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. 23var kMath; 24 25const INVPIO2 = kMath[0]; 26const PIO2_1 = kMath[1]; 27const PIO2_1T = kMath[2]; 28const PIO2_2 = kMath[3]; 29const PIO2_2T = kMath[4]; 30const PIO2_3 = kMath[5]; 31const PIO2_3T = kMath[6]; 32const PIO4 = kMath[32]; 33const 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. 38macro 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 } 110endmacro 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// 138macro KSIN(x) 139kMath[7+x] 140endmacro 141 142macro 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; 148endmacro 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// 182macro KCOS(x) 183kMath[13+x] 184endmacro 185 186macro 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 } 203endmacro 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// 241macro KTAN(x) 242kMath[19+x] 243endmacro 244 245function 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 310function 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 320function 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 332function 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 342function 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 352function 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// 425const LN2_HI = kMath[34]; 426const LN2_LO = kMath[35]; 427const TWO54 = kMath[36]; 428const TWO_THIRD = kMath[37]; 429macro KLOG1P(x) 430(kMath[38+x]) 431endmacro 432 433function 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// 610const KEXPM1_OVERFLOW = kMath[45]; 611const INVLN2 = kMath[46]; 612macro KEXPM1(x) 613(kMath[47+x]) 614endmacro 615 616function 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// 733const KSINH_OVERFLOW = kMath[52]; 734const TWO_M28 = 3.725290298461914e-9; // 2^-28, empty lower half 735const LOG_MAXD = 709.7822265625; // 0x40862e42 00000000, empty lower half 736 737function 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// 785const KCOSH_OVERFLOW = kMath[52]; 786 787function 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} 815