Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 1 | // The following is adapted from fdlibm (http://www.netlib.org/fdlibm), |
| 2 | // |
| 3 | // ==================================================== |
| 4 | // Copyright (C) 1993-2004 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 2014 the V8 project authors. All rights reserved. |
| 15 | // |
| 16 | // The following is a straightforward translation of fdlibm routines |
| 17 | // by Raymond Toy (rtoy@google.com). |
| 18 | |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 19 | // rempio2result is used as a container for return values of %RemPiO2. It is |
| 20 | // initialized to a two-element Float64Array during genesis. |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 21 | |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 22 | (function(global, utils) { |
| 23 | |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 24 | "use strict"; |
| 25 | |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 26 | %CheckIsBootstrapping(); |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 27 | |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 28 | // ------------------------------------------------------------------- |
| 29 | // Imports |
| 30 | |
| 31 | var GlobalFloat64Array = global.Float64Array; |
| 32 | var GlobalMath = global.Math; |
| 33 | var MathAbs; |
| 34 | var MathExp; |
| 35 | var NaN = %GetRootNaN(); |
| 36 | var rempio2result; |
| 37 | |
| 38 | utils.Import(function(from) { |
| 39 | MathAbs = from.MathAbs; |
| 40 | MathExp = from.MathExp; |
| 41 | }); |
| 42 | |
| 43 | utils.CreateDoubleResultArray = function(global) { |
| 44 | rempio2result = new GlobalFloat64Array(2); |
| 45 | }; |
| 46 | |
| 47 | // ------------------------------------------------------------------- |
| 48 | |
| 49 | define INVPIO2 = 6.36619772367581382433e-01; |
| 50 | define PIO2_1 = 1.57079632673412561417; |
| 51 | define PIO2_1T = 6.07710050650619224932e-11; |
| 52 | define PIO2_2 = 6.07710050630396597660e-11; |
| 53 | define PIO2_2T = 2.02226624879595063154e-21; |
| 54 | define PIO2_3 = 2.02226624871116645580e-21; |
| 55 | define PIO2_3T = 8.47842766036889956997e-32; |
| 56 | define PIO4 = 7.85398163397448278999e-01; |
| 57 | define PIO4LO = 3.06161699786838301793e-17; |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 58 | |
| 59 | // Compute k and r such that x - k*pi/2 = r where |r| < pi/4. For |
| 60 | // precision, r is returned as two values y0 and y1 such that r = y0 + y1 |
| 61 | // to more than double precision. |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 62 | |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 63 | macro REMPIO2(X) |
| 64 | var n, y0, y1; |
| 65 | var hx = %_DoubleHi(X); |
| 66 | var ix = hx & 0x7fffffff; |
| 67 | |
| 68 | if (ix < 0x4002d97c) { |
| 69 | // |X| ~< 3*pi/4, special case with n = +/- 1 |
| 70 | if (hx > 0) { |
| 71 | var z = X - PIO2_1; |
| 72 | if (ix != 0x3ff921fb) { |
| 73 | // 33+53 bit pi is good enough |
| 74 | y0 = z - PIO2_1T; |
| 75 | y1 = (z - y0) - PIO2_1T; |
| 76 | } else { |
| 77 | // near pi/2, use 33+33+53 bit pi |
| 78 | z -= PIO2_2; |
| 79 | y0 = z - PIO2_2T; |
| 80 | y1 = (z - y0) - PIO2_2T; |
| 81 | } |
| 82 | n = 1; |
| 83 | } else { |
| 84 | // Negative X |
| 85 | var z = X + PIO2_1; |
| 86 | if (ix != 0x3ff921fb) { |
| 87 | // 33+53 bit pi is good enough |
| 88 | y0 = z + PIO2_1T; |
| 89 | y1 = (z - y0) + PIO2_1T; |
| 90 | } else { |
| 91 | // near pi/2, use 33+33+53 bit pi |
| 92 | z += PIO2_2; |
| 93 | y0 = z + PIO2_2T; |
| 94 | y1 = (z - y0) + PIO2_2T; |
| 95 | } |
| 96 | n = -1; |
| 97 | } |
| 98 | } else if (ix <= 0x413921fb) { |
| 99 | // |X| ~<= 2^19*(pi/2), medium size |
| 100 | var t = MathAbs(X); |
| 101 | n = (t * INVPIO2 + 0.5) | 0; |
| 102 | var r = t - n * PIO2_1; |
| 103 | var w = n * PIO2_1T; |
| 104 | // First round good to 85 bit |
| 105 | y0 = r - w; |
| 106 | if (ix - (%_DoubleHi(y0) & 0x7ff00000) > 0x1000000) { |
| 107 | // 2nd iteration needed, good to 118 |
| 108 | t = r; |
| 109 | w = n * PIO2_2; |
| 110 | r = t - w; |
| 111 | w = n * PIO2_2T - ((t - r) - w); |
| 112 | y0 = r - w; |
| 113 | if (ix - (%_DoubleHi(y0) & 0x7ff00000) > 0x3100000) { |
| 114 | // 3rd iteration needed. 151 bits accuracy |
| 115 | t = r; |
| 116 | w = n * PIO2_3; |
| 117 | r = t - w; |
| 118 | w = n * PIO2_3T - ((t - r) - w); |
| 119 | y0 = r - w; |
| 120 | } |
| 121 | } |
| 122 | y1 = (r - y0) - w; |
| 123 | if (hx < 0) { |
| 124 | n = -n; |
| 125 | y0 = -y0; |
| 126 | y1 = -y1; |
| 127 | } |
| 128 | } else { |
| 129 | // Need to do full Payne-Hanek reduction here. |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 130 | n = %RemPiO2(X, rempio2result); |
| 131 | y0 = rempio2result[0]; |
| 132 | y1 = rempio2result[1]; |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 133 | } |
| 134 | endmacro |
| 135 | |
| 136 | |
| 137 | // __kernel_sin(X, Y, IY) |
| 138 | // kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 |
| 139 | // Input X is assumed to be bounded by ~pi/4 in magnitude. |
| 140 | // Input Y is the tail of X so that x = X + Y. |
| 141 | // |
| 142 | // Algorithm |
| 143 | // 1. Since ieee_sin(-x) = -ieee_sin(x), we need only to consider positive x. |
| 144 | // 2. ieee_sin(x) is approximated by a polynomial of degree 13 on |
| 145 | // [0,pi/4] |
| 146 | // 3 13 |
| 147 | // sin(x) ~ x + S1*x + ... + S6*x |
| 148 | // where |
| 149 | // |
| 150 | // |ieee_sin(x) 2 4 6 8 10 12 | -58 |
| 151 | // |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 |
| 152 | // | x | |
| 153 | // |
| 154 | // 3. ieee_sin(X+Y) = ieee_sin(X) + sin'(X')*Y |
| 155 | // ~ ieee_sin(X) + (1-X*X/2)*Y |
| 156 | // For better accuracy, let |
| 157 | // 3 2 2 2 2 |
| 158 | // r = X *(S2+X *(S3+X *(S4+X *(S5+X *S6)))) |
| 159 | // then 3 2 |
| 160 | // sin(x) = X + (S1*X + (X *(r-Y/2)+Y)) |
| 161 | // |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 162 | define S1 = -1.66666666666666324348e-01; |
| 163 | define S2 = 8.33333333332248946124e-03; |
| 164 | define S3 = -1.98412698298579493134e-04; |
| 165 | define S4 = 2.75573137070700676789e-06; |
| 166 | define S5 = -2.50507602534068634195e-08; |
| 167 | define S6 = 1.58969099521155010221e-10; |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 168 | |
| 169 | macro RETURN_KERNELSIN(X, Y, SIGN) |
| 170 | var z = X * X; |
| 171 | var v = z * X; |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 172 | var r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6))); |
| 173 | return (X - ((z * (0.5 * Y - v * r) - Y) - v * S1)) SIGN; |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 174 | endmacro |
| 175 | |
| 176 | // __kernel_cos(X, Y) |
| 177 | // kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 |
| 178 | // Input X is assumed to be bounded by ~pi/4 in magnitude. |
| 179 | // Input Y is the tail of X so that x = X + Y. |
| 180 | // |
| 181 | // Algorithm |
| 182 | // 1. Since ieee_cos(-x) = ieee_cos(x), we need only to consider positive x. |
| 183 | // 2. ieee_cos(x) is approximated by a polynomial of degree 14 on |
| 184 | // [0,pi/4] |
| 185 | // 4 14 |
| 186 | // cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x |
| 187 | // where the remez error is |
| 188 | // |
| 189 | // | 2 4 6 8 10 12 14 | -58 |
| 190 | // |ieee_cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 |
| 191 | // | | |
| 192 | // |
| 193 | // 4 6 8 10 12 14 |
| 194 | // 3. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then |
| 195 | // ieee_cos(x) = 1 - x*x/2 + r |
| 196 | // since ieee_cos(X+Y) ~ ieee_cos(X) - ieee_sin(X)*Y |
| 197 | // ~ ieee_cos(X) - X*Y, |
| 198 | // a correction term is necessary in ieee_cos(x) and hence |
| 199 | // cos(X+Y) = 1 - (X*X/2 - (r - X*Y)) |
| 200 | // For better accuracy when x > 0.3, let qx = |x|/4 with |
| 201 | // the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. |
| 202 | // Then |
| 203 | // cos(X+Y) = (1-qx) - ((X*X/2-qx) - (r-X*Y)). |
| 204 | // Note that 1-qx and (X*X/2-qx) is EXACT here, and the |
| 205 | // magnitude of the latter is at least a quarter of X*X/2, |
| 206 | // thus, reducing the rounding error in the subtraction. |
| 207 | // |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 208 | define C1 = 4.16666666666666019037e-02; |
| 209 | define C2 = -1.38888888888741095749e-03; |
| 210 | define C3 = 2.48015872894767294178e-05; |
| 211 | define C4 = -2.75573143513906633035e-07; |
| 212 | define C5 = 2.08757232129817482790e-09; |
| 213 | define C6 = -1.13596475577881948265e-11; |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 214 | |
| 215 | macro RETURN_KERNELCOS(X, Y, SIGN) |
| 216 | var ix = %_DoubleHi(X) & 0x7fffffff; |
| 217 | var z = X * X; |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 218 | var r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6))))); |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 219 | if (ix < 0x3fd33333) { // |x| ~< 0.3 |
| 220 | return (1 - (0.5 * z - (z * r - X * Y))) SIGN; |
| 221 | } else { |
| 222 | var qx; |
| 223 | if (ix > 0x3fe90000) { // |x| > 0.78125 |
| 224 | qx = 0.28125; |
| 225 | } else { |
| 226 | qx = %_ConstructDouble(%_DoubleHi(0.25 * X), 0); |
| 227 | } |
| 228 | var hz = 0.5 * z - qx; |
| 229 | return (1 - qx - (hz - (z * r - X * Y))) SIGN; |
| 230 | } |
| 231 | endmacro |
| 232 | |
| 233 | |
| 234 | // kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 |
| 235 | // Input x is assumed to be bounded by ~pi/4 in magnitude. |
| 236 | // Input y is the tail of x. |
| 237 | // Input k indicates whether ieee_tan (if k = 1) or -1/tan (if k = -1) |
| 238 | // is returned. |
| 239 | // |
| 240 | // Algorithm |
| 241 | // 1. Since ieee_tan(-x) = -ieee_tan(x), we need only to consider positive x. |
| 242 | // 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. |
| 243 | // 3. ieee_tan(x) is approximated by a odd polynomial of degree 27 on |
| 244 | // [0,0.67434] |
| 245 | // 3 27 |
| 246 | // tan(x) ~ x + T1*x + ... + T13*x |
| 247 | // where |
| 248 | // |
| 249 | // |ieee_tan(x) 2 4 26 | -59.2 |
| 250 | // |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 |
| 251 | // | x | |
| 252 | // |
| 253 | // Note: ieee_tan(x+y) = ieee_tan(x) + tan'(x)*y |
| 254 | // ~ ieee_tan(x) + (1+x*x)*y |
| 255 | // Therefore, for better accuracy in computing ieee_tan(x+y), let |
| 256 | // 3 2 2 2 2 |
| 257 | // r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) |
| 258 | // then |
| 259 | // 3 2 |
| 260 | // tan(x+y) = x + (T1*x + (x *(r+y)+y)) |
| 261 | // |
| 262 | // 4. For x in [0.67434,pi/4], let y = pi/4 - x, then |
| 263 | // tan(x) = ieee_tan(pi/4-y) = (1-ieee_tan(y))/(1+ieee_tan(y)) |
| 264 | // = 1 - 2*(ieee_tan(y) - (ieee_tan(y)^2)/(1+ieee_tan(y))) |
| 265 | // |
| 266 | // Set returnTan to 1 for tan; -1 for cot. Anything else is illegal |
| 267 | // and will cause incorrect results. |
| 268 | // |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 269 | define T00 = 3.33333333333334091986e-01; |
| 270 | define T01 = 1.33333333333201242699e-01; |
| 271 | define T02 = 5.39682539762260521377e-02; |
| 272 | define T03 = 2.18694882948595424599e-02; |
| 273 | define T04 = 8.86323982359930005737e-03; |
| 274 | define T05 = 3.59207910759131235356e-03; |
| 275 | define T06 = 1.45620945432529025516e-03; |
| 276 | define T07 = 5.88041240820264096874e-04; |
| 277 | define T08 = 2.46463134818469906812e-04; |
| 278 | define T09 = 7.81794442939557092300e-05; |
| 279 | define T10 = 7.14072491382608190305e-05; |
| 280 | define T11 = -1.85586374855275456654e-05; |
| 281 | define T12 = 2.59073051863633712884e-05; |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 282 | |
| 283 | function KernelTan(x, y, returnTan) { |
| 284 | var z; |
| 285 | var w; |
| 286 | var hx = %_DoubleHi(x); |
| 287 | var ix = hx & 0x7fffffff; |
| 288 | |
| 289 | if (ix < 0x3e300000) { // |x| < 2^-28 |
| 290 | if (((ix | %_DoubleLo(x)) | (returnTan + 1)) == 0) { |
| 291 | // x == 0 && returnTan = -1 |
| 292 | return 1 / MathAbs(x); |
| 293 | } else { |
| 294 | if (returnTan == 1) { |
| 295 | return x; |
| 296 | } else { |
| 297 | // Compute -1/(x + y) carefully |
| 298 | var w = x + y; |
| 299 | var z = %_ConstructDouble(%_DoubleHi(w), 0); |
| 300 | var v = y - (z - x); |
| 301 | var a = -1 / w; |
| 302 | var t = %_ConstructDouble(%_DoubleHi(a), 0); |
| 303 | var s = 1 + t * z; |
| 304 | return t + a * (s + t * v); |
| 305 | } |
| 306 | } |
| 307 | } |
| 308 | if (ix >= 0x3fe59428) { // |x| > .6744 |
| 309 | if (x < 0) { |
| 310 | x = -x; |
| 311 | y = -y; |
| 312 | } |
| 313 | z = PIO4 - x; |
| 314 | w = PIO4LO - y; |
| 315 | x = z + w; |
| 316 | y = 0; |
| 317 | } |
| 318 | z = x * x; |
| 319 | w = z * z; |
| 320 | |
| 321 | // Break x^5 * (T1 + x^2*T2 + ...) into |
| 322 | // x^5 * (T1 + x^4*T3 + ... + x^20*T11) + |
| 323 | // x^5 * (x^2 * (T2 + x^4*T4 + ... + x^22*T12)) |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 324 | var r = T01 + w * (T03 + w * (T05 + |
| 325 | w * (T07 + w * (T09 + w * T11)))); |
| 326 | var v = z * (T02 + w * (T04 + w * (T06 + |
| 327 | w * (T08 + w * (T10 + w * T12))))); |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 328 | var s = z * x; |
| 329 | r = y + z * (s * (r + v) + y); |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 330 | r = r + T00 * s; |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 331 | w = x + r; |
| 332 | if (ix >= 0x3fe59428) { |
| 333 | return (1 - ((hx >> 30) & 2)) * |
| 334 | (returnTan - 2.0 * (x - (w * w / (w + returnTan) - r))); |
| 335 | } |
| 336 | if (returnTan == 1) { |
| 337 | return w; |
| 338 | } else { |
| 339 | z = %_ConstructDouble(%_DoubleHi(w), 0); |
| 340 | v = r - (z - x); |
| 341 | var a = -1 / w; |
| 342 | var t = %_ConstructDouble(%_DoubleHi(a), 0); |
| 343 | s = 1 + t * z; |
| 344 | return t + a * (s + t * v); |
| 345 | } |
| 346 | } |
| 347 | |
| 348 | function MathSinSlow(x) { |
| 349 | REMPIO2(x); |
| 350 | var sign = 1 - (n & 2); |
| 351 | if (n & 1) { |
| 352 | RETURN_KERNELCOS(y0, y1, * sign); |
| 353 | } else { |
| 354 | RETURN_KERNELSIN(y0, y1, * sign); |
| 355 | } |
| 356 | } |
| 357 | |
| 358 | function MathCosSlow(x) { |
| 359 | REMPIO2(x); |
| 360 | if (n & 1) { |
| 361 | var sign = (n & 2) - 1; |
| 362 | RETURN_KERNELSIN(y0, y1, * sign); |
| 363 | } else { |
| 364 | var sign = 1 - (n & 2); |
| 365 | RETURN_KERNELCOS(y0, y1, * sign); |
| 366 | } |
| 367 | } |
| 368 | |
| 369 | // ECMA 262 - 15.8.2.16 |
| 370 | function MathSin(x) { |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 371 | x = +x; // Convert to number. |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 372 | if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) { |
| 373 | // |x| < pi/4, approximately. No reduction needed. |
| 374 | RETURN_KERNELSIN(x, 0, /* empty */); |
| 375 | } |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 376 | return +MathSinSlow(x); |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 377 | } |
| 378 | |
| 379 | // ECMA 262 - 15.8.2.7 |
| 380 | function MathCos(x) { |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 381 | x = +x; // Convert to number. |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 382 | if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) { |
| 383 | // |x| < pi/4, approximately. No reduction needed. |
| 384 | RETURN_KERNELCOS(x, 0, /* empty */); |
| 385 | } |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 386 | return +MathCosSlow(x); |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 387 | } |
| 388 | |
| 389 | // ECMA 262 - 15.8.2.18 |
| 390 | function MathTan(x) { |
| 391 | x = x * 1; // Convert to number. |
| 392 | if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) { |
| 393 | // |x| < pi/4, approximately. No reduction needed. |
| 394 | return KernelTan(x, 0, 1); |
| 395 | } |
| 396 | REMPIO2(x); |
| 397 | return KernelTan(y0, y1, (n & 1) ? -1 : 1); |
| 398 | } |
| 399 | |
| 400 | // ES6 draft 09-27-13, section 20.2.2.20. |
| 401 | // Math.log1p |
| 402 | // |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 403 | // Method : |
| 404 | // 1. Argument Reduction: find k and f such that |
| 405 | // 1+x = 2^k * (1+f), |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 406 | // where sqrt(2)/2 < 1+f < sqrt(2) . |
| 407 | // |
| 408 | // Note. If k=0, then f=x is exact. However, if k!=0, then f |
| 409 | // may not be representable exactly. In that case, a correction |
| 410 | // term is need. Let u=1+x rounded. Let c = (1+x)-u, then |
| 411 | // log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), |
| 412 | // and add back the correction term c/u. |
| 413 | // (Note: when x > 2**53, one can simply return log(x)) |
| 414 | // |
| 415 | // 2. Approximation of log1p(f). |
| 416 | // Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) |
| 417 | // = 2s + 2/3 s**3 + 2/5 s**5 + ....., |
| 418 | // = 2s + s*R |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 419 | // We use a special Reme algorithm on [0,0.1716] to generate |
| 420 | // a polynomial of degree 14 to approximate R The maximum error |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 421 | // of this polynomial approximation is bounded by 2**-58.45. In |
| 422 | // other words, |
| 423 | // 2 4 6 8 10 12 14 |
| 424 | // R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s |
| 425 | // (the values of Lp1 to Lp7 are listed in the program) |
| 426 | // and |
| 427 | // | 2 14 | -58.45 |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 428 | // | Lp1*s +...+Lp7*s - R(z) | <= 2 |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 429 | // | | |
| 430 | // Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. |
| 431 | // In order to guarantee error in log below 1ulp, we compute log |
| 432 | // by |
| 433 | // log1p(f) = f - (hfsq - s*(hfsq+R)). |
| 434 | // |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 435 | // 3. Finally, log1p(x) = k*ln2 + log1p(f). |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 436 | // = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 437 | // Here ln2 is split into two floating point number: |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 438 | // ln2_hi + ln2_lo, |
| 439 | // where n*ln2_hi is always exact for |n| < 2000. |
| 440 | // |
| 441 | // Special cases: |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 442 | // log1p(x) is NaN with signal if x < -1 (including -INF) ; |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 443 | // log1p(+INF) is +INF; log1p(-1) is -INF with signal; |
| 444 | // log1p(NaN) is that NaN with no signal. |
| 445 | // |
| 446 | // Accuracy: |
| 447 | // according to an error analysis, the error is always less than |
| 448 | // 1 ulp (unit in the last place). |
| 449 | // |
| 450 | // Constants: |
| 451 | // Constants are found in fdlibm.cc. We assume the C++ compiler to convert |
| 452 | // from decimal to binary accurately enough to produce the intended values. |
| 453 | // |
| 454 | // Note: Assuming log() return accurate answer, the following |
| 455 | // algorithm can be used to compute log1p(x) to within a few ULP: |
| 456 | // |
| 457 | // u = 1+x; |
| 458 | // if (u==1.0) return x ; else |
| 459 | // return log(u)*(x/(u-1.0)); |
| 460 | // |
| 461 | // See HP-15C Advanced Functions Handbook, p.193. |
| 462 | // |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 463 | define LN2_HI = 6.93147180369123816490e-01; |
| 464 | define LN2_LO = 1.90821492927058770002e-10; |
| 465 | define TWO_THIRD = 6.666666666666666666e-01; |
| 466 | define LP1 = 6.666666666666735130e-01; |
| 467 | define LP2 = 3.999999999940941908e-01; |
| 468 | define LP3 = 2.857142874366239149e-01; |
| 469 | define LP4 = 2.222219843214978396e-01; |
| 470 | define LP5 = 1.818357216161805012e-01; |
| 471 | define LP6 = 1.531383769920937332e-01; |
| 472 | define LP7 = 1.479819860511658591e-01; |
| 473 | |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 474 | // 2^54 |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 475 | define TWO54 = 18014398509481984; |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 476 | |
| 477 | function MathLog1p(x) { |
| 478 | x = x * 1; // Convert to number. |
| 479 | var hx = %_DoubleHi(x); |
| 480 | var ax = hx & 0x7fffffff; |
| 481 | var k = 1; |
| 482 | var f = x; |
| 483 | var hu = 1; |
| 484 | var c = 0; |
| 485 | var u = x; |
| 486 | |
| 487 | if (hx < 0x3fda827a) { |
| 488 | // x < 0.41422 |
| 489 | if (ax >= 0x3ff00000) { // |x| >= 1 |
| 490 | if (x === -1) { |
| 491 | return -INFINITY; // log1p(-1) = -inf |
| 492 | } else { |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 493 | return NaN; // log1p(x<-1) = NaN |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 494 | } |
| 495 | } else if (ax < 0x3c900000) { |
| 496 | // For |x| < 2^-54 we can return x. |
| 497 | return x; |
| 498 | } else if (ax < 0x3e200000) { |
| 499 | // For |x| < 2^-29 we can use a simple two-term Taylor series. |
| 500 | return x - x * x * 0.5; |
| 501 | } |
| 502 | |
| 503 | if ((hx > 0) || (hx <= -0x402D413D)) { // (int) 0xbfd2bec3 = -0x402d413d |
| 504 | // -.2929 < x < 0.41422 |
| 505 | k = 0; |
| 506 | } |
| 507 | } |
| 508 | |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 509 | // Handle Infinity and NaN |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 510 | if (hx >= 0x7ff00000) return x; |
| 511 | |
| 512 | if (k !== 0) { |
| 513 | if (hx < 0x43400000) { |
| 514 | // x < 2^53 |
| 515 | u = 1 + x; |
| 516 | hu = %_DoubleHi(u); |
| 517 | k = (hu >> 20) - 1023; |
| 518 | c = (k > 0) ? 1 - (u - x) : x - (u - 1); |
| 519 | c = c / u; |
| 520 | } else { |
| 521 | hu = %_DoubleHi(u); |
| 522 | k = (hu >> 20) - 1023; |
| 523 | } |
| 524 | hu = hu & 0xfffff; |
| 525 | if (hu < 0x6a09e) { |
| 526 | u = %_ConstructDouble(hu | 0x3ff00000, %_DoubleLo(u)); // Normalize u. |
| 527 | } else { |
| 528 | ++k; |
| 529 | u = %_ConstructDouble(hu | 0x3fe00000, %_DoubleLo(u)); // Normalize u/2. |
| 530 | hu = (0x00100000 - hu) >> 2; |
| 531 | } |
| 532 | f = u - 1; |
| 533 | } |
| 534 | |
| 535 | var hfsq = 0.5 * f * f; |
| 536 | if (hu === 0) { |
| 537 | // |f| < 2^-20; |
| 538 | if (f === 0) { |
| 539 | if (k === 0) { |
| 540 | return 0.0; |
| 541 | } else { |
| 542 | return k * LN2_HI + (c + k * LN2_LO); |
| 543 | } |
| 544 | } |
| 545 | var R = hfsq * (1 - TWO_THIRD * f); |
| 546 | if (k === 0) { |
| 547 | return f - R; |
| 548 | } else { |
| 549 | return k * LN2_HI - ((R - (k * LN2_LO + c)) - f); |
| 550 | } |
| 551 | } |
| 552 | |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 553 | var s = f / (2 + f); |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 554 | var z = s * s; |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 555 | var R = z * (LP1 + z * (LP2 + z * (LP3 + z * (LP4 + |
| 556 | z * (LP5 + z * (LP6 + z * LP7)))))); |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 557 | if (k === 0) { |
| 558 | return f - (hfsq - s * (hfsq + R)); |
| 559 | } else { |
| 560 | return k * LN2_HI - ((hfsq - (s * (hfsq + R) + (k * LN2_LO + c))) - f); |
| 561 | } |
| 562 | } |
| 563 | |
| 564 | // ES6 draft 09-27-13, section 20.2.2.14. |
| 565 | // Math.expm1 |
| 566 | // Returns exp(x)-1, the exponential of x minus 1. |
| 567 | // |
| 568 | // Method |
| 569 | // 1. Argument reduction: |
| 570 | // Given x, find r and integer k such that |
| 571 | // |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 572 | // x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 573 | // |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 574 | // Here a correction term c will be computed to compensate |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 575 | // the error in r when rounded to a floating-point number. |
| 576 | // |
| 577 | // 2. Approximating expm1(r) by a special rational function on |
| 578 | // the interval [0,0.34658]: |
| 579 | // Since |
| 580 | // r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... |
| 581 | // we define R1(r*r) by |
| 582 | // r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) |
| 583 | // That is, |
| 584 | // R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) |
| 585 | // = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) |
| 586 | // = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 587 | // We use a special Remes algorithm on [0,0.347] to generate |
| 588 | // a polynomial of degree 5 in r*r to approximate R1. The |
| 589 | // maximum error of this polynomial approximation is bounded |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 590 | // by 2**-61. In other words, |
| 591 | // R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 |
| 592 | // where Q1 = -1.6666666666666567384E-2, |
| 593 | // Q2 = 3.9682539681370365873E-4, |
| 594 | // Q3 = -9.9206344733435987357E-6, |
| 595 | // Q4 = 2.5051361420808517002E-7, |
| 596 | // Q5 = -6.2843505682382617102E-9; |
| 597 | // (where z=r*r, and the values of Q1 to Q5 are listed below) |
| 598 | // with error bounded by |
| 599 | // | 5 | -61 |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 600 | // | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 601 | // | | |
| 602 | // |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 603 | // expm1(r) = exp(r)-1 is then computed by the following |
| 604 | // specific way which minimize the accumulation rounding error: |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 605 | // 2 3 |
| 606 | // r r [ 3 - (R1 + R1*r/2) ] |
| 607 | // expm1(r) = r + --- + --- * [--------------------] |
| 608 | // 2 2 [ 6 - r*(3 - R1*r/2) ] |
| 609 | // |
| 610 | // To compensate the error in the argument reduction, we use |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 611 | // expm1(r+c) = expm1(r) + c + expm1(r)*c |
| 612 | // ~ expm1(r) + c + r*c |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 613 | // Thus c+r*c will be added in as the correction terms for |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 614 | // expm1(r+c). Now rearrange the term to avoid optimization |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 615 | // screw up: |
| 616 | // ( 2 2 ) |
| 617 | // ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) |
| 618 | // expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) |
| 619 | // ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) |
| 620 | // ( ) |
| 621 | // |
| 622 | // = r - E |
| 623 | // 3. Scale back to obtain expm1(x): |
| 624 | // From step 1, we have |
| 625 | // expm1(x) = either 2^k*[expm1(r)+1] - 1 |
| 626 | // = or 2^k*[expm1(r) + (1-2^-k)] |
| 627 | // 4. Implementation notes: |
| 628 | // (A). To save one multiplication, we scale the coefficient Qi |
| 629 | // to Qi*2^i, and replace z by (x^2)/2. |
| 630 | // (B). To achieve maximum accuracy, we compute expm1(x) by |
| 631 | // (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) |
| 632 | // (ii) if k=0, return r-E |
| 633 | // (iii) if k=-1, return 0.5*(r-E)-0.5 |
| 634 | // (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) |
| 635 | // else return 1.0+2.0*(r-E); |
| 636 | // (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) |
| 637 | // (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 638 | // (vii) return 2^k(1-((E+2^-k)-r)) |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 639 | // |
| 640 | // Special cases: |
| 641 | // expm1(INF) is INF, expm1(NaN) is NaN; |
| 642 | // expm1(-INF) is -1, and |
| 643 | // for finite argument, only expm1(0)=0 is exact. |
| 644 | // |
| 645 | // Accuracy: |
| 646 | // according to an error analysis, the error is always less than |
| 647 | // 1 ulp (unit in the last place). |
| 648 | // |
| 649 | // Misc. info. |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 650 | // For IEEE double |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 651 | // if x > 7.09782712893383973096e+02 then expm1(x) overflow |
| 652 | // |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 653 | define KEXPM1_OVERFLOW = 7.09782712893383973096e+02; |
| 654 | define INVLN2 = 1.44269504088896338700; |
| 655 | define EXPM1_1 = -3.33333333333331316428e-02; |
| 656 | define EXPM1_2 = 1.58730158725481460165e-03; |
| 657 | define EXPM1_3 = -7.93650757867487942473e-05; |
| 658 | define EXPM1_4 = 4.00821782732936239552e-06; |
| 659 | define EXPM1_5 = -2.01099218183624371326e-07; |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 660 | |
| 661 | function MathExpm1(x) { |
| 662 | x = x * 1; // Convert to number. |
| 663 | var y; |
| 664 | var hi; |
| 665 | var lo; |
| 666 | var k; |
| 667 | var t; |
| 668 | var c; |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 669 | |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 670 | var hx = %_DoubleHi(x); |
| 671 | var xsb = hx & 0x80000000; // Sign bit of x |
| 672 | var y = (xsb === 0) ? x : -x; // y = |x| |
| 673 | hx &= 0x7fffffff; // High word of |x| |
| 674 | |
| 675 | // Filter out huge and non-finite argument |
| 676 | if (hx >= 0x4043687a) { // if |x| ~=> 56 * ln2 |
| 677 | if (hx >= 0x40862e42) { // if |x| >= 709.78 |
| 678 | if (hx >= 0x7ff00000) { |
| 679 | // expm1(inf) = inf; expm1(-inf) = -1; expm1(nan) = nan; |
| 680 | return (x === -INFINITY) ? -1 : x; |
| 681 | } |
| 682 | if (x > KEXPM1_OVERFLOW) return INFINITY; // Overflow |
| 683 | } |
| 684 | if (xsb != 0) return -1; // x < -56 * ln2, return -1. |
| 685 | } |
| 686 | |
| 687 | // Argument reduction |
| 688 | if (hx > 0x3fd62e42) { // if |x| > 0.5 * ln2 |
| 689 | if (hx < 0x3ff0a2b2) { // and |x| < 1.5 * ln2 |
| 690 | if (xsb === 0) { |
| 691 | hi = x - LN2_HI; |
| 692 | lo = LN2_LO; |
| 693 | k = 1; |
| 694 | } else { |
| 695 | hi = x + LN2_HI; |
| 696 | lo = -LN2_LO; |
| 697 | k = -1; |
| 698 | } |
| 699 | } else { |
| 700 | k = (INVLN2 * x + ((xsb === 0) ? 0.5 : -0.5)) | 0; |
| 701 | t = k; |
| 702 | // t * ln2_hi is exact here. |
| 703 | hi = x - t * LN2_HI; |
| 704 | lo = t * LN2_LO; |
| 705 | } |
| 706 | x = hi - lo; |
| 707 | c = (hi - x) - lo; |
| 708 | } else if (hx < 0x3c900000) { |
| 709 | // When |x| < 2^-54, we can return x. |
| 710 | return x; |
| 711 | } else { |
| 712 | // Fall through. |
| 713 | k = 0; |
| 714 | } |
| 715 | |
| 716 | // x is now in primary range |
| 717 | var hfx = 0.5 * x; |
| 718 | var hxs = x * hfx; |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 719 | var r1 = 1 + hxs * (EXPM1_1 + hxs * (EXPM1_2 + hxs * |
| 720 | (EXPM1_3 + hxs * (EXPM1_4 + hxs * EXPM1_5)))); |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 721 | t = 3 - r1 * hfx; |
| 722 | var e = hxs * ((r1 - t) / (6 - x * t)); |
| 723 | if (k === 0) { // c is 0 |
| 724 | return x - (x*e - hxs); |
| 725 | } else { |
| 726 | e = (x * (e - c) - c); |
| 727 | e -= hxs; |
| 728 | if (k === -1) return 0.5 * (x - e) - 0.5; |
| 729 | if (k === 1) { |
| 730 | if (x < -0.25) return -2 * (e - (x + 0.5)); |
| 731 | return 1 + 2 * (x - e); |
| 732 | } |
| 733 | |
| 734 | if (k <= -2 || k > 56) { |
| 735 | // suffice to return exp(x) + 1 |
| 736 | y = 1 - (e - x); |
| 737 | // Add k to y's exponent |
| 738 | y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y)); |
| 739 | return y - 1; |
| 740 | } |
| 741 | if (k < 20) { |
| 742 | // t = 1 - 2^k |
| 743 | t = %_ConstructDouble(0x3ff00000 - (0x200000 >> k), 0); |
| 744 | y = t - (e - x); |
| 745 | // Add k to y's exponent |
| 746 | y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y)); |
| 747 | } else { |
| 748 | // t = 2^-k |
| 749 | t = %_ConstructDouble((0x3ff - k) << 20, 0); |
| 750 | y = x - (e + t); |
| 751 | y += 1; |
| 752 | // Add k to y's exponent |
| 753 | y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y)); |
| 754 | } |
| 755 | } |
| 756 | return y; |
| 757 | } |
| 758 | |
| 759 | |
| 760 | // ES6 draft 09-27-13, section 20.2.2.30. |
| 761 | // Math.sinh |
| 762 | // Method : |
| 763 | // mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2 |
| 764 | // 1. Replace x by |x| (sinh(-x) = -sinh(x)). |
| 765 | // 2. |
| 766 | // E + E/(E+1) |
| 767 | // 0 <= x <= 22 : sinh(x) := --------------, E=expm1(x) |
| 768 | // 2 |
| 769 | // |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 770 | // 22 <= x <= lnovft : sinh(x) := exp(x)/2 |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 771 | // lnovft <= x <= ln2ovft: sinh(x) := exp(x/2)/2 * exp(x/2) |
| 772 | // ln2ovft < x : sinh(x) := x*shuge (overflow) |
| 773 | // |
| 774 | // Special cases: |
| 775 | // sinh(x) is |x| if x is +Infinity, -Infinity, or NaN. |
| 776 | // only sinh(0)=0 is exact for finite x. |
| 777 | // |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 778 | define KSINH_OVERFLOW = 710.4758600739439; |
| 779 | define TWO_M28 = 3.725290298461914e-9; // 2^-28, empty lower half |
| 780 | define LOG_MAXD = 709.7822265625; // 0x40862e42 00000000, empty lower half |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 781 | |
| 782 | function MathSinh(x) { |
| 783 | x = x * 1; // Convert to number. |
| 784 | var h = (x < 0) ? -0.5 : 0.5; |
| 785 | // |x| in [0, 22]. return sign(x)*0.5*(E+E/(E+1)) |
| 786 | var ax = MathAbs(x); |
| 787 | if (ax < 22) { |
| 788 | // For |x| < 2^-28, sinh(x) = x |
| 789 | if (ax < TWO_M28) return x; |
| 790 | var t = MathExpm1(ax); |
| 791 | if (ax < 1) return h * (2 * t - t * t / (t + 1)); |
| 792 | return h * (t + t / (t + 1)); |
| 793 | } |
| 794 | // |x| in [22, log(maxdouble)], return 0.5 * exp(|x|) |
| 795 | if (ax < LOG_MAXD) return h * MathExp(ax); |
| 796 | // |x| in [log(maxdouble), overflowthreshold] |
| 797 | // overflowthreshold = 710.4758600739426 |
| 798 | if (ax <= KSINH_OVERFLOW) { |
| 799 | var w = MathExp(0.5 * ax); |
| 800 | var t = h * w; |
| 801 | return t * w; |
| 802 | } |
| 803 | // |x| > overflowthreshold or is NaN. |
| 804 | // Return Infinity of the appropriate sign or NaN. |
| 805 | return x * INFINITY; |
| 806 | } |
| 807 | |
| 808 | |
| 809 | // ES6 draft 09-27-13, section 20.2.2.12. |
| 810 | // Math.cosh |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 811 | // Method : |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 812 | // mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2 |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 813 | // 1. Replace x by |x| (cosh(x) = cosh(-x)). |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 814 | // 2. |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 815 | // [ exp(x) - 1 ]^2 |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 816 | // 0 <= x <= ln2/2 : cosh(x) := 1 + ------------------- |
| 817 | // 2*exp(x) |
| 818 | // |
| 819 | // exp(x) + 1/exp(x) |
| 820 | // ln2/2 <= x <= 22 : cosh(x) := ------------------- |
| 821 | // 2 |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 822 | // 22 <= x <= lnovft : cosh(x) := exp(x)/2 |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 823 | // lnovft <= x <= ln2ovft: cosh(x) := exp(x/2)/2 * exp(x/2) |
| 824 | // ln2ovft < x : cosh(x) := huge*huge (overflow) |
| 825 | // |
| 826 | // Special cases: |
| 827 | // cosh(x) is |x| if x is +INF, -INF, or NaN. |
| 828 | // only cosh(0)=1 is exact for finite x. |
| 829 | // |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 830 | define KCOSH_OVERFLOW = 710.4758600739439; |
Ben Murdoch | b8a8cc1 | 2014-11-26 15:28:44 +0000 | [diff] [blame] | 831 | |
| 832 | function MathCosh(x) { |
| 833 | x = x * 1; // Convert to number. |
| 834 | var ix = %_DoubleHi(x) & 0x7fffffff; |
| 835 | // |x| in [0,0.5*log2], return 1+expm1(|x|)^2/(2*exp(|x|)) |
| 836 | if (ix < 0x3fd62e43) { |
| 837 | var t = MathExpm1(MathAbs(x)); |
| 838 | var w = 1 + t; |
| 839 | // For |x| < 2^-55, cosh(x) = 1 |
| 840 | if (ix < 0x3c800000) return w; |
| 841 | return 1 + (t * t) / (w + w); |
| 842 | } |
| 843 | // |x| in [0.5*log2, 22], return (exp(|x|)+1/exp(|x|)/2 |
| 844 | if (ix < 0x40360000) { |
| 845 | var t = MathExp(MathAbs(x)); |
| 846 | return 0.5 * t + 0.5 / t; |
| 847 | } |
| 848 | // |x| in [22, log(maxdouble)], return half*exp(|x|) |
| 849 | if (ix < 0x40862e42) return 0.5 * MathExp(MathAbs(x)); |
| 850 | // |x| in [log(maxdouble), overflowthreshold] |
| 851 | if (MathAbs(x) <= KCOSH_OVERFLOW) { |
| 852 | var w = MathExp(0.5 * MathAbs(x)); |
| 853 | var t = 0.5 * w; |
| 854 | return t * w; |
| 855 | } |
| 856 | if (NUMBER_IS_NAN(x)) return x; |
| 857 | // |x| > overflowthreshold. |
| 858 | return INFINITY; |
| 859 | } |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 860 | |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 861 | // ES6 draft 09-27-13, section 20.2.2.33. |
| 862 | // Math.tanh(x) |
| 863 | // Method : |
| 864 | // x -x |
| 865 | // e - e |
| 866 | // 0. tanh(x) is defined to be ----------- |
| 867 | // x -x |
| 868 | // e + e |
| 869 | // 1. reduce x to non-negative by tanh(-x) = -tanh(x). |
| 870 | // 2. 0 <= x <= 2**-55 : tanh(x) := x*(one+x) |
| 871 | // -t |
| 872 | // 2**-55 < x <= 1 : tanh(x) := -----; t = expm1(-2x) |
| 873 | // t + 2 |
| 874 | // 2 |
| 875 | // 1 <= x <= 22.0 : tanh(x) := 1- ----- ; t = expm1(2x) |
| 876 | // t + 2 |
| 877 | // 22.0 < x <= INF : tanh(x) := 1. |
| 878 | // |
| 879 | // Special cases: |
| 880 | // tanh(NaN) is NaN; |
| 881 | // only tanh(0) = 0 is exact for finite argument. |
| 882 | // |
| 883 | |
| 884 | define TWO_M55 = 2.77555756156289135105e-17; // 2^-55, empty lower half |
| 885 | |
| 886 | function MathTanh(x) { |
| 887 | x = x * 1; // Convert to number. |
| 888 | // x is Infinity or NaN |
| 889 | if (!NUMBER_IS_FINITE(x)) { |
| 890 | if (x > 0) return 1; |
| 891 | if (x < 0) return -1; |
| 892 | return x; |
| 893 | } |
| 894 | |
| 895 | var ax = MathAbs(x); |
| 896 | var z; |
| 897 | // |x| < 22 |
| 898 | if (ax < 22) { |
| 899 | if (ax < TWO_M55) { |
| 900 | // |x| < 2^-55, tanh(small) = small. |
| 901 | return x; |
| 902 | } |
| 903 | if (ax >= 1) { |
| 904 | // |x| >= 1 |
| 905 | var t = MathExpm1(2 * ax); |
| 906 | z = 1 - 2 / (t + 2); |
| 907 | } else { |
| 908 | var t = MathExpm1(-2 * ax); |
| 909 | z = -t / (t + 2); |
| 910 | } |
| 911 | } else { |
| 912 | // |x| > 22, return +/- 1 |
| 913 | z = 1; |
| 914 | } |
| 915 | return (x >= 0) ? z : -z; |
| 916 | } |
| 917 | |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 918 | // ES6 draft 09-27-13, section 20.2.2.21. |
| 919 | // Return the base 10 logarithm of x |
| 920 | // |
| 921 | // Method : |
| 922 | // Let log10_2hi = leading 40 bits of log10(2) and |
| 923 | // log10_2lo = log10(2) - log10_2hi, |
| 924 | // ivln10 = 1/log(10) rounded. |
| 925 | // Then |
| 926 | // n = ilogb(x), |
| 927 | // if(n<0) n = n+1; |
| 928 | // x = scalbn(x,-n); |
| 929 | // log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x)) |
| 930 | // |
| 931 | // Note 1: |
| 932 | // To guarantee log10(10**n)=n, where 10**n is normal, the rounding |
| 933 | // mode must set to Round-to-Nearest. |
| 934 | // Note 2: |
| 935 | // [1/log(10)] rounded to 53 bits has error .198 ulps; |
| 936 | // log10 is monotonic at all binary break points. |
| 937 | // |
| 938 | // Special cases: |
| 939 | // log10(x) is NaN if x < 0; |
| 940 | // log10(+INF) is +INF; log10(0) is -INF; |
| 941 | // log10(NaN) is that NaN; |
| 942 | // log10(10**N) = N for N=0,1,...,22. |
| 943 | // |
| 944 | |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 945 | define IVLN10 = 4.34294481903251816668e-01; |
| 946 | define LOG10_2HI = 3.01029995663611771306e-01; |
| 947 | define LOG10_2LO = 3.69423907715893078616e-13; |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 948 | |
| 949 | function MathLog10(x) { |
| 950 | x = x * 1; // Convert to number. |
| 951 | var hx = %_DoubleHi(x); |
| 952 | var lx = %_DoubleLo(x); |
| 953 | var k = 0; |
| 954 | |
| 955 | if (hx < 0x00100000) { |
| 956 | // x < 2^-1022 |
| 957 | // log10(+/- 0) = -Infinity. |
| 958 | if (((hx & 0x7fffffff) | lx) === 0) return -INFINITY; |
| 959 | // log10 of negative number is NaN. |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 960 | if (hx < 0) return NaN; |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 961 | // Subnormal number. Scale up x. |
| 962 | k -= 54; |
| 963 | x *= TWO54; |
| 964 | hx = %_DoubleHi(x); |
| 965 | lx = %_DoubleLo(x); |
| 966 | } |
| 967 | |
| 968 | // Infinity or NaN. |
| 969 | if (hx >= 0x7ff00000) return x; |
| 970 | |
| 971 | k += (hx >> 20) - 1023; |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 972 | var i = (k & 0x80000000) >>> 31; |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 973 | hx = (hx & 0x000fffff) | ((0x3ff - i) << 20); |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 974 | var y = k + i; |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 975 | x = %_ConstructDouble(hx, lx); |
| 976 | |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 977 | var z = y * LOG10_2LO + IVLN10 * %_MathLogRT(x); |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 978 | return z + y * LOG10_2HI; |
| 979 | } |
| 980 | |
| 981 | |
| 982 | // ES6 draft 09-27-13, section 20.2.2.22. |
| 983 | // Return the base 2 logarithm of x |
| 984 | // |
| 985 | // fdlibm does not have an explicit log2 function, but fdlibm's pow |
| 986 | // function does implement an accurate log2 function as part of the |
| 987 | // pow implementation. This extracts the core parts of that as a |
| 988 | // separate log2 function. |
| 989 | |
| 990 | // Method: |
| 991 | // Compute log2(x) in two pieces: |
| 992 | // log2(x) = w1 + w2 |
| 993 | // where w1 has 53-24 = 29 bits of trailing zeroes. |
| 994 | |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 995 | define DP_H = 5.84962487220764160156e-01; |
| 996 | define DP_L = 1.35003920212974897128e-08; |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 997 | |
| 998 | // Polynomial coefficients for (3/2)*(log2(x) - 2*s - 2/3*s^3) |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 999 | define LOG2_1 = 5.99999999999994648725e-01; |
| 1000 | define LOG2_2 = 4.28571428578550184252e-01; |
| 1001 | define LOG2_3 = 3.33333329818377432918e-01; |
| 1002 | define LOG2_4 = 2.72728123808534006489e-01; |
| 1003 | define LOG2_5 = 2.30660745775561754067e-01; |
| 1004 | define LOG2_6 = 2.06975017800338417784e-01; |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 1005 | |
| 1006 | // cp = 2/(3*ln(2)). Note that cp_h + cp_l is cp, but with more accuracy. |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 1007 | define CP = 9.61796693925975554329e-01; |
| 1008 | define CP_H = 9.61796700954437255859e-01; |
| 1009 | define CP_L = -7.02846165095275826516e-09; |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 1010 | // 2^53 |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 1011 | define TWO53 = 9007199254740992; |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 1012 | |
| 1013 | function MathLog2(x) { |
| 1014 | x = x * 1; // Convert to number. |
| 1015 | var ax = MathAbs(x); |
| 1016 | var hx = %_DoubleHi(x); |
| 1017 | var lx = %_DoubleLo(x); |
| 1018 | var ix = hx & 0x7fffffff; |
| 1019 | |
| 1020 | // Handle special cases. |
| 1021 | // log2(+/- 0) = -Infinity |
| 1022 | if ((ix | lx) == 0) return -INFINITY; |
| 1023 | |
| 1024 | // log(x) = NaN, if x < 0 |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 1025 | if (hx < 0) return NaN; |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 1026 | |
| 1027 | // log2(Infinity) = Infinity, log2(NaN) = NaN |
| 1028 | if (ix >= 0x7ff00000) return x; |
| 1029 | |
| 1030 | var n = 0; |
| 1031 | |
| 1032 | // Take care of subnormal number. |
| 1033 | if (ix < 0x00100000) { |
| 1034 | ax *= TWO53; |
| 1035 | n -= 53; |
| 1036 | ix = %_DoubleHi(ax); |
| 1037 | } |
| 1038 | |
| 1039 | n += (ix >> 20) - 0x3ff; |
| 1040 | var j = ix & 0x000fffff; |
| 1041 | |
| 1042 | // Determine interval. |
| 1043 | ix = j | 0x3ff00000; // normalize ix. |
| 1044 | |
| 1045 | var bp = 1; |
| 1046 | var dp_h = 0; |
| 1047 | var dp_l = 0; |
| 1048 | if (j > 0x3988e) { // |x| > sqrt(3/2) |
| 1049 | if (j < 0xbb67a) { // |x| < sqrt(3) |
| 1050 | bp = 1.5; |
| 1051 | dp_h = DP_H; |
| 1052 | dp_l = DP_L; |
| 1053 | } else { |
| 1054 | n += 1; |
| 1055 | ix -= 0x00100000; |
| 1056 | } |
| 1057 | } |
| 1058 | |
| 1059 | ax = %_ConstructDouble(ix, %_DoubleLo(ax)); |
| 1060 | |
| 1061 | // Compute ss = s_h + s_l = (x - 1)/(x+1) or (x - 1.5)/(x + 1.5) |
| 1062 | var u = ax - bp; |
| 1063 | var v = 1 / (ax + bp); |
| 1064 | var ss = u * v; |
| 1065 | var s_h = %_ConstructDouble(%_DoubleHi(ss), 0); |
| 1066 | |
| 1067 | // t_h = ax + bp[k] High |
| 1068 | var t_h = %_ConstructDouble(%_DoubleHi(ax + bp), 0) |
| 1069 | var t_l = ax - (t_h - bp); |
| 1070 | var s_l = v * ((u - s_h * t_h) - s_h * t_l); |
| 1071 | |
| 1072 | // Compute log2(ax) |
| 1073 | var s2 = ss * ss; |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 1074 | var r = s2 * s2 * (LOG2_1 + s2 * (LOG2_2 + s2 * (LOG2_3 + s2 * ( |
| 1075 | LOG2_4 + s2 * (LOG2_5 + s2 * LOG2_6))))); |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 1076 | r += s_l * (s_h + ss); |
| 1077 | s2 = s_h * s_h; |
| 1078 | t_h = %_ConstructDouble(%_DoubleHi(3.0 + s2 + r), 0); |
| 1079 | t_l = r - ((t_h - 3.0) - s2); |
| 1080 | // u + v = ss * (1 + ...) |
| 1081 | u = s_h * t_h; |
| 1082 | v = s_l * t_h + t_l * ss; |
| 1083 | |
| 1084 | // 2 / (3 * log(2)) * (ss + ...) |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 1085 | var p_h = %_ConstructDouble(%_DoubleHi(u + v), 0); |
| 1086 | var p_l = v - (p_h - u); |
| 1087 | var z_h = CP_H * p_h; |
| 1088 | var z_l = CP_L * p_h + p_l * CP + dp_l; |
Emily Bernier | d0a1eb7 | 2015-03-24 16:35:39 -0400 | [diff] [blame] | 1089 | |
| 1090 | // log2(ax) = (ss + ...) * 2 / (3 * log(2)) = n + dp_h + z_h + z_l |
| 1091 | var t = n; |
| 1092 | var t1 = %_ConstructDouble(%_DoubleHi(((z_h + z_l) + dp_h) + t), 0); |
| 1093 | var t2 = z_l - (((t1 - t) - dp_h) - z_h); |
| 1094 | |
| 1095 | // t1 + t2 = log2(ax), sum up because we do not care about extra precision. |
| 1096 | return t1 + t2; |
| 1097 | } |
Ben Murdoch | 4a90d5f | 2016-03-22 12:00:34 +0000 | [diff] [blame^] | 1098 | |
| 1099 | //------------------------------------------------------------------- |
| 1100 | |
| 1101 | utils.InstallFunctions(GlobalMath, DONT_ENUM, [ |
| 1102 | "cos", MathCos, |
| 1103 | "sin", MathSin, |
| 1104 | "tan", MathTan, |
| 1105 | "sinh", MathSinh, |
| 1106 | "cosh", MathCosh, |
| 1107 | "tanh", MathTanh, |
| 1108 | "log10", MathLog10, |
| 1109 | "log2", MathLog2, |
| 1110 | "log1p", MathLog1p, |
| 1111 | "expm1", MathExpm1 |
| 1112 | ]); |
| 1113 | |
| 1114 | %SetForceInlineFlag(MathSin); |
| 1115 | %SetForceInlineFlag(MathCos); |
| 1116 | |
| 1117 | }) |