 = -9.9206344733435987357E-6, |
| 1931 | * Q4 = 2.5051361420808517002E-7, |
| 1932 | * Q5 = -6.2843505682382617102E-9; |
| 1933 | * z = r*r, |
| 1934 | * with error bounded by |
| 1935 | * | 5 | -61 |
| 1936 | * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 |
| 1937 | * | | |
| 1938 | * |
| 1939 | * expm1(r) = exp(r)-1 is then computed by the following |
| 1940 | * specific way which minimize the accumulation rounding error: |
| 1941 | * 2 3 |
| 1942 | * r r [ 3 - (R1 + R1*r/2) ] |
| 1943 | * expm1(r) = r + --- + --- * [--------------------] |
| 1944 | * 2 2 [ 6 - r*(3 - R1*r/2) ] |
| 1945 | * |
| 1946 | * To compensate the error in the argument reduction, we use |
| 1947 | * expm1(r+c) = expm1(r) + c + expm1(r)*c |
| 1948 | * ~ expm1(r) + c + r*c |
| 1949 | * Thus c+r*c will be added in as the correction terms for |
| 1950 | * expm1(r+c). Now rearrange the term to avoid optimization |
| 1951 | * screw up: |
| 1952 | * ( 2 2 ) |
| 1953 | * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) |
| 1954 | * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) |
| 1955 | * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) |
| 1956 | * ( ) |
| 1957 | * |
| 1958 | * = r - E |
| 1959 | * 3. Scale back to obtain expm1(x): |
| 1960 | * From step 1, we have |
| 1961 | * expm1(x) = either 2^k*[expm1(r)+1] - 1 |
| 1962 | * = or 2^k*[expm1(r) + (1-2^-k)] |
| 1963 | * 4. Implementation notes: |
| 1964 | * (A). To save one multiplication, we scale the coefficient Qi |
| 1965 | * to Qi*2^i, and replace z by (x^2)/2. |
| 1966 | * (B). To achieve maximum accuracy, we compute expm1(x) by |
| 1967 | * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) |
| 1968 | * (ii) if k=0, return r-E |
| 1969 | * (iii) if k=-1, return 0.5*(r-E)-0.5 |
| 1970 | * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) |
| 1971 | * else return 1.0+2.0*(r-E); |
| 1972 | * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) |
| 1973 | * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else |
| 1974 | * (vii) return 2^k(1-((E+2^-k)-r)) |
| 1975 | * |
| 1976 | * Special cases: |
| 1977 | * expm1(INF) is INF, expm1(NaN) is NaN; |
| 1978 | * expm1(-INF) is -1, and |
| 1979 | * for finite argument, only expm1(0)=0 is exact. |
| 1980 | * |
| 1981 | * Accuracy: |
| 1982 | * according to an error analysis, the error is always less than |
| 1983 | * 1 ulp (unit in the last place). |
| 1984 | * |
| 1985 | * Misc. info. |
| 1986 | * For IEEE double |
| 1987 | * if x > 7.09782712893383973096e+02 then expm1(x) overflow |
| 1988 | * |
| 1989 | * Constants: |
| 1990 | * The hexadecimal values are the intended ones for the following |
| 1991 | * constants. The decimal values may be used, provided that the |
| 1992 | * compiler will convert from decimal to binary accurately enough |
| 1993 | * to produce the hexadecimal values shown. |
| 1994 | */ |
| 1995 | double expm1(double x) { |
| 1996 | static const double |
| 1997 | one = 1.0, |
| 1998 | tiny = 1.0e-300, |
| 1999 | o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ |
| 2000 | ln2_hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */ |
| 2001 | ln2_lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */ |
| 2002 | invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */ |
| 2003 | /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = |
| 2004 | x*x/2: */ |
| 2005 | Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ |
| 2006 | Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ |
| 2007 | Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ |
| 2008 | Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ |
| 2009 | Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ |
| 2010 | |
| 2011 | static volatile double huge = 1.0e+300; |
| 2012 | |
| 2013 | double y, hi, lo, c, t, e, hxs, hfx, r1, twopk; |
| 2014 | int32_t k, xsb; |
| 2015 | uint32_t hx; |
| 2016 | |
| 2017 | GET_HIGH_WORD(hx, x); |
| 2018 | xsb = hx & 0x80000000; /* sign bit of x */ |
| 2019 | hx &= 0x7fffffff; /* high word of |x| */ |
| 2020 | |
| 2021 | /* filter out huge and non-finite argument */ |
| 2022 | if (hx >= 0x4043687A) { /* if |x|>=56*ln2 */ |
| 2023 | if (hx >= 0x40862E42) { /* if |x|>=709.78... */ |
| 2024 | if (hx >= 0x7ff00000) { |
| 2025 | uint32_t low; |
| 2026 | GET_LOW_WORD(low, x); |
| 2027 | if (((hx & 0xfffff) | low) != 0) |
| 2028 | return x + x; /* NaN */ |
| 2029 | else |
| 2030 | return (xsb == 0) ? x : -1.0; /* exp(+-inf)={inf,-1} */ |
| 2031 | } |
| 2032 | if (x > o_threshold) return huge * huge; /* overflow */ |
| 2033 | } |
| 2034 | if (xsb != 0) { /* x < -56*ln2, return -1.0 with inexact */ |
| 2035 | if (x + tiny < 0.0) /* raise inexact */ |
| 2036 | return tiny - one; /* return -1 */ |
| 2037 | } |
| 2038 | } |
| 2039 | |
| 2040 | /* argument reduction */ |
| 2041 | if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ |
| 2042 | if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ |
| 2043 | if (xsb == 0) { |
| 2044 | hi = x - ln2_hi; |
| 2045 | lo = ln2_lo; |
| 2046 | k = 1; |
| 2047 | } else { |
| 2048 | hi = x + ln2_hi; |
| 2049 | lo = -ln2_lo; |
| 2050 | k = -1; |
| 2051 | } |
| 2052 | } else { |
| 2053 | k = invln2 * x + ((xsb == 0) ? 0.5 : -0.5); |
| 2054 | t = k; |
| 2055 | hi = x - t * ln2_hi; /* t*ln2_hi is exact here */ |
| 2056 | lo = t * ln2_lo; |
| 2057 | } |
| 2058 | STRICT_ASSIGN(double, x, hi - lo); |
| 2059 | c = (hi - x) - lo; |
| 2060 | } else if (hx < 0x3c900000) { /* when |x|<2**-54, return x */ |
| 2061 | t = huge + x; /* return x with inexact flags when x!=0 */ |
| 2062 | return x - (t - (huge + x)); |
| 2063 | } else { |
| 2064 | k = 0; |
| 2065 | } |
| 2066 | |
| 2067 | /* x is now in primary range */ |
| 2068 | hfx = 0.5 * x; |
| 2069 | hxs = x * hfx; |
| 2070 | r1 = one + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5)))); |
| 2071 | t = 3.0 - r1 * hfx; |
| 2072 | e = hxs * ((r1 - t) / (6.0 - x * t)); |
| 2073 | if (k == 0) { |
| 2074 | return x - (x * e - hxs); /* c is 0 */ |
| 2075 | } else { |
| 2076 | INSERT_WORDS(twopk, 0x3ff00000 + (k << 20), 0); /* 2^k */ |
| 2077 | e = (x * (e - c) - c); |
| 2078 | e -= hxs; |
| 2079 | if (k == -1) return 0.5 * (x - e) - 0.5; |
| 2080 | if (k == 1) { |
| 2081 | if (x < -0.25) |
| 2082 | return -2.0 * (e - (x + 0.5)); |
| 2083 | else |
| 2084 | return one + 2.0 * (x - e); |
| 2085 | } |
| 2086 | if (k <= -2 || k > 56) { /* suffice to return exp(x)-1 */ |
| 2087 | y = one - (e - x); |
| 2088 | // TODO(mvstanton): is this replacement for the hex float |
| 2089 | // sufficient? |
| 2090 | // if (k == 1024) y = y*2.0*0x1p1023; |
| 2091 | if (k == 1024) |
| 2092 | y = y * 2.0 * 8.98846567431158e+307; |
| 2093 | else |
| 2094 | y = y * twopk; |
| 2095 | return y - one; |
| 2096 | } |
| 2097 | t = one; |
| 2098 | if (k < 20) { |
| 2099 | SET_HIGH_WORD(t, 0x3ff00000 - (0x200000 >> k)); /* t=1-2^-k */ |
| 2100 | y = t - (e - x); |
| 2101 | y = y * twopk; |
| 2102 | } else { |
| 2103 | SET_HIGH_WORD(t, ((0x3ff - k) << 20)); /* 2^-k */ |
| 2104 | y = x - (e + t); |
| 2105 | y += one; |
| 2106 | y = y * twopk; |
| 2107 | } |
| 2108 | } |
| 2109 | return y; |
| 2110 | } |
| 2111 | |
| 2112 | double cbrt(double x) { |
| 2113 | static const uint32_t |
| 2114 | B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */ |
| 2115 | B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */ |
| 2116 | |
| 2117 | /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */ |
| 2118 | static const double P0 = 1.87595182427177009643, /* 0x3ffe03e6, 0x0f61e692 */ |
| 2119 | P1 = -1.88497979543377169875, /* 0xbffe28e0, 0x92f02420 */ |
| 2120 | P2 = 1.621429720105354466140, /* 0x3ff9f160, 0x4a49d6c2 */ |
| 2121 | P3 = -0.758397934778766047437, /* 0xbfe844cb, 0xbee751d9 */ |
| 2122 | P4 = 0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */ |
| 2123 | |
| 2124 | int32_t hx; |
| 2125 | union { |
| 2126 | double value; |
| 2127 | uint64_t bits; |
| 2128 | } u; |
| 2129 | double r, s, t = 0.0, w; |
| 2130 | uint32_t sign; |
| 2131 | uint32_t high, low; |
| 2132 | |
| 2133 | EXTRACT_WORDS(hx, low, x); |
| 2134 | sign = hx & 0x80000000; /* sign= sign(x) */ |
| 2135 | hx ^= sign; |
| 2136 | if (hx >= 0x7ff00000) return (x + x); /* cbrt(NaN,INF) is itself */ |
| 2137 | |
| 2138 | /* |
| 2139 | * Rough cbrt to 5 bits: |
| 2140 | * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3) |
| 2141 | * where e is integral and >= 0, m is real and in [0, 1), and "/" and |
| 2142 | * "%" are integer division and modulus with rounding towards minus |
| 2143 | * infinity. The RHS is always >= the LHS and has a maximum relative |
| 2144 | * error of about 1 in 16. Adding a bias of -0.03306235651 to the |
| 2145 | * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE |
| 2146 | * floating point representation, for finite positive normal values, |
| 2147 | * ordinary integer divison of the value in bits magically gives |
| 2148 | * almost exactly the RHS of the above provided we first subtract the |
| 2149 | * exponent bias (1023 for doubles) and later add it back. We do the |
| 2150 | * subtraction virtually to keep e >= 0 so that ordinary integer |
| 2151 | * division rounds towards minus infinity; this is also efficient. |
| 2152 | */ |
| 2153 | if (hx < 0x00100000) { /* zero or subnormal? */ |
| 2154 | if ((hx | low) == 0) return (x); /* cbrt(0) is itself */ |
| 2155 | SET_HIGH_WORD(t, 0x43500000); /* set t= 2**54 */ |
| 2156 | t *= x; |
| 2157 | GET_HIGH_WORD(high, t); |
| 2158 | INSERT_WORDS(t, sign | ((high & 0x7fffffff) / 3 + B2), 0); |
| 2159 | } else { |
| 2160 | INSERT_WORDS(t, sign | (hx / 3 + B1), 0); |
| 2161 | } |
| 2162 | |
| 2163 | /* |
| 2164 | * New cbrt to 23 bits: |
| 2165 | * cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x) |
| 2166 | * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r) |
| 2167 | * to within 2**-23.5 when |r - 1| < 1/10. The rough approximation |
| 2168 | * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this |
| 2169 | * gives us bounds for r = t**3/x. |
| 2170 | * |
| 2171 | * Try to optimize for parallel evaluation as in k_tanf.c. |
| 2172 | */ |
| 2173 | r = (t * t) * (t / x); |
| 2174 | t = t * ((P0 + r * (P1 + r * P2)) + ((r * r) * r) * (P3 + r * P4)); |
| 2175 | |
| 2176 | /* |
| 2177 | * Round t away from zero to 23 bits (sloppily except for ensuring that |
| 2178 | * the result is larger in magnitude than cbrt(x) but not much more than |
| 2179 | * 2 23-bit ulps larger). With rounding towards zero, the error bound |
| 2180 | * would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps |
| 2181 | * in the rounded t, the infinite-precision error in the Newton |
| 2182 | * approximation barely affects third digit in the final error |
| 2183 | * 0.667; the error in the rounded t can be up to about 3 23-bit ulps |
| 2184 | * before the final error is larger than 0.667 ulps. |
| 2185 | */ |
| 2186 | u.value = t; |
| 2187 | u.bits = (u.bits + 0x80000000) & 0xffffffffc0000000ULL; |
| 2188 | t = u.value; |
| 2189 | |
| 2190 | /* one step Newton iteration to 53 bits with error < 0.667 ulps */ |
| 2191 | s = t * t; /* t*t is exact */ |
| 2192 | r = x / s; /* error <= 0.5 ulps; |r| < |t| */ |
| 2193 | w = t + t; /* t+t is exact */ |
| 2194 | r = (r - t) / (w + r); /* r-t is exact; w+r ~= 3*t */ |
| 2195 | t = t + t * r; /* error <= 0.5 + 0.5/3 + epsilon */ |
| 2196 | |
| 2197 | return (t); |
| 2198 | } |
| 2199 | |
| 2200 | /* sin(x) |
| 2201 | * Return sine function of x. |
| 2202 | * |
| 2203 | * kernel function: |
| 2204 | * __kernel_sin ... sine function on [-pi/4,pi/4] |
| 2205 | * __kernel_cos ... cose function on [-pi/4,pi/4] |
| 2206 | * __ieee754_rem_pio2 ... argument reduction routine |
| 2207 | * |
| 2208 | * Method. |
| 2209 | * Let S,C and T denote the sin, cos and tan respectively on |
| 2210 | * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 |
| 2211 | * in [-pi/4 , +pi/4], and let n = k mod 4. |
| 2212 | * We have |
| 2213 | * |
| 2214 | * n sin(x) cos(x) tan(x) |
| 2215 | * ---------------------------------------------------------- |
| 2216 | * 0 S C T |
| 2217 | * 1 C -S -1/T |
| 2218 | * 2 -S -C T |
| 2219 | * 3 -C S -1/T |
| 2220 | * ---------------------------------------------------------- |
| 2221 | * |
| 2222 | * Special cases: |
| 2223 | * Let trig be any of sin, cos, or tan. |
| 2224 | * trig(+-INF) is NaN, with signals; |
| 2225 | * trig(NaN) is that NaN; |
| 2226 | * |
| 2227 | * Accuracy: |
| 2228 | * TRIG(x) returns trig(x) nearly rounded |
| 2229 | */ |
| 2230 | double sin(double x) { |
| 2231 | double y[2], z = 0.0; |
| 2232 | int32_t n, ix; |
| 2233 | |
| 2234 | /* High word of x. */ |
| 2235 | GET_HIGH_WORD(ix, x); |
| 2236 | |
| 2237 | /* |x| ~< pi/4 */ |
| 2238 | ix &= 0x7fffffff; |
| 2239 | if (ix <= 0x3fe921fb) { |
| 2240 | return __kernel_sin(x, z, 0); |
| 2241 | } else if (ix >= 0x7ff00000) { |
| 2242 | /* sin(Inf or NaN) is NaN */ |
| 2243 | return x - x; |
| 2244 | } else { |
| 2245 | /* argument reduction needed */ |
| 2246 | n = __ieee754_rem_pio2(x, y); |
| 2247 | switch (n & 3) { |
| 2248 | case 0: |
| 2249 | return __kernel_sin(y[0], y[1], 1); |
| 2250 | case 1: |
| 2251 | return __kernel_cos(y[0], y[1]); |
| 2252 | case 2: |
| 2253 | return -__kernel_sin(y[0], y[1], 1); |
| 2254 | default: |
| 2255 | return -__kernel_cos(y[0], y[1]); |
| 2256 | } |
| 2257 | } |
| 2258 | } |
| 2259 | |
| 2260 | /* tan(x) |
| 2261 | * Return tangent function of x. |
| 2262 | * |
| 2263 | * kernel function: |
| 2264 | * __kernel_tan ... tangent function on [-pi/4,pi/4] |
| 2265 | * __ieee754_rem_pio2 ... argument reduction routine |
| 2266 | * |
| 2267 | * Method. |
| 2268 | * Let S,C and T denote the sin, cos and tan respectively on |
| 2269 | * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 |
| 2270 | * in [-pi/4 , +pi/4], and let n = k mod 4. |
| 2271 | * We have |
| 2272 | * |
| 2273 | * n sin(x) cos(x) tan(x) |
| 2274 | * ---------------------------------------------------------- |
| 2275 | * 0 S C T |
| 2276 | * 1 C -S -1/T |
| 2277 | * 2 -S -C T |
| 2278 | * 3 -C S -1/T |
| 2279 | * ---------------------------------------------------------- |
| 2280 | * |
| 2281 | * Special cases: |
| 2282 | * Let trig be any of sin, cos, or tan. |
| 2283 | * trig(+-INF) is NaN, with signals; |
| 2284 | * trig(NaN) is that NaN; |
| 2285 | * |
| 2286 | * Accuracy: |
| 2287 | * TRIG(x) returns trig(x) nearly rounded |
| 2288 | */ |
| 2289 | double tan(double x) { |
| 2290 | double y[2], z = 0.0; |
| 2291 | int32_t n, ix; |
| 2292 | |
| 2293 | /* High word of x. */ |
| 2294 | GET_HIGH_WORD(ix, x); |
| 2295 | |
| 2296 | /* |x| ~< pi/4 */ |
| 2297 | ix &= 0x7fffffff; |
| 2298 | if (ix <= 0x3fe921fb) { |
| 2299 | return __kernel_tan(x, z, 1); |
| 2300 | } else if (ix >= 0x7ff00000) { |
| 2301 | /* tan(Inf or NaN) is NaN */ |
| 2302 | return x - x; /* NaN */ |
| 2303 | } else { |
| 2304 | /* argument reduction needed */ |
| 2305 | n = __ieee754_rem_pio2(x, y); |
| 2306 | /* 1 -> n even, -1 -> n odd */ |
| 2307 | return __kernel_tan(y[0], y[1], 1 - ((n & 1) << 1)); |
| 2308 | } |
| 2309 | } |
| 2310 | |
| 2311 | } // namespace ieee754 |
| 2312 | } // namespace base |
| 2313 | } // namespace v8 |