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 | |
| 19 | // Double constants that do not have empty lower 32 bits are found in fdlibm.cc |
| 20 | // and exposed through kMath as typed array. We assume the compiler to convert |
| 21 | // from decimal to binary accurately enough to produce the intended values. |
| 22 | // kMath is initialized to a Float64Array during genesis and not writable. |
| 23 | var kMath; |
| 24 | |
| 25 | const INVPIO2 = kMath[0]; |
| 26 | const PIO2_1 = kMath[1]; |
| 27 | const PIO2_1T = kMath[2]; |
| 28 | const PIO2_2 = kMath[3]; |
| 29 | const PIO2_2T = kMath[4]; |
| 30 | const PIO2_3 = kMath[5]; |
| 31 | const PIO2_3T = kMath[6]; |
| 32 | const PIO4 = kMath[32]; |
| 33 | const PIO4LO = kMath[33]; |
| 34 | |
| 35 | // Compute k and r such that x - k*pi/2 = r where |r| < pi/4. For |
| 36 | // precision, r is returned as two values y0 and y1 such that r = y0 + y1 |
| 37 | // to more than double precision. |
| 38 | macro REMPIO2(X) |
| 39 | var n, y0, y1; |
| 40 | var hx = %_DoubleHi(X); |
| 41 | var ix = hx & 0x7fffffff; |
| 42 | |
| 43 | if (ix < 0x4002d97c) { |
| 44 | // |X| ~< 3*pi/4, special case with n = +/- 1 |
| 45 | if (hx > 0) { |
| 46 | var z = X - PIO2_1; |
| 47 | if (ix != 0x3ff921fb) { |
| 48 | // 33+53 bit pi is good enough |
| 49 | y0 = z - PIO2_1T; |
| 50 | y1 = (z - y0) - PIO2_1T; |
| 51 | } else { |
| 52 | // near pi/2, use 33+33+53 bit pi |
| 53 | z -= PIO2_2; |
| 54 | y0 = z - PIO2_2T; |
| 55 | y1 = (z - y0) - PIO2_2T; |
| 56 | } |
| 57 | n = 1; |
| 58 | } else { |
| 59 | // Negative X |
| 60 | var z = X + PIO2_1; |
| 61 | if (ix != 0x3ff921fb) { |
| 62 | // 33+53 bit pi is good enough |
| 63 | y0 = z + PIO2_1T; |
| 64 | y1 = (z - y0) + PIO2_1T; |
| 65 | } else { |
| 66 | // near pi/2, use 33+33+53 bit pi |
| 67 | z += PIO2_2; |
| 68 | y0 = z + PIO2_2T; |
| 69 | y1 = (z - y0) + PIO2_2T; |
| 70 | } |
| 71 | n = -1; |
| 72 | } |
| 73 | } else if (ix <= 0x413921fb) { |
| 74 | // |X| ~<= 2^19*(pi/2), medium size |
| 75 | var t = MathAbs(X); |
| 76 | n = (t * INVPIO2 + 0.5) | 0; |
| 77 | var r = t - n * PIO2_1; |
| 78 | var w = n * PIO2_1T; |
| 79 | // First round good to 85 bit |
| 80 | y0 = r - w; |
| 81 | if (ix - (%_DoubleHi(y0) & 0x7ff00000) > 0x1000000) { |
| 82 | // 2nd iteration needed, good to 118 |
| 83 | t = r; |
| 84 | w = n * PIO2_2; |
| 85 | r = t - w; |
| 86 | w = n * PIO2_2T - ((t - r) - w); |
| 87 | y0 = r - w; |
| 88 | if (ix - (%_DoubleHi(y0) & 0x7ff00000) > 0x3100000) { |
| 89 | // 3rd iteration needed. 151 bits accuracy |
| 90 | t = r; |
| 91 | w = n * PIO2_3; |
| 92 | r = t - w; |
| 93 | w = n * PIO2_3T - ((t - r) - w); |
| 94 | y0 = r - w; |
| 95 | } |
| 96 | } |
| 97 | y1 = (r - y0) - w; |
| 98 | if (hx < 0) { |
| 99 | n = -n; |
| 100 | y0 = -y0; |
| 101 | y1 = -y1; |
| 102 | } |
| 103 | } else { |
| 104 | // Need to do full Payne-Hanek reduction here. |
| 105 | var r = %RemPiO2(X); |
| 106 | n = r[0]; |
| 107 | y0 = r[1]; |
| 108 | y1 = r[2]; |
| 109 | } |
| 110 | endmacro |
| 111 | |
| 112 | |
| 113 | // __kernel_sin(X, Y, IY) |
| 114 | // kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 |
| 115 | // Input X is assumed to be bounded by ~pi/4 in magnitude. |
| 116 | // Input Y is the tail of X so that x = X + Y. |
| 117 | // |
| 118 | // Algorithm |
| 119 | // 1. Since ieee_sin(-x) = -ieee_sin(x), we need only to consider positive x. |
| 120 | // 2. ieee_sin(x) is approximated by a polynomial of degree 13 on |
| 121 | // [0,pi/4] |
| 122 | // 3 13 |
| 123 | // sin(x) ~ x + S1*x + ... + S6*x |
| 124 | // where |
| 125 | // |
| 126 | // |ieee_sin(x) 2 4 6 8 10 12 | -58 |
| 127 | // |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 |
| 128 | // | x | |
| 129 | // |
| 130 | // 3. ieee_sin(X+Y) = ieee_sin(X) + sin'(X')*Y |
| 131 | // ~ ieee_sin(X) + (1-X*X/2)*Y |
| 132 | // For better accuracy, let |
| 133 | // 3 2 2 2 2 |
| 134 | // r = X *(S2+X *(S3+X *(S4+X *(S5+X *S6)))) |
| 135 | // then 3 2 |
| 136 | // sin(x) = X + (S1*X + (X *(r-Y/2)+Y)) |
| 137 | // |
| 138 | macro KSIN(x) |
| 139 | kMath[7+x] |
| 140 | endmacro |
| 141 | |
| 142 | macro RETURN_KERNELSIN(X, Y, SIGN) |
| 143 | var z = X * X; |
| 144 | var v = z * X; |
| 145 | var r = KSIN(1) + z * (KSIN(2) + z * (KSIN(3) + |
| 146 | z * (KSIN(4) + z * KSIN(5)))); |
| 147 | return (X - ((z * (0.5 * Y - v * r) - Y) - v * KSIN(0))) SIGN; |
| 148 | endmacro |
| 149 | |
| 150 | // __kernel_cos(X, Y) |
| 151 | // kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 |
| 152 | // Input X is assumed to be bounded by ~pi/4 in magnitude. |
| 153 | // Input Y is the tail of X so that x = X + Y. |
| 154 | // |
| 155 | // Algorithm |
| 156 | // 1. Since ieee_cos(-x) = ieee_cos(x), we need only to consider positive x. |
| 157 | // 2. ieee_cos(x) is approximated by a polynomial of degree 14 on |
| 158 | // [0,pi/4] |
| 159 | // 4 14 |
| 160 | // cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x |
| 161 | // where the remez error is |
| 162 | // |
| 163 | // | 2 4 6 8 10 12 14 | -58 |
| 164 | // |ieee_cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 |
| 165 | // | | |
| 166 | // |
| 167 | // 4 6 8 10 12 14 |
| 168 | // 3. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then |
| 169 | // ieee_cos(x) = 1 - x*x/2 + r |
| 170 | // since ieee_cos(X+Y) ~ ieee_cos(X) - ieee_sin(X)*Y |
| 171 | // ~ ieee_cos(X) - X*Y, |
| 172 | // a correction term is necessary in ieee_cos(x) and hence |
| 173 | // cos(X+Y) = 1 - (X*X/2 - (r - X*Y)) |
| 174 | // For better accuracy when x > 0.3, let qx = |x|/4 with |
| 175 | // the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. |
| 176 | // Then |
| 177 | // cos(X+Y) = (1-qx) - ((X*X/2-qx) - (r-X*Y)). |
| 178 | // Note that 1-qx and (X*X/2-qx) is EXACT here, and the |
| 179 | // magnitude of the latter is at least a quarter of X*X/2, |
| 180 | // thus, reducing the rounding error in the subtraction. |
| 181 | // |
| 182 | macro KCOS(x) |
| 183 | kMath[13+x] |
| 184 | endmacro |
| 185 | |
| 186 | macro RETURN_KERNELCOS(X, Y, SIGN) |
| 187 | var ix = %_DoubleHi(X) & 0x7fffffff; |
| 188 | var z = X * X; |
| 189 | var r = z * (KCOS(0) + z * (KCOS(1) + z * (KCOS(2)+ |
| 190 | z * (KCOS(3) + z * (KCOS(4) + z * KCOS(5)))))); |
| 191 | if (ix < 0x3fd33333) { // |x| ~< 0.3 |
| 192 | return (1 - (0.5 * z - (z * r - X * Y))) SIGN; |
| 193 | } else { |
| 194 | var qx; |
| 195 | if (ix > 0x3fe90000) { // |x| > 0.78125 |
| 196 | qx = 0.28125; |
| 197 | } else { |
| 198 | qx = %_ConstructDouble(%_DoubleHi(0.25 * X), 0); |
| 199 | } |
| 200 | var hz = 0.5 * z - qx; |
| 201 | return (1 - qx - (hz - (z * r - X * Y))) SIGN; |
| 202 | } |
| 203 | endmacro |
| 204 | |
| 205 | |
| 206 | // kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 |
| 207 | // Input x is assumed to be bounded by ~pi/4 in magnitude. |
| 208 | // Input y is the tail of x. |
| 209 | // Input k indicates whether ieee_tan (if k = 1) or -1/tan (if k = -1) |
| 210 | // is returned. |
| 211 | // |
| 212 | // Algorithm |
| 213 | // 1. Since ieee_tan(-x) = -ieee_tan(x), we need only to consider positive x. |
| 214 | // 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. |
| 215 | // 3. ieee_tan(x) is approximated by a odd polynomial of degree 27 on |
| 216 | // [0,0.67434] |
| 217 | // 3 27 |
| 218 | // tan(x) ~ x + T1*x + ... + T13*x |
| 219 | // where |
| 220 | // |
| 221 | // |ieee_tan(x) 2 4 26 | -59.2 |
| 222 | // |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 |
| 223 | // | x | |
| 224 | // |
| 225 | // Note: ieee_tan(x+y) = ieee_tan(x) + tan'(x)*y |
| 226 | // ~ ieee_tan(x) + (1+x*x)*y |
| 227 | // Therefore, for better accuracy in computing ieee_tan(x+y), let |
| 228 | // 3 2 2 2 2 |
| 229 | // r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) |
| 230 | // then |
| 231 | // 3 2 |
| 232 | // tan(x+y) = x + (T1*x + (x *(r+y)+y)) |
| 233 | // |
| 234 | // 4. For x in [0.67434,pi/4], let y = pi/4 - x, then |
| 235 | // tan(x) = ieee_tan(pi/4-y) = (1-ieee_tan(y))/(1+ieee_tan(y)) |
| 236 | // = 1 - 2*(ieee_tan(y) - (ieee_tan(y)^2)/(1+ieee_tan(y))) |
| 237 | // |
| 238 | // Set returnTan to 1 for tan; -1 for cot. Anything else is illegal |
| 239 | // and will cause incorrect results. |
| 240 | // |
| 241 | macro KTAN(x) |
| 242 | kMath[19+x] |
| 243 | endmacro |
| 244 | |
| 245 | function KernelTan(x, y, returnTan) { |
| 246 | var z; |
| 247 | var w; |
| 248 | var hx = %_DoubleHi(x); |
| 249 | var ix = hx & 0x7fffffff; |
| 250 | |
| 251 | if (ix < 0x3e300000) { // |x| < 2^-28 |
| 252 | if (((ix | %_DoubleLo(x)) | (returnTan + 1)) == 0) { |
| 253 | // x == 0 && returnTan = -1 |
| 254 | return 1 / MathAbs(x); |
| 255 | } else { |
| 256 | if (returnTan == 1) { |
| 257 | return x; |
| 258 | } else { |
| 259 | // Compute -1/(x + y) carefully |
| 260 | var w = x + y; |
| 261 | var z = %_ConstructDouble(%_DoubleHi(w), 0); |
| 262 | var v = y - (z - x); |
| 263 | var a = -1 / w; |
| 264 | var t = %_ConstructDouble(%_DoubleHi(a), 0); |
| 265 | var s = 1 + t * z; |
| 266 | return t + a * (s + t * v); |
| 267 | } |
| 268 | } |
| 269 | } |
| 270 | if (ix >= 0x3fe59428) { // |x| > .6744 |
| 271 | if (x < 0) { |
| 272 | x = -x; |
| 273 | y = -y; |
| 274 | } |
| 275 | z = PIO4 - x; |
| 276 | w = PIO4LO - y; |
| 277 | x = z + w; |
| 278 | y = 0; |
| 279 | } |
| 280 | z = x * x; |
| 281 | w = z * z; |
| 282 | |
| 283 | // Break x^5 * (T1 + x^2*T2 + ...) into |
| 284 | // x^5 * (T1 + x^4*T3 + ... + x^20*T11) + |
| 285 | // x^5 * (x^2 * (T2 + x^4*T4 + ... + x^22*T12)) |
| 286 | var r = KTAN(1) + w * (KTAN(3) + w * (KTAN(5) + |
| 287 | w * (KTAN(7) + w * (KTAN(9) + w * KTAN(11))))); |
| 288 | var v = z * (KTAN(2) + w * (KTAN(4) + w * (KTAN(6) + |
| 289 | w * (KTAN(8) + w * (KTAN(10) + w * KTAN(12)))))); |
| 290 | var s = z * x; |
| 291 | r = y + z * (s * (r + v) + y); |
| 292 | r = r + KTAN(0) * s; |
| 293 | w = x + r; |
| 294 | if (ix >= 0x3fe59428) { |
| 295 | return (1 - ((hx >> 30) & 2)) * |
| 296 | (returnTan - 2.0 * (x - (w * w / (w + returnTan) - r))); |
| 297 | } |
| 298 | if (returnTan == 1) { |
| 299 | return w; |
| 300 | } else { |
| 301 | z = %_ConstructDouble(%_DoubleHi(w), 0); |
| 302 | v = r - (z - x); |
| 303 | var a = -1 / w; |
| 304 | var t = %_ConstructDouble(%_DoubleHi(a), 0); |
| 305 | s = 1 + t * z; |
| 306 | return t + a * (s + t * v); |
| 307 | } |
| 308 | } |
| 309 | |
| 310 | function MathSinSlow(x) { |
| 311 | REMPIO2(x); |
| 312 | var sign = 1 - (n & 2); |
| 313 | if (n & 1) { |
| 314 | RETURN_KERNELCOS(y0, y1, * sign); |
| 315 | } else { |
| 316 | RETURN_KERNELSIN(y0, y1, * sign); |
| 317 | } |
| 318 | } |
| 319 | |
| 320 | function MathCosSlow(x) { |
| 321 | REMPIO2(x); |
| 322 | if (n & 1) { |
| 323 | var sign = (n & 2) - 1; |
| 324 | RETURN_KERNELSIN(y0, y1, * sign); |
| 325 | } else { |
| 326 | var sign = 1 - (n & 2); |
| 327 | RETURN_KERNELCOS(y0, y1, * sign); |
| 328 | } |
| 329 | } |
| 330 | |
| 331 | // ECMA 262 - 15.8.2.16 |
| 332 | function MathSin(x) { |
| 333 | x = x * 1; // Convert to number. |
| 334 | if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) { |
| 335 | // |x| < pi/4, approximately. No reduction needed. |
| 336 | RETURN_KERNELSIN(x, 0, /* empty */); |
| 337 | } |
| 338 | return MathSinSlow(x); |
| 339 | } |
| 340 | |
| 341 | // ECMA 262 - 15.8.2.7 |
| 342 | function MathCos(x) { |
| 343 | x = x * 1; // Convert to number. |
| 344 | if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) { |
| 345 | // |x| < pi/4, approximately. No reduction needed. |
| 346 | RETURN_KERNELCOS(x, 0, /* empty */); |
| 347 | } |
| 348 | return MathCosSlow(x); |
| 349 | } |
| 350 | |
| 351 | // ECMA 262 - 15.8.2.18 |
| 352 | function MathTan(x) { |
| 353 | x = x * 1; // Convert to number. |
| 354 | if ((%_DoubleHi(x) & 0x7fffffff) <= 0x3fe921fb) { |
| 355 | // |x| < pi/4, approximately. No reduction needed. |
| 356 | return KernelTan(x, 0, 1); |
| 357 | } |
| 358 | REMPIO2(x); |
| 359 | return KernelTan(y0, y1, (n & 1) ? -1 : 1); |
| 360 | } |
| 361 | |
| 362 | // ES6 draft 09-27-13, section 20.2.2.20. |
| 363 | // Math.log1p |
| 364 | // |
| 365 | // Method : |
| 366 | // 1. Argument Reduction: find k and f such that |
| 367 | // 1+x = 2^k * (1+f), |
| 368 | // where sqrt(2)/2 < 1+f < sqrt(2) . |
| 369 | // |
| 370 | // Note. If k=0, then f=x is exact. However, if k!=0, then f |
| 371 | // may not be representable exactly. In that case, a correction |
| 372 | // term is need. Let u=1+x rounded. Let c = (1+x)-u, then |
| 373 | // log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), |
| 374 | // and add back the correction term c/u. |
| 375 | // (Note: when x > 2**53, one can simply return log(x)) |
| 376 | // |
| 377 | // 2. Approximation of log1p(f). |
| 378 | // Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) |
| 379 | // = 2s + 2/3 s**3 + 2/5 s**5 + ....., |
| 380 | // = 2s + s*R |
| 381 | // We use a special Reme algorithm on [0,0.1716] to generate |
| 382 | // a polynomial of degree 14 to approximate R The maximum error |
| 383 | // of this polynomial approximation is bounded by 2**-58.45. In |
| 384 | // other words, |
| 385 | // 2 4 6 8 10 12 14 |
| 386 | // R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s |
| 387 | // (the values of Lp1 to Lp7 are listed in the program) |
| 388 | // and |
| 389 | // | 2 14 | -58.45 |
| 390 | // | Lp1*s +...+Lp7*s - R(z) | <= 2 |
| 391 | // | | |
| 392 | // Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. |
| 393 | // In order to guarantee error in log below 1ulp, we compute log |
| 394 | // by |
| 395 | // log1p(f) = f - (hfsq - s*(hfsq+R)). |
| 396 | // |
| 397 | // 3. Finally, log1p(x) = k*ln2 + log1p(f). |
| 398 | // = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) |
| 399 | // Here ln2 is split into two floating point number: |
| 400 | // ln2_hi + ln2_lo, |
| 401 | // where n*ln2_hi is always exact for |n| < 2000. |
| 402 | // |
| 403 | // Special cases: |
| 404 | // log1p(x) is NaN with signal if x < -1 (including -INF) ; |
| 405 | // log1p(+INF) is +INF; log1p(-1) is -INF with signal; |
| 406 | // log1p(NaN) is that NaN with no signal. |
| 407 | // |
| 408 | // Accuracy: |
| 409 | // according to an error analysis, the error is always less than |
| 410 | // 1 ulp (unit in the last place). |
| 411 | // |
| 412 | // Constants: |
| 413 | // Constants are found in fdlibm.cc. We assume the C++ compiler to convert |
| 414 | // from decimal to binary accurately enough to produce the intended values. |
| 415 | // |
| 416 | // Note: Assuming log() return accurate answer, the following |
| 417 | // algorithm can be used to compute log1p(x) to within a few ULP: |
| 418 | // |
| 419 | // u = 1+x; |
| 420 | // if (u==1.0) return x ; else |
| 421 | // return log(u)*(x/(u-1.0)); |
| 422 | // |
| 423 | // See HP-15C Advanced Functions Handbook, p.193. |
| 424 | // |
| 425 | const LN2_HI = kMath[34]; |
| 426 | const LN2_LO = kMath[35]; |
| 427 | const TWO54 = kMath[36]; |
| 428 | const TWO_THIRD = kMath[37]; |
| 429 | macro KLOG1P(x) |
| 430 | (kMath[38+x]) |
| 431 | endmacro |
| 432 | |
| 433 | function MathLog1p(x) { |
| 434 | x = x * 1; // Convert to number. |
| 435 | var hx = %_DoubleHi(x); |
| 436 | var ax = hx & 0x7fffffff; |
| 437 | var k = 1; |
| 438 | var f = x; |
| 439 | var hu = 1; |
| 440 | var c = 0; |
| 441 | var u = x; |
| 442 | |
| 443 | if (hx < 0x3fda827a) { |
| 444 | // x < 0.41422 |
| 445 | if (ax >= 0x3ff00000) { // |x| >= 1 |
| 446 | if (x === -1) { |
| 447 | return -INFINITY; // log1p(-1) = -inf |
| 448 | } else { |
| 449 | return NAN; // log1p(x<-1) = NaN |
| 450 | } |
| 451 | } else if (ax < 0x3c900000) { |
| 452 | // For |x| < 2^-54 we can return x. |
| 453 | return x; |
| 454 | } else if (ax < 0x3e200000) { |
| 455 | // For |x| < 2^-29 we can use a simple two-term Taylor series. |
| 456 | return x - x * x * 0.5; |
| 457 | } |
| 458 | |
| 459 | if ((hx > 0) || (hx <= -0x402D413D)) { // (int) 0xbfd2bec3 = -0x402d413d |
| 460 | // -.2929 < x < 0.41422 |
| 461 | k = 0; |
| 462 | } |
| 463 | } |
| 464 | |
| 465 | // Handle Infinity and NAN |
| 466 | if (hx >= 0x7ff00000) return x; |
| 467 | |
| 468 | if (k !== 0) { |
| 469 | if (hx < 0x43400000) { |
| 470 | // x < 2^53 |
| 471 | u = 1 + x; |
| 472 | hu = %_DoubleHi(u); |
| 473 | k = (hu >> 20) - 1023; |
| 474 | c = (k > 0) ? 1 - (u - x) : x - (u - 1); |
| 475 | c = c / u; |
| 476 | } else { |
| 477 | hu = %_DoubleHi(u); |
| 478 | k = (hu >> 20) - 1023; |
| 479 | } |
| 480 | hu = hu & 0xfffff; |
| 481 | if (hu < 0x6a09e) { |
| 482 | u = %_ConstructDouble(hu | 0x3ff00000, %_DoubleLo(u)); // Normalize u. |
| 483 | } else { |
| 484 | ++k; |
| 485 | u = %_ConstructDouble(hu | 0x3fe00000, %_DoubleLo(u)); // Normalize u/2. |
| 486 | hu = (0x00100000 - hu) >> 2; |
| 487 | } |
| 488 | f = u - 1; |
| 489 | } |
| 490 | |
| 491 | var hfsq = 0.5 * f * f; |
| 492 | if (hu === 0) { |
| 493 | // |f| < 2^-20; |
| 494 | if (f === 0) { |
| 495 | if (k === 0) { |
| 496 | return 0.0; |
| 497 | } else { |
| 498 | return k * LN2_HI + (c + k * LN2_LO); |
| 499 | } |
| 500 | } |
| 501 | var R = hfsq * (1 - TWO_THIRD * f); |
| 502 | if (k === 0) { |
| 503 | return f - R; |
| 504 | } else { |
| 505 | return k * LN2_HI - ((R - (k * LN2_LO + c)) - f); |
| 506 | } |
| 507 | } |
| 508 | |
| 509 | var s = f / (2 + f); |
| 510 | var z = s * s; |
| 511 | var R = z * (KLOG1P(0) + z * (KLOG1P(1) + z * |
| 512 | (KLOG1P(2) + z * (KLOG1P(3) + z * |
| 513 | (KLOG1P(4) + z * (KLOG1P(5) + z * KLOG1P(6))))))); |
| 514 | if (k === 0) { |
| 515 | return f - (hfsq - s * (hfsq + R)); |
| 516 | } else { |
| 517 | return k * LN2_HI - ((hfsq - (s * (hfsq + R) + (k * LN2_LO + c))) - f); |
| 518 | } |
| 519 | } |
| 520 | |
| 521 | // ES6 draft 09-27-13, section 20.2.2.14. |
| 522 | // Math.expm1 |
| 523 | // Returns exp(x)-1, the exponential of x minus 1. |
| 524 | // |
| 525 | // Method |
| 526 | // 1. Argument reduction: |
| 527 | // Given x, find r and integer k such that |
| 528 | // |
| 529 | // x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 |
| 530 | // |
| 531 | // Here a correction term c will be computed to compensate |
| 532 | // the error in r when rounded to a floating-point number. |
| 533 | // |
| 534 | // 2. Approximating expm1(r) by a special rational function on |
| 535 | // the interval [0,0.34658]: |
| 536 | // Since |
| 537 | // r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... |
| 538 | // we define R1(r*r) by |
| 539 | // r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) |
| 540 | // That is, |
| 541 | // R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) |
| 542 | // = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) |
| 543 | // = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... |
| 544 | // We use a special Remes algorithm on [0,0.347] to generate |
| 545 | // a polynomial of degree 5 in r*r to approximate R1. The |
| 546 | // maximum error of this polynomial approximation is bounded |
| 547 | // by 2**-61. In other words, |
| 548 | // R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 |
| 549 | // where Q1 = -1.6666666666666567384E-2, |
| 550 | // Q2 = 3.9682539681370365873E-4, |
| 551 | // Q3 = -9.9206344733435987357E-6, |
| 552 | // Q4 = 2.5051361420808517002E-7, |
| 553 | // Q5 = -6.2843505682382617102E-9; |
| 554 | // (where z=r*r, and the values of Q1 to Q5 are listed below) |
| 555 | // with error bounded by |
| 556 | // | 5 | -61 |
| 557 | // | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 |
| 558 | // | | |
| 559 | // |
| 560 | // expm1(r) = exp(r)-1 is then computed by the following |
| 561 | // specific way which minimize the accumulation rounding error: |
| 562 | // 2 3 |
| 563 | // r r [ 3 - (R1 + R1*r/2) ] |
| 564 | // expm1(r) = r + --- + --- * [--------------------] |
| 565 | // 2 2 [ 6 - r*(3 - R1*r/2) ] |
| 566 | // |
| 567 | // To compensate the error in the argument reduction, we use |
| 568 | // expm1(r+c) = expm1(r) + c + expm1(r)*c |
| 569 | // ~ expm1(r) + c + r*c |
| 570 | // Thus c+r*c will be added in as the correction terms for |
| 571 | // expm1(r+c). Now rearrange the term to avoid optimization |
| 572 | // screw up: |
| 573 | // ( 2 2 ) |
| 574 | // ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) |
| 575 | // expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) |
| 576 | // ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) |
| 577 | // ( ) |
| 578 | // |
| 579 | // = r - E |
| 580 | // 3. Scale back to obtain expm1(x): |
| 581 | // From step 1, we have |
| 582 | // expm1(x) = either 2^k*[expm1(r)+1] - 1 |
| 583 | // = or 2^k*[expm1(r) + (1-2^-k)] |
| 584 | // 4. Implementation notes: |
| 585 | // (A). To save one multiplication, we scale the coefficient Qi |
| 586 | // to Qi*2^i, and replace z by (x^2)/2. |
| 587 | // (B). To achieve maximum accuracy, we compute expm1(x) by |
| 588 | // (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) |
| 589 | // (ii) if k=0, return r-E |
| 590 | // (iii) if k=-1, return 0.5*(r-E)-0.5 |
| 591 | // (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) |
| 592 | // else return 1.0+2.0*(r-E); |
| 593 | // (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) |
| 594 | // (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else |
| 595 | // (vii) return 2^k(1-((E+2^-k)-r)) |
| 596 | // |
| 597 | // Special cases: |
| 598 | // expm1(INF) is INF, expm1(NaN) is NaN; |
| 599 | // expm1(-INF) is -1, and |
| 600 | // for finite argument, only expm1(0)=0 is exact. |
| 601 | // |
| 602 | // Accuracy: |
| 603 | // according to an error analysis, the error is always less than |
| 604 | // 1 ulp (unit in the last place). |
| 605 | // |
| 606 | // Misc. info. |
| 607 | // For IEEE double |
| 608 | // if x > 7.09782712893383973096e+02 then expm1(x) overflow |
| 609 | // |
| 610 | const KEXPM1_OVERFLOW = kMath[45]; |
| 611 | const INVLN2 = kMath[46]; |
| 612 | macro KEXPM1(x) |
| 613 | (kMath[47+x]) |
| 614 | endmacro |
| 615 | |
| 616 | function MathExpm1(x) { |
| 617 | x = x * 1; // Convert to number. |
| 618 | var y; |
| 619 | var hi; |
| 620 | var lo; |
| 621 | var k; |
| 622 | var t; |
| 623 | var c; |
| 624 | |
| 625 | var hx = %_DoubleHi(x); |
| 626 | var xsb = hx & 0x80000000; // Sign bit of x |
| 627 | var y = (xsb === 0) ? x : -x; // y = |x| |
| 628 | hx &= 0x7fffffff; // High word of |x| |
| 629 | |
| 630 | // Filter out huge and non-finite argument |
| 631 | if (hx >= 0x4043687a) { // if |x| ~=> 56 * ln2 |
| 632 | if (hx >= 0x40862e42) { // if |x| >= 709.78 |
| 633 | if (hx >= 0x7ff00000) { |
| 634 | // expm1(inf) = inf; expm1(-inf) = -1; expm1(nan) = nan; |
| 635 | return (x === -INFINITY) ? -1 : x; |
| 636 | } |
| 637 | if (x > KEXPM1_OVERFLOW) return INFINITY; // Overflow |
| 638 | } |
| 639 | if (xsb != 0) return -1; // x < -56 * ln2, return -1. |
| 640 | } |
| 641 | |
| 642 | // Argument reduction |
| 643 | if (hx > 0x3fd62e42) { // if |x| > 0.5 * ln2 |
| 644 | if (hx < 0x3ff0a2b2) { // and |x| < 1.5 * ln2 |
| 645 | if (xsb === 0) { |
| 646 | hi = x - LN2_HI; |
| 647 | lo = LN2_LO; |
| 648 | k = 1; |
| 649 | } else { |
| 650 | hi = x + LN2_HI; |
| 651 | lo = -LN2_LO; |
| 652 | k = -1; |
| 653 | } |
| 654 | } else { |
| 655 | k = (INVLN2 * x + ((xsb === 0) ? 0.5 : -0.5)) | 0; |
| 656 | t = k; |
| 657 | // t * ln2_hi is exact here. |
| 658 | hi = x - t * LN2_HI; |
| 659 | lo = t * LN2_LO; |
| 660 | } |
| 661 | x = hi - lo; |
| 662 | c = (hi - x) - lo; |
| 663 | } else if (hx < 0x3c900000) { |
| 664 | // When |x| < 2^-54, we can return x. |
| 665 | return x; |
| 666 | } else { |
| 667 | // Fall through. |
| 668 | k = 0; |
| 669 | } |
| 670 | |
| 671 | // x is now in primary range |
| 672 | var hfx = 0.5 * x; |
| 673 | var hxs = x * hfx; |
| 674 | var r1 = 1 + hxs * (KEXPM1(0) + hxs * (KEXPM1(1) + hxs * |
| 675 | (KEXPM1(2) + hxs * (KEXPM1(3) + hxs * KEXPM1(4))))); |
| 676 | t = 3 - r1 * hfx; |
| 677 | var e = hxs * ((r1 - t) / (6 - x * t)); |
| 678 | if (k === 0) { // c is 0 |
| 679 | return x - (x*e - hxs); |
| 680 | } else { |
| 681 | e = (x * (e - c) - c); |
| 682 | e -= hxs; |
| 683 | if (k === -1) return 0.5 * (x - e) - 0.5; |
| 684 | if (k === 1) { |
| 685 | if (x < -0.25) return -2 * (e - (x + 0.5)); |
| 686 | return 1 + 2 * (x - e); |
| 687 | } |
| 688 | |
| 689 | if (k <= -2 || k > 56) { |
| 690 | // suffice to return exp(x) + 1 |
| 691 | y = 1 - (e - x); |
| 692 | // Add k to y's exponent |
| 693 | y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y)); |
| 694 | return y - 1; |
| 695 | } |
| 696 | if (k < 20) { |
| 697 | // t = 1 - 2^k |
| 698 | t = %_ConstructDouble(0x3ff00000 - (0x200000 >> k), 0); |
| 699 | y = t - (e - x); |
| 700 | // Add k to y's exponent |
| 701 | y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y)); |
| 702 | } else { |
| 703 | // t = 2^-k |
| 704 | t = %_ConstructDouble((0x3ff - k) << 20, 0); |
| 705 | y = x - (e + t); |
| 706 | y += 1; |
| 707 | // Add k to y's exponent |
| 708 | y = %_ConstructDouble(%_DoubleHi(y) + (k << 20), %_DoubleLo(y)); |
| 709 | } |
| 710 | } |
| 711 | return y; |
| 712 | } |
| 713 | |
| 714 | |
| 715 | // ES6 draft 09-27-13, section 20.2.2.30. |
| 716 | // Math.sinh |
| 717 | // Method : |
| 718 | // mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2 |
| 719 | // 1. Replace x by |x| (sinh(-x) = -sinh(x)). |
| 720 | // 2. |
| 721 | // E + E/(E+1) |
| 722 | // 0 <= x <= 22 : sinh(x) := --------------, E=expm1(x) |
| 723 | // 2 |
| 724 | // |
| 725 | // 22 <= x <= lnovft : sinh(x) := exp(x)/2 |
| 726 | // lnovft <= x <= ln2ovft: sinh(x) := exp(x/2)/2 * exp(x/2) |
| 727 | // ln2ovft < x : sinh(x) := x*shuge (overflow) |
| 728 | // |
| 729 | // Special cases: |
| 730 | // sinh(x) is |x| if x is +Infinity, -Infinity, or NaN. |
| 731 | // only sinh(0)=0 is exact for finite x. |
| 732 | // |
| 733 | const KSINH_OVERFLOW = kMath[52]; |
| 734 | const TWO_M28 = 3.725290298461914e-9; // 2^-28, empty lower half |
| 735 | const LOG_MAXD = 709.7822265625; // 0x40862e42 00000000, empty lower half |
| 736 | |
| 737 | function MathSinh(x) { |
| 738 | x = x * 1; // Convert to number. |
| 739 | var h = (x < 0) ? -0.5 : 0.5; |
| 740 | // |x| in [0, 22]. return sign(x)*0.5*(E+E/(E+1)) |
| 741 | var ax = MathAbs(x); |
| 742 | if (ax < 22) { |
| 743 | // For |x| < 2^-28, sinh(x) = x |
| 744 | if (ax < TWO_M28) return x; |
| 745 | var t = MathExpm1(ax); |
| 746 | if (ax < 1) return h * (2 * t - t * t / (t + 1)); |
| 747 | return h * (t + t / (t + 1)); |
| 748 | } |
| 749 | // |x| in [22, log(maxdouble)], return 0.5 * exp(|x|) |
| 750 | if (ax < LOG_MAXD) return h * MathExp(ax); |
| 751 | // |x| in [log(maxdouble), overflowthreshold] |
| 752 | // overflowthreshold = 710.4758600739426 |
| 753 | if (ax <= KSINH_OVERFLOW) { |
| 754 | var w = MathExp(0.5 * ax); |
| 755 | var t = h * w; |
| 756 | return t * w; |
| 757 | } |
| 758 | // |x| > overflowthreshold or is NaN. |
| 759 | // Return Infinity of the appropriate sign or NaN. |
| 760 | return x * INFINITY; |
| 761 | } |
| 762 | |
| 763 | |
| 764 | // ES6 draft 09-27-13, section 20.2.2.12. |
| 765 | // Math.cosh |
| 766 | // Method : |
| 767 | // mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2 |
| 768 | // 1. Replace x by |x| (cosh(x) = cosh(-x)). |
| 769 | // 2. |
| 770 | // [ exp(x) - 1 ]^2 |
| 771 | // 0 <= x <= ln2/2 : cosh(x) := 1 + ------------------- |
| 772 | // 2*exp(x) |
| 773 | // |
| 774 | // exp(x) + 1/exp(x) |
| 775 | // ln2/2 <= x <= 22 : cosh(x) := ------------------- |
| 776 | // 2 |
| 777 | // 22 <= x <= lnovft : cosh(x) := exp(x)/2 |
| 778 | // lnovft <= x <= ln2ovft: cosh(x) := exp(x/2)/2 * exp(x/2) |
| 779 | // ln2ovft < x : cosh(x) := huge*huge (overflow) |
| 780 | // |
| 781 | // Special cases: |
| 782 | // cosh(x) is |x| if x is +INF, -INF, or NaN. |
| 783 | // only cosh(0)=1 is exact for finite x. |
| 784 | // |
| 785 | const KCOSH_OVERFLOW = kMath[52]; |
| 786 | |
| 787 | function MathCosh(x) { |
| 788 | x = x * 1; // Convert to number. |
| 789 | var ix = %_DoubleHi(x) & 0x7fffffff; |
| 790 | // |x| in [0,0.5*log2], return 1+expm1(|x|)^2/(2*exp(|x|)) |
| 791 | if (ix < 0x3fd62e43) { |
| 792 | var t = MathExpm1(MathAbs(x)); |
| 793 | var w = 1 + t; |
| 794 | // For |x| < 2^-55, cosh(x) = 1 |
| 795 | if (ix < 0x3c800000) return w; |
| 796 | return 1 + (t * t) / (w + w); |
| 797 | } |
| 798 | // |x| in [0.5*log2, 22], return (exp(|x|)+1/exp(|x|)/2 |
| 799 | if (ix < 0x40360000) { |
| 800 | var t = MathExp(MathAbs(x)); |
| 801 | return 0.5 * t + 0.5 / t; |
| 802 | } |
| 803 | // |x| in [22, log(maxdouble)], return half*exp(|x|) |
| 804 | if (ix < 0x40862e42) return 0.5 * MathExp(MathAbs(x)); |
| 805 | // |x| in [log(maxdouble), overflowthreshold] |
| 806 | if (MathAbs(x) <= KCOSH_OVERFLOW) { |
| 807 | var w = MathExp(0.5 * MathAbs(x)); |
| 808 | var t = 0.5 * w; |
| 809 | return t * w; |
| 810 | } |
| 811 | if (NUMBER_IS_NAN(x)) return x; |
| 812 | // |x| > overflowthreshold. |
| 813 | return INFINITY; |
| 814 | } |