Line data Source code
1 : // The following is adapted from fdlibm (http://www.netlib.org/fdlibm).
2 : //
3 : // ====================================================
4 : // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 : //
6 : // Developed at SunSoft, a Sun Microsystems, Inc. business.
7 : // Permission to use, copy, modify, and distribute this
8 : // software is freely granted, provided that this notice
9 : // is preserved.
10 : // ====================================================
11 : //
12 : // The original source code covered by the above license above has been
13 : // modified significantly by Google Inc.
14 : // Copyright 2016 the V8 project authors. All rights reserved.
15 :
16 : #include "src/base/ieee754.h"
17 :
18 : #include <cmath>
19 : #include <limits>
20 :
21 : #include "src/base/build_config.h"
22 : #include "src/base/macros.h"
23 :
24 : namespace v8 {
25 : namespace base {
26 : namespace ieee754 {
27 :
28 : namespace {
29 :
30 : /* Disable "potential divide by 0" warning in Visual Studio compiler. */
31 :
32 : #if V8_CC_MSVC
33 :
34 : #pragma warning(disable : 4723)
35 :
36 : #endif
37 :
38 : /*
39 : * The original fdlibm code used statements like:
40 : * n0 = ((*(int*)&one)>>29)^1; * index of high word *
41 : * ix0 = *(n0+(int*)&x); * high word of x *
42 : * ix1 = *((1-n0)+(int*)&x); * low word of x *
43 : * to dig two 32 bit words out of the 64 bit IEEE floating point
44 : * value. That is non-ANSI, and, moreover, the gcc instruction
45 : * scheduler gets it wrong. We instead use the following macros.
46 : * Unlike the original code, we determine the endianness at compile
47 : * time, not at run time; I don't see much benefit to selecting
48 : * endianness at run time.
49 : */
50 :
51 : /*
52 : * A union which permits us to convert between a double and two 32 bit
53 : * ints.
54 : */
55 :
56 : #if V8_TARGET_LITTLE_ENDIAN
57 :
58 : typedef union {
59 : double value;
60 : struct {
61 : uint32_t lsw;
62 : uint32_t msw;
63 : } parts;
64 : struct {
65 : uint64_t w;
66 : } xparts;
67 : } ieee_double_shape_type;
68 :
69 : #else
70 :
71 : typedef union {
72 : double value;
73 : struct {
74 : uint32_t msw;
75 : uint32_t lsw;
76 : } parts;
77 : struct {
78 : uint64_t w;
79 : } xparts;
80 : } ieee_double_shape_type;
81 :
82 : #endif
83 :
84 : /* Get two 32 bit ints from a double. */
85 :
86 : #define EXTRACT_WORDS(ix0, ix1, d) \
87 : do { \
88 : ieee_double_shape_type ew_u; \
89 : ew_u.value = (d); \
90 : (ix0) = ew_u.parts.msw; \
91 : (ix1) = ew_u.parts.lsw; \
92 : } while (0)
93 :
94 : /* Get a 64-bit int from a double. */
95 : #define EXTRACT_WORD64(ix, d) \
96 : do { \
97 : ieee_double_shape_type ew_u; \
98 : ew_u.value = (d); \
99 : (ix) = ew_u.xparts.w; \
100 : } while (0)
101 :
102 : /* Get the more significant 32 bit int from a double. */
103 :
104 : #define GET_HIGH_WORD(i, d) \
105 : do { \
106 : ieee_double_shape_type gh_u; \
107 : gh_u.value = (d); \
108 : (i) = gh_u.parts.msw; \
109 : } while (0)
110 :
111 : /* Get the less significant 32 bit int from a double. */
112 :
113 : #define GET_LOW_WORD(i, d) \
114 : do { \
115 : ieee_double_shape_type gl_u; \
116 : gl_u.value = (d); \
117 : (i) = gl_u.parts.lsw; \
118 : } while (0)
119 :
120 : /* Set a double from two 32 bit ints. */
121 :
122 : #define INSERT_WORDS(d, ix0, ix1) \
123 : do { \
124 : ieee_double_shape_type iw_u; \
125 : iw_u.parts.msw = (ix0); \
126 : iw_u.parts.lsw = (ix1); \
127 : (d) = iw_u.value; \
128 : } while (0)
129 :
130 : /* Set a double from a 64-bit int. */
131 : #define INSERT_WORD64(d, ix) \
132 : do { \
133 : ieee_double_shape_type iw_u; \
134 : iw_u.xparts.w = (ix); \
135 : (d) = iw_u.value; \
136 : } while (0)
137 :
138 : /* Set the more significant 32 bits of a double from an int. */
139 :
140 : #define SET_HIGH_WORD(d, v) \
141 : do { \
142 : ieee_double_shape_type sh_u; \
143 : sh_u.value = (d); \
144 : sh_u.parts.msw = (v); \
145 : (d) = sh_u.value; \
146 : } while (0)
147 :
148 : /* Set the less significant 32 bits of a double from an int. */
149 :
150 : #define SET_LOW_WORD(d, v) \
151 : do { \
152 : ieee_double_shape_type sl_u; \
153 : sl_u.value = (d); \
154 : sl_u.parts.lsw = (v); \
155 : (d) = sl_u.value; \
156 : } while (0)
157 :
158 : /* Support macro. */
159 :
160 : #define STRICT_ASSIGN(type, lval, rval) ((lval) = (rval))
161 :
162 : int32_t __ieee754_rem_pio2(double x, double *y) WARN_UNUSED_RESULT;
163 : double __kernel_cos(double x, double y) WARN_UNUSED_RESULT;
164 : int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec,
165 : const int32_t *ipio2) WARN_UNUSED_RESULT;
166 : double __kernel_sin(double x, double y, int iy) WARN_UNUSED_RESULT;
167 :
168 : /* __ieee754_rem_pio2(x,y)
169 : *
170 : * return the remainder of x rem pi/2 in y[0]+y[1]
171 : * use __kernel_rem_pio2()
172 : */
173 9333683 : int32_t __ieee754_rem_pio2(double x, double *y) {
174 : /*
175 : * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
176 : */
177 : static const int32_t two_over_pi[] = {
178 : 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, 0x95993C,
179 : 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, 0x424DD2, 0xE00649,
180 : 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, 0xA73EE8, 0x8235F5, 0x2EBB44,
181 : 0x84E99C, 0x7026B4, 0x5F7E41, 0x3991D6, 0x398353, 0x39F49C, 0x845F8B,
182 : 0xBDF928, 0x3B1FF8, 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D,
183 : 0x367ECF, 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
184 : 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, 0x560330,
185 : 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, 0x91615E, 0xE61B08,
186 : 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, 0x4D7327, 0x310606, 0x1556CA,
187 : 0x73A8C9, 0x60E27B, 0xC08C6B,
188 : };
189 :
190 : static const int32_t npio2_hw[] = {
191 : 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
192 : 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
193 : 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
194 : 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
195 : 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
196 : 0x404858EB, 0x404921FB,
197 : };
198 :
199 : /*
200 : * invpio2: 53 bits of 2/pi
201 : * pio2_1: first 33 bit of pi/2
202 : * pio2_1t: pi/2 - pio2_1
203 : * pio2_2: second 33 bit of pi/2
204 : * pio2_2t: pi/2 - (pio2_1+pio2_2)
205 : * pio2_3: third 33 bit of pi/2
206 : * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
207 : */
208 :
209 : static const double
210 : zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
211 : half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
212 : two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
213 : invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
214 : pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
215 : pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
216 : pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
217 : pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
218 : pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
219 : pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
220 :
221 : double z, w, t, r, fn;
222 : double tx[3];
223 : int32_t e0, i, j, nx, n, ix, hx;
224 : uint32_t low;
225 :
226 : z = 0;
227 9333683 : GET_HIGH_WORD(hx, x); /* high word of x */
228 9333683 : ix = hx & 0x7fffffff;
229 9333683 : if (ix <= 0x3fe921fb) { /* |x| ~<= pi/4 , no need for reduction */
230 0 : y[0] = x;
231 0 : y[1] = 0;
232 0 : return 0;
233 : }
234 9333683 : if (ix < 0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */
235 6001594 : if (hx > 0) {
236 5996415 : z = x - pio2_1;
237 5996415 : if (ix != 0x3ff921fb) { /* 33+53 bit pi is good enough */
238 5996189 : y[0] = z - pio2_1t;
239 5996189 : y[1] = (z - y[0]) - pio2_1t;
240 : } else { /* near pi/2, use 33+33+53 bit pi */
241 226 : z -= pio2_2;
242 226 : y[0] = z - pio2_2t;
243 226 : y[1] = (z - y[0]) - pio2_2t;
244 : }
245 : return 1;
246 : } else { /* negative x */
247 5179 : z = x + pio2_1;
248 5179 : if (ix != 0x3ff921fb) { /* 33+53 bit pi is good enough */
249 5082 : y[0] = z + pio2_1t;
250 5082 : y[1] = (z - y[0]) + pio2_1t;
251 : } else { /* near pi/2, use 33+33+53 bit pi */
252 97 : z += pio2_2;
253 97 : y[0] = z + pio2_2t;
254 97 : y[1] = (z - y[0]) + pio2_2t;
255 : }
256 : return -1;
257 : }
258 : }
259 3332089 : if (ix <= 0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
260 3293146 : t = fabs(x);
261 3293146 : n = static_cast<int32_t>(t * invpio2 + half);
262 3293146 : fn = static_cast<double>(n);
263 3293146 : r = t - fn * pio2_1;
264 3293146 : w = fn * pio2_1t; /* 1st round good to 85 bit */
265 3293146 : if (n < 32 && ix != npio2_hw[n - 1]) {
266 82159 : y[0] = r - w; /* quick check no cancellation */
267 : } else {
268 : uint32_t high;
269 3210987 : j = ix >> 20;
270 3210987 : y[0] = r - w;
271 3210987 : GET_HIGH_WORD(high, y[0]);
272 3210987 : i = j - ((high >> 20) & 0x7ff);
273 3210987 : if (i > 16) { /* 2nd iteration needed, good to 118 */
274 : t = r;
275 1895991 : w = fn * pio2_2;
276 1895991 : r = t - w;
277 1895991 : w = fn * pio2_2t - ((t - r) - w);
278 1895991 : y[0] = r - w;
279 1895991 : GET_HIGH_WORD(high, y[0]);
280 1895991 : i = j - ((high >> 20) & 0x7ff);
281 1895991 : if (i > 49) { /* 3rd iteration need, 151 bits acc */
282 : t = r; /* will cover all possible cases */
283 288 : w = fn * pio2_3;
284 288 : r = t - w;
285 288 : w = fn * pio2_3t - ((t - r) - w);
286 288 : y[0] = r - w;
287 : }
288 : }
289 : }
290 3293146 : y[1] = (r - y[0]) - w;
291 3293146 : if (hx < 0) {
292 11490 : y[0] = -y[0];
293 11490 : y[1] = -y[1];
294 11490 : return -n;
295 : } else {
296 : return n;
297 : }
298 : }
299 : /*
300 : * all other (large) arguments
301 : */
302 38943 : if (ix >= 0x7ff00000) { /* x is inf or NaN */
303 0 : y[0] = y[1] = x - x;
304 0 : return 0;
305 : }
306 : /* set z = scalbn(|x|,ilogb(x)-23) */
307 38943 : GET_LOW_WORD(low, x);
308 38943 : SET_LOW_WORD(z, low);
309 38943 : e0 = (ix >> 20) - 1046; /* e0 = ilogb(z)-23; */
310 38943 : SET_HIGH_WORD(z, ix - static_cast<int32_t>(e0 << 20));
311 116829 : for (i = 0; i < 2; i++) {
312 77886 : tx[i] = static_cast<double>(static_cast<int32_t>(z));
313 77886 : z = (z - tx[i]) * two24;
314 : }
315 38943 : tx[2] = z;
316 : nx = 3;
317 38943 : while (tx[nx - 1] == zero) nx--; /* skip zero term */
318 38943 : n = __kernel_rem_pio2(tx, y, e0, nx, 2, two_over_pi);
319 38943 : if (hx < 0) {
320 9180 : y[0] = -y[0];
321 9180 : y[1] = -y[1];
322 9180 : return -n;
323 : }
324 : return n;
325 : }
326 :
327 : /* __kernel_cos( x, y )
328 : * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
329 : * Input x is assumed to be bounded by ~pi/4 in magnitude.
330 : * Input y is the tail of x.
331 : *
332 : * Algorithm
333 : * 1. Since cos(-x) = cos(x), we need only to consider positive x.
334 : * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
335 : * 3. cos(x) is approximated by a polynomial of degree 14 on
336 : * [0,pi/4]
337 : * 4 14
338 : * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
339 : * where the remez error is
340 : *
341 : * | 2 4 6 8 10 12 14 | -58
342 : * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
343 : * | |
344 : *
345 : * 4 6 8 10 12 14
346 : * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
347 : * cos(x) = 1 - x*x/2 + r
348 : * since cos(x+y) ~ cos(x) - sin(x)*y
349 : * ~ cos(x) - x*y,
350 : * a correction term is necessary in cos(x) and hence
351 : * cos(x+y) = 1 - (x*x/2 - (r - x*y))
352 : * For better accuracy when x > 0.3, let qx = |x|/4 with
353 : * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
354 : * Then
355 : * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
356 : * Note that 1-qx and (x*x/2-qx) is EXACT here, and the
357 : * magnitude of the latter is at least a quarter of x*x/2,
358 : * thus, reducing the rounding error in the subtraction.
359 : */
360 : V8_INLINE double __kernel_cos(double x, double y) {
361 : static const double
362 : one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
363 : C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
364 : C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
365 : C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
366 : C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
367 : C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
368 : C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
369 :
370 : double a, iz, z, r, qx;
371 : int32_t ix;
372 22780090 : GET_HIGH_WORD(ix, x);
373 22780090 : ix &= 0x7fffffff; /* ix = |x|'s high word*/
374 22780090 : if (ix < 0x3e400000) { /* if x < 2**27 */
375 525708 : if (static_cast<int>(x) == 0) return one; /* generate inexact */
376 : }
377 22254382 : z = x * x;
378 22254382 : r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6)))));
379 22254382 : if (ix < 0x3FD33333) { /* if |x| < 0.3 */
380 15707833 : return one - (0.5 * z - (z * r - x * y));
381 : } else {
382 6546549 : if (ix > 0x3fe90000) { /* x > 0.78125 */
383 : qx = 0.28125;
384 : } else {
385 6425336 : INSERT_WORDS(qx, ix - 0x00200000, 0); /* x/4 */
386 : }
387 6546549 : iz = 0.5 * z - qx;
388 6546549 : a = one - qx;
389 6546549 : return a - (iz - (z * r - x * y));
390 : }
391 : }
392 :
393 : /* __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
394 : * double x[],y[]; int e0,nx,prec; int ipio2[];
395 : *
396 : * __kernel_rem_pio2 return the last three digits of N with
397 : * y = x - N*pi/2
398 : * so that |y| < pi/2.
399 : *
400 : * The method is to compute the integer (mod 8) and fraction parts of
401 : * (2/pi)*x without doing the full multiplication. In general we
402 : * skip the part of the product that are known to be a huge integer (
403 : * more accurately, = 0 mod 8 ). Thus the number of operations are
404 : * independent of the exponent of the input.
405 : *
406 : * (2/pi) is represented by an array of 24-bit integers in ipio2[].
407 : *
408 : * Input parameters:
409 : * x[] The input value (must be positive) is broken into nx
410 : * pieces of 24-bit integers in double precision format.
411 : * x[i] will be the i-th 24 bit of x. The scaled exponent
412 : * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
413 : * match x's up to 24 bits.
414 : *
415 : * Example of breaking a double positive z into x[0]+x[1]+x[2]:
416 : * e0 = ilogb(z)-23
417 : * z = scalbn(z,-e0)
418 : * for i = 0,1,2
419 : * x[i] = floor(z)
420 : * z = (z-x[i])*2**24
421 : *
422 : *
423 : * y[] output result in an array of double precision numbers.
424 : * The dimension of y[] is:
425 : * 24-bit precision 1
426 : * 53-bit precision 2
427 : * 64-bit precision 2
428 : * 113-bit precision 3
429 : * The actual value is the sum of them. Thus for 113-bit
430 : * precison, one may have to do something like:
431 : *
432 : * long double t,w,r_head, r_tail;
433 : * t = (long double)y[2] + (long double)y[1];
434 : * w = (long double)y[0];
435 : * r_head = t+w;
436 : * r_tail = w - (r_head - t);
437 : *
438 : * e0 The exponent of x[0]
439 : *
440 : * nx dimension of x[]
441 : *
442 : * prec an integer indicating the precision:
443 : * 0 24 bits (single)
444 : * 1 53 bits (double)
445 : * 2 64 bits (extended)
446 : * 3 113 bits (quad)
447 : *
448 : * ipio2[]
449 : * integer array, contains the (24*i)-th to (24*i+23)-th
450 : * bit of 2/pi after binary point. The corresponding
451 : * floating value is
452 : *
453 : * ipio2[i] * 2^(-24(i+1)).
454 : *
455 : * External function:
456 : * double scalbn(), floor();
457 : *
458 : *
459 : * Here is the description of some local variables:
460 : *
461 : * jk jk+1 is the initial number of terms of ipio2[] needed
462 : * in the computation. The recommended value is 2,3,4,
463 : * 6 for single, double, extended,and quad.
464 : *
465 : * jz local integer variable indicating the number of
466 : * terms of ipio2[] used.
467 : *
468 : * jx nx - 1
469 : *
470 : * jv index for pointing to the suitable ipio2[] for the
471 : * computation. In general, we want
472 : * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
473 : * is an integer. Thus
474 : * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
475 : * Hence jv = max(0,(e0-3)/24).
476 : *
477 : * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
478 : *
479 : * q[] double array with integral value, representing the
480 : * 24-bits chunk of the product of x and 2/pi.
481 : *
482 : * q0 the corresponding exponent of q[0]. Note that the
483 : * exponent for q[i] would be q0-24*i.
484 : *
485 : * PIo2[] double precision array, obtained by cutting pi/2
486 : * into 24 bits chunks.
487 : *
488 : * f[] ipio2[] in floating point
489 : *
490 : * iq[] integer array by breaking up q[] in 24-bits chunk.
491 : *
492 : * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
493 : *
494 : * ih integer. If >0 it indicates q[] is >= 0.5, hence
495 : * it also indicates the *sign* of the result.
496 : *
497 : */
498 38943 : int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec,
499 : const int32_t *ipio2) {
500 : /* Constants:
501 : * The hexadecimal values are the intended ones for the following
502 : * constants. The decimal values may be used, provided that the
503 : * compiler will convert from decimal to binary accurately enough
504 : * to produce the hexadecimal values shown.
505 : */
506 : static const int init_jk[] = {2, 3, 4, 6}; /* initial value for jk */
507 :
508 : static const double PIo2[] = {
509 : 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
510 : 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
511 : 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
512 : 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
513 : 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
514 : 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
515 : 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
516 : 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
517 : };
518 :
519 : static const double
520 : zero = 0.0,
521 : one = 1.0,
522 : two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
523 : twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
524 :
525 : int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
526 : double z, fw, f[20], fq[20], q[20];
527 :
528 : /* initialize jk*/
529 38943 : jk = init_jk[prec];
530 : jp = jk;
531 :
532 : /* determine jx,jv,q0, note that 3>q0 */
533 38943 : jx = nx - 1;
534 38943 : jv = (e0 - 3) / 24;
535 38943 : if (jv < 0) jv = 0;
536 38943 : q0 = e0 - 24 * (jv + 1);
537 :
538 : /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
539 38943 : j = jv - jx;
540 38943 : m = jx + jk;
541 262572 : for (i = 0; i <= m; i++, j++) {
542 223629 : f[i] = (j < 0) ? zero : static_cast<double>(ipio2[j]);
543 : }
544 :
545 : /* compute q[0],q[1],...q[jk] */
546 194715 : for (i = 0; i <= jk; i++) {
547 339285 : for (j = 0, fw = 0.0; j <= jx; j++) fw += x[j] * f[jx + i - j];
548 194715 : q[i] = fw;
549 : }
550 :
551 : jz = jk;
552 : recompute:
553 : /* distill q[] into iq[] reversingly */
554 213111 : for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) {
555 171102 : fw = static_cast<double>(static_cast<int32_t>(twon24 * z));
556 171102 : iq[i] = static_cast<int32_t>(z - two24 * fw);
557 171102 : z = q[j - 1] + fw;
558 : }
559 :
560 : /* compute n */
561 42009 : z = scalbn(z, q0); /* actual value of z */
562 42009 : z -= 8.0 * floor(z * 0.125); /* trim off integer >= 8 */
563 42009 : n = static_cast<int32_t>(z);
564 42009 : z -= static_cast<double>(n);
565 : ih = 0;
566 42009 : if (q0 > 0) { /* need iq[jz-1] to determine n */
567 2664 : i = (iq[jz - 1] >> (24 - q0));
568 2664 : n += i;
569 2664 : iq[jz - 1] -= i << (24 - q0);
570 2664 : ih = iq[jz - 1] >> (23 - q0);
571 39345 : } else if (q0 == 0) {
572 1242 : ih = iq[jz - 1] >> 23;
573 38103 : } else if (z >= 0.5) {
574 : ih = 2;
575 : }
576 :
577 42009 : if (ih > 0) { /* q > 0.5 */
578 21375 : n += 1;
579 : carry = 0;
580 108309 : for (i = 0; i < jz; i++) { /* compute 1-q */
581 86934 : j = iq[i];
582 86934 : if (carry == 0) {
583 25758 : if (j != 0) {
584 : carry = 1;
585 21375 : iq[i] = 0x1000000 - j;
586 : }
587 : } else {
588 61176 : iq[i] = 0xffffff - j;
589 : }
590 : }
591 21375 : if (q0 > 0) { /* rare case: chance is 1 in 12 */
592 1302 : switch (q0) {
593 : case 1:
594 792 : iq[jz - 1] &= 0x7fffff;
595 792 : break;
596 : case 2:
597 510 : iq[jz - 1] &= 0x3fffff;
598 510 : break;
599 : }
600 : }
601 21375 : if (ih == 2) {
602 19431 : z = one - z;
603 19431 : if (carry != 0) z -= scalbn(one, q0);
604 : }
605 : }
606 :
607 : /* check if recomputation is needed */
608 42009 : if (z == zero) {
609 : j = 0;
610 6132 : for (i = jz - 1; i >= jk; i--) j |= iq[i];
611 6132 : if (j == 0) { /* need recomputation */
612 0 : for (k = 1; jk >= k && iq[jk - k] == 0; k++) {
613 : /* k = no. of terms needed */
614 : }
615 :
616 6132 : for (i = jz + 1; i <= jz + k; i++) { /* add q[jz+1] to q[jz+k] */
617 3066 : f[jx + i] = ipio2[jv + i];
618 3066 : for (j = 0, fw = 0.0; j <= jx; j++) fw += x[j] * f[jx + i - j];
619 3066 : q[i] = fw;
620 : }
621 : jz += k;
622 : goto recompute;
623 : }
624 : }
625 :
626 : /* chop off zero terms */
627 38943 : if (z == 0.0) {
628 3066 : jz -= 1;
629 3066 : q0 -= 24;
630 6132 : while (iq[jz] == 0) {
631 0 : jz--;
632 0 : q0 -= 24;
633 : }
634 : } else { /* break z into 24-bit if necessary */
635 35877 : z = scalbn(z, -q0);
636 35877 : if (z >= two24) {
637 555 : fw = static_cast<double>(static_cast<int32_t>(twon24 * z));
638 555 : iq[jz] = z - two24 * fw;
639 555 : jz += 1;
640 555 : q0 += 24;
641 555 : iq[jz] = fw;
642 : } else {
643 35322 : iq[jz] = z;
644 : }
645 : }
646 :
647 : /* convert integer "bit" chunk to floating-point value */
648 38943 : fw = scalbn(one, q0);
649 234213 : for (i = jz; i >= 0; i--) {
650 195270 : q[i] = fw * iq[i];
651 195270 : fw *= twon24;
652 : }
653 :
654 : /* compute PIo2[0,...,jp]*q[jz,...,0] */
655 195270 : for (i = jz; i >= 0; i--) {
656 586920 : for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++) fw += PIo2[k] * q[i + k];
657 195270 : fq[jz - i] = fw;
658 : }
659 :
660 : /* compress fq[] into y[] */
661 38943 : switch (prec) {
662 : case 0:
663 : fw = 0.0;
664 0 : for (i = jz; i >= 0; i--) fw += fq[i];
665 0 : y[0] = (ih == 0) ? fw : -fw;
666 0 : break;
667 : case 1:
668 : case 2:
669 : fw = 0.0;
670 195270 : for (i = jz; i >= 0; i--) fw += fq[i];
671 38943 : y[0] = (ih == 0) ? fw : -fw;
672 38943 : fw = fq[0] - fw;
673 38943 : for (i = 1; i <= jz; i++) fw += fq[i];
674 38943 : y[1] = (ih == 0) ? fw : -fw;
675 38943 : break;
676 : case 3: /* painful */
677 0 : for (i = jz; i > 0; i--) {
678 0 : fw = fq[i - 1] + fq[i];
679 0 : fq[i] += fq[i - 1] - fw;
680 0 : fq[i - 1] = fw;
681 : }
682 0 : for (i = jz; i > 1; i--) {
683 0 : fw = fq[i - 1] + fq[i];
684 0 : fq[i] += fq[i - 1] - fw;
685 0 : fq[i - 1] = fw;
686 : }
687 0 : for (fw = 0.0, i = jz; i >= 2; i--) fw += fq[i];
688 0 : if (ih == 0) {
689 0 : y[0] = fq[0];
690 0 : y[1] = fq[1];
691 0 : y[2] = fw;
692 : } else {
693 0 : y[0] = -fq[0];
694 0 : y[1] = -fq[1];
695 0 : y[2] = -fw;
696 : }
697 : }
698 38943 : return n & 7;
699 : }
700 :
701 : /* __kernel_sin( x, y, iy)
702 : * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854
703 : * Input x is assumed to be bounded by ~pi/4 in magnitude.
704 : * Input y is the tail of x.
705 : * Input iy indicates whether y is 0. (if iy=0, y assume to be 0).
706 : *
707 : * Algorithm
708 : * 1. Since sin(-x) = -sin(x), we need only to consider positive x.
709 : * 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0.
710 : * 3. sin(x) is approximated by a polynomial of degree 13 on
711 : * [0,pi/4]
712 : * 3 13
713 : * sin(x) ~ x + S1*x + ... + S6*x
714 : * where
715 : *
716 : * |sin(x) 2 4 6 8 10 12 | -58
717 : * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2
718 : * | x |
719 : *
720 : * 4. sin(x+y) = sin(x) + sin'(x')*y
721 : * ~ sin(x) + (1-x*x/2)*y
722 : * For better accuracy, let
723 : * 3 2 2 2 2
724 : * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6))))
725 : * then 3 2
726 : * sin(x) = x + (S1*x + (x *(r-y/2)+y))
727 : */
728 : V8_INLINE double __kernel_sin(double x, double y, int iy) {
729 : static const double
730 : half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
731 : S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */
732 : S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */
733 : S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */
734 : S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */
735 : S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */
736 : S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */
737 :
738 : double z, r, v;
739 : int32_t ix;
740 38756128 : GET_HIGH_WORD(ix, x);
741 38756128 : ix &= 0x7fffffff; /* high word of x */
742 38756128 : if (ix < 0x3e400000) { /* |x| < 2**-27 */
743 600787 : if (static_cast<int>(x) == 0) return x;
744 : } /* generate inexact */
745 38155341 : z = x * x;
746 38155341 : v = z * x;
747 38155341 : r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6)));
748 : if (iy == 0) {
749 37242225 : return x + v * (S1 + z * r);
750 : } else {
751 913116 : return x - ((z * (half * y - v * r) - y) - v * S1);
752 : }
753 : }
754 :
755 : /* __kernel_tan( x, y, k )
756 : * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
757 : * Input x is assumed to be bounded by ~pi/4 in magnitude.
758 : * Input y is the tail of x.
759 : * Input k indicates whether tan (if k=1) or
760 : * -1/tan (if k= -1) is returned.
761 : *
762 : * Algorithm
763 : * 1. Since tan(-x) = -tan(x), we need only to consider positive x.
764 : * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
765 : * 3. tan(x) is approximated by a odd polynomial of degree 27 on
766 : * [0,0.67434]
767 : * 3 27
768 : * tan(x) ~ x + T1*x + ... + T13*x
769 : * where
770 : *
771 : * |tan(x) 2 4 26 | -59.2
772 : * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
773 : * | x |
774 : *
775 : * Note: tan(x+y) = tan(x) + tan'(x)*y
776 : * ~ tan(x) + (1+x*x)*y
777 : * Therefore, for better accuracy in computing tan(x+y), let
778 : * 3 2 2 2 2
779 : * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
780 : * then
781 : * 3 2
782 : * tan(x+y) = x + (T1*x + (x *(r+y)+y))
783 : *
784 : * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
785 : * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
786 : * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
787 : */
788 1526857 : double __kernel_tan(double x, double y, int iy) {
789 : static const double xxx[] = {
790 : 3.33333333333334091986e-01, /* 3FD55555, 55555563 */
791 : 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
792 : 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
793 : 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
794 : 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
795 : 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
796 : 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
797 : 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
798 : 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
799 : 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
800 : 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
801 : -1.85586374855275456654e-05, /* BEF375CB, DB605373 */
802 : 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
803 : /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */
804 : /* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
805 : /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */
806 : };
807 : #define one xxx[13]
808 : #define pio4 xxx[14]
809 : #define pio4lo xxx[15]
810 : #define T xxx
811 :
812 : double z, r, v, w, s;
813 : int32_t ix, hx;
814 :
815 1526857 : GET_HIGH_WORD(hx, x); /* high word of x */
816 1526857 : ix = hx & 0x7fffffff; /* high word of |x| */
817 1526857 : if (ix < 0x3e300000) { /* x < 2**-28 */
818 2875 : if (static_cast<int>(x) == 0) { /* generate inexact */
819 : uint32_t low;
820 2875 : GET_LOW_WORD(low, x);
821 2875 : if (((ix | low) | (iy + 1)) == 0) {
822 0 : return one / fabs(x);
823 : } else {
824 2875 : if (iy == 1) {
825 : return x;
826 : } else { /* compute -1 / (x+y) carefully */
827 : double a, t;
828 :
829 116 : z = w = x + y;
830 116 : SET_LOW_WORD(z, 0);
831 116 : v = y - (z - x);
832 116 : t = a = -one / w;
833 116 : SET_LOW_WORD(t, 0);
834 116 : s = one + t * z;
835 116 : return t + a * (s + t * v);
836 : }
837 : }
838 : }
839 : }
840 1523982 : if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */
841 214692 : if (hx < 0) {
842 107806 : x = -x;
843 107806 : y = -y;
844 : }
845 214692 : z = pio4 - x;
846 214692 : w = pio4lo - y;
847 214692 : x = z + w;
848 : y = 0.0;
849 : }
850 1523982 : z = x * x;
851 1523982 : w = z * z;
852 : /*
853 : * Break x^5*(T[1]+x^2*T[2]+...) into
854 : * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
855 : * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
856 : */
857 1523982 : r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + w * T[11]))));
858 : v = z *
859 1523982 : (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + w * T[12])))));
860 1523982 : s = z * x;
861 1523982 : r = y + z * (s * (r + v) + y);
862 1523982 : r += T[0] * s;
863 1523982 : w = x + r;
864 1523982 : if (ix >= 0x3FE59428) {
865 214692 : v = iy;
866 214692 : return (1 - ((hx >> 30) & 2)) * (v - 2.0 * (x - (w * w / (w + v) - r)));
867 : }
868 1309290 : if (iy == 1) {
869 : return w;
870 : } else {
871 : /*
872 : * if allow error up to 2 ulp, simply return
873 : * -1.0 / (x+r) here
874 : */
875 : /* compute -1.0 / (x+r) accurately */
876 : double a, t;
877 : z = w;
878 654572 : SET_LOW_WORD(z, 0);
879 654572 : v = r - (z - x); /* z+v = r+x */
880 654572 : t = a = -1.0 / w; /* a = -1.0/w */
881 654572 : SET_LOW_WORD(t, 0);
882 654572 : s = 1.0 + t * z;
883 654572 : return t + a * (s + t * v);
884 : }
885 :
886 : #undef one
887 : #undef pio4
888 : #undef pio4lo
889 : #undef T
890 : }
891 :
892 : } // namespace
893 :
894 : /* acos(x)
895 : * Method :
896 : * acos(x) = pi/2 - asin(x)
897 : * acos(-x) = pi/2 + asin(x)
898 : * For |x|<=0.5
899 : * acos(x) = pi/2 - (x + x*x^2*R(x^2)) (see asin.c)
900 : * For x>0.5
901 : * acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2)))
902 : * = 2asin(sqrt((1-x)/2))
903 : * = 2s + 2s*z*R(z) ...z=(1-x)/2, s=sqrt(z)
904 : * = 2f + (2c + 2s*z*R(z))
905 : * where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term
906 : * for f so that f+c ~ sqrt(z).
907 : * For x<-0.5
908 : * acos(x) = pi - 2asin(sqrt((1-|x|)/2))
909 : * = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z)
910 : *
911 : * Special cases:
912 : * if x is NaN, return x itself;
913 : * if |x|>1, return NaN with invalid signal.
914 : *
915 : * Function needed: sqrt
916 : */
917 28966 : double acos(double x) {
918 : static const double
919 : one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
920 : pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
921 : pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
922 : pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
923 : pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
924 : pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
925 : pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
926 : pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
927 : pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
928 : pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
929 : qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
930 : qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
931 : qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
932 : qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
933 :
934 : double z, p, q, r, w, s, c, df;
935 : int32_t hx, ix;
936 28966 : GET_HIGH_WORD(hx, x);
937 28966 : ix = hx & 0x7fffffff;
938 28966 : if (ix >= 0x3ff00000) { /* |x| >= 1 */
939 : uint32_t lx;
940 20239 : GET_LOW_WORD(lx, x);
941 20239 : if (((ix - 0x3ff00000) | lx) == 0) { /* |x|==1 */
942 1373 : if (hx > 0)
943 : return 0.0; /* acos(1) = 0 */
944 : else
945 672 : return pi + 2.0 * pio2_lo; /* acos(-1)= pi */
946 : }
947 18866 : return (x - x) / (x - x); /* acos(|x|>1) is NaN */
948 : }
949 8727 : if (ix < 0x3fe00000) { /* |x| < 0.5 */
950 6965 : if (ix <= 0x3c600000) return pio2_hi + pio2_lo; /*if|x|<2**-57*/
951 5233 : z = x * x;
952 5233 : p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
953 5233 : q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
954 5233 : r = p / q;
955 5233 : return pio2_hi - (x - (pio2_lo - x * r));
956 1762 : } else if (hx < 0) { /* x < -0.5 */
957 852 : z = (one + x) * 0.5;
958 852 : p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
959 852 : q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
960 852 : s = sqrt(z);
961 852 : r = p / q;
962 852 : w = r * s - pio2_lo;
963 852 : return pi - 2.0 * (s + w);
964 : } else { /* x > 0.5 */
965 910 : z = (one - x) * 0.5;
966 910 : s = sqrt(z);
967 : df = s;
968 910 : SET_LOW_WORD(df, 0);
969 910 : c = (z - df * df) / (s + df);
970 910 : p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
971 910 : q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
972 910 : r = p / q;
973 910 : w = r * s + c;
974 910 : return 2.0 * (df + w);
975 : }
976 : }
977 :
978 : /* acosh(x)
979 : * Method :
980 : * Based on
981 : * acosh(x) = log [ x + sqrt(x*x-1) ]
982 : * we have
983 : * acosh(x) := log(x)+ln2, if x is large; else
984 : * acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else
985 : * acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1.
986 : *
987 : * Special cases:
988 : * acosh(x) is NaN with signal if x<1.
989 : * acosh(NaN) is NaN without signal.
990 : */
991 1246 : double acosh(double x) {
992 : static const double
993 : one = 1.0,
994 : ln2 = 6.93147180559945286227e-01; /* 0x3FE62E42, 0xFEFA39EF */
995 : double t;
996 : int32_t hx;
997 : uint32_t lx;
998 1246 : EXTRACT_WORDS(hx, lx, x);
999 1246 : if (hx < 0x3ff00000) { /* x < 1 */
1000 642 : return (x - x) / (x - x);
1001 604 : } else if (hx >= 0x41b00000) { /* x > 2**28 */
1002 265 : if (hx >= 0x7ff00000) { /* x is inf of NaN */
1003 99 : return x + x;
1004 : } else {
1005 166 : return log(x) + ln2; /* acosh(huge)=log(2x) */
1006 : }
1007 339 : } else if (((hx - 0x3ff00000) | lx) == 0) {
1008 : return 0.0; /* acosh(1) = 0 */
1009 312 : } else if (hx > 0x40000000) { /* 2**28 > x > 2 */
1010 163 : t = x * x;
1011 163 : return log(2.0 * x - one / (x + sqrt(t - one)));
1012 : } else { /* 1<x<2 */
1013 149 : t = x - one;
1014 149 : return log1p(t + sqrt(2.0 * t + t * t));
1015 : }
1016 : }
1017 :
1018 : /* asin(x)
1019 : * Method :
1020 : * Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
1021 : * we approximate asin(x) on [0,0.5] by
1022 : * asin(x) = x + x*x^2*R(x^2)
1023 : * where
1024 : * R(x^2) is a rational approximation of (asin(x)-x)/x^3
1025 : * and its remez error is bounded by
1026 : * |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
1027 : *
1028 : * For x in [0.5,1]
1029 : * asin(x) = pi/2-2*asin(sqrt((1-x)/2))
1030 : * Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
1031 : * then for x>0.98
1032 : * asin(x) = pi/2 - 2*(s+s*z*R(z))
1033 : * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
1034 : * For x<=0.98, let pio4_hi = pio2_hi/2, then
1035 : * f = hi part of s;
1036 : * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
1037 : * and
1038 : * asin(x) = pi/2 - 2*(s+s*z*R(z))
1039 : * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
1040 : * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
1041 : *
1042 : * Special cases:
1043 : * if x is NaN, return x itself;
1044 : * if |x|>1, return NaN with invalid signal.
1045 : */
1046 28955 : double asin(double x) {
1047 : static const double
1048 : one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
1049 : huge = 1.000e+300,
1050 : pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
1051 : pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
1052 : pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
1053 : /* coefficient for R(x^2) */
1054 : pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
1055 : pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
1056 : pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
1057 : pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
1058 : pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
1059 : pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
1060 : qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
1061 : qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
1062 : qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
1063 : qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
1064 :
1065 : double t, w, p, q, c, r, s;
1066 : int32_t hx, ix;
1067 :
1068 : t = 0;
1069 28955 : GET_HIGH_WORD(hx, x);
1070 28955 : ix = hx & 0x7fffffff;
1071 28955 : if (ix >= 0x3ff00000) { /* |x|>= 1 */
1072 : uint32_t lx;
1073 20240 : GET_LOW_WORD(lx, x);
1074 20240 : if (((ix - 0x3ff00000) | lx) == 0) /* asin(1)=+-pi/2 with inexact */
1075 1372 : return x * pio2_hi + x * pio2_lo;
1076 18868 : return (x - x) / (x - x); /* asin(|x|>1) is NaN */
1077 8715 : } else if (ix < 0x3fe00000) { /* |x|<0.5 */
1078 6953 : if (ix < 0x3e400000) { /* if |x| < 2**-27 */
1079 2636 : if (huge + x > one) return x; /* return x with inexact if x!=0*/
1080 : } else {
1081 4317 : t = x * x;
1082 : }
1083 4317 : p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
1084 4317 : q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
1085 4317 : w = p / q;
1086 4317 : return x + x * w;
1087 : }
1088 : /* 1> |x|>= 0.5 */
1089 1762 : w = one - fabs(x);
1090 1762 : t = w * 0.5;
1091 1762 : p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
1092 1762 : q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
1093 1762 : s = sqrt(t);
1094 1762 : if (ix >= 0x3FEF3333) { /* if |x| > 0.975 */
1095 120 : w = p / q;
1096 120 : t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
1097 : } else {
1098 : w = s;
1099 1642 : SET_LOW_WORD(w, 0);
1100 1642 : c = (t - w * w) / (s + w);
1101 1642 : r = p / q;
1102 1642 : p = 2.0 * s * r - (pio2_lo - 2.0 * c);
1103 1642 : q = pio4_hi - 2.0 * w;
1104 1642 : t = pio4_hi - (p - q);
1105 : }
1106 1762 : if (hx > 0)
1107 : return t;
1108 : else
1109 852 : return -t;
1110 : }
1111 : /* asinh(x)
1112 : * Method :
1113 : * Based on
1114 : * asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
1115 : * we have
1116 : * asinh(x) := x if 1+x*x=1,
1117 : * := sign(x)*(log(x)+ln2)) for large |x|, else
1118 : * := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
1119 : * := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2)))
1120 : */
1121 1504 : double asinh(double x) {
1122 : static const double
1123 : one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
1124 : ln2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
1125 : huge = 1.00000000000000000000e+300;
1126 :
1127 : double t, w;
1128 : int32_t hx, ix;
1129 1504 : GET_HIGH_WORD(hx, x);
1130 1504 : ix = hx & 0x7fffffff;
1131 1504 : if (ix >= 0x7ff00000) return x + x; /* x is inf or NaN */
1132 1338 : if (ix < 0x3e300000) { /* |x|<2**-28 */
1133 290 : if (huge + x > one) return x; /* return x inexact except 0 */
1134 : }
1135 1048 : if (ix > 0x41b00000) { /* |x| > 2**28 */
1136 430 : w = log(fabs(x)) + ln2;
1137 618 : } else if (ix > 0x40000000) { /* 2**28 > |x| > 2.0 */
1138 282 : t = fabs(x);
1139 282 : w = log(2.0 * t + one / (sqrt(x * x + one) + t));
1140 : } else { /* 2.0 > |x| > 2**-28 */
1141 336 : t = x * x;
1142 336 : w = log1p(fabs(x) + t / (one + sqrt(one + t)));
1143 : }
1144 1048 : if (hx > 0) {
1145 : return w;
1146 : } else {
1147 451 : return -w;
1148 : }
1149 : }
1150 :
1151 : /* atan(x)
1152 : * Method
1153 : * 1. Reduce x to positive by atan(x) = -atan(-x).
1154 : * 2. According to the integer k=4t+0.25 chopped, t=x, the argument
1155 : * is further reduced to one of the following intervals and the
1156 : * arctangent of t is evaluated by the corresponding formula:
1157 : *
1158 : * [0,7/16] atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
1159 : * [7/16,11/16] atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
1160 : * [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
1161 : * [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
1162 : * [39/16,INF] atan(x) = atan(INF) + atan( -1/t )
1163 : *
1164 : * Constants:
1165 : * The hexadecimal values are the intended ones for the following
1166 : * constants. The decimal values may be used, provided that the
1167 : * compiler will convert from decimal to binary accurately enough
1168 : * to produce the hexadecimal values shown.
1169 : */
1170 1234342 : double atan(double x) {
1171 : static const double atanhi[] = {
1172 : 4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
1173 : 7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
1174 : 9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
1175 : 1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
1176 : };
1177 :
1178 : static const double atanlo[] = {
1179 : 2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
1180 : 3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
1181 : 1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
1182 : 6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
1183 : };
1184 :
1185 : static const double aT[] = {
1186 : 3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */
1187 : -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
1188 : 1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */
1189 : -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
1190 : 9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */
1191 : -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
1192 : 6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */
1193 : -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
1194 : 4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */
1195 : -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
1196 : 1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */
1197 : };
1198 :
1199 : static const double one = 1.0, huge = 1.0e300;
1200 :
1201 : double w, s1, s2, z;
1202 : int32_t ix, hx, id;
1203 :
1204 1234342 : GET_HIGH_WORD(hx, x);
1205 1234342 : ix = hx & 0x7fffffff;
1206 1234342 : if (ix >= 0x44100000) { /* if |x| >= 2^66 */
1207 : uint32_t low;
1208 2968 : GET_LOW_WORD(low, x);
1209 2968 : if (ix > 0x7ff00000 || (ix == 0x7ff00000 && (low != 0)))
1210 832 : return x + x; /* NaN */
1211 2136 : if (hx > 0)
1212 1063 : return atanhi[3] + *(volatile double *)&atanlo[3];
1213 : else
1214 1073 : return -atanhi[3] - *(volatile double *)&atanlo[3];
1215 : }
1216 1231374 : if (ix < 0x3fdc0000) { /* |x| < 0.4375 */
1217 598869 : if (ix < 0x3e400000) { /* |x| < 2^-27 */
1218 171378 : if (huge + x > one) return x; /* raise inexact */
1219 : }
1220 : id = -1;
1221 : } else {
1222 632505 : x = fabs(x);
1223 632505 : if (ix < 0x3ff30000) { /* |x| < 1.1875 */
1224 59727 : if (ix < 0x3fe60000) { /* 7/16 <=|x|<11/16 */
1225 : id = 0;
1226 20186 : x = (2.0 * x - one) / (2.0 + x);
1227 : } else { /* 11/16<=|x|< 19/16 */
1228 : id = 1;
1229 39541 : x = (x - one) / (x + one);
1230 : }
1231 : } else {
1232 572778 : if (ix < 0x40038000) { /* |x| < 2.4375 */
1233 : id = 2;
1234 25328 : x = (x - 1.5) / (one + 1.5 * x);
1235 : } else { /* 2.4375 <= |x| < 2^66 */
1236 : id = 3;
1237 547450 : x = -1.0 / x;
1238 : }
1239 : }
1240 : }
1241 : /* end of argument reduction */
1242 1059996 : z = x * x;
1243 1059996 : w = z * z;
1244 : /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
1245 1059996 : s1 = z * (aT[0] +
1246 2119992 : w * (aT[2] + w * (aT[4] + w * (aT[6] + w * (aT[8] + w * aT[10])))));
1247 1059996 : s2 = w * (aT[1] + w * (aT[3] + w * (aT[5] + w * (aT[7] + w * aT[9]))));
1248 1059996 : if (id < 0) {
1249 427491 : return x - x * (s1 + s2);
1250 : } else {
1251 632505 : z = atanhi[id] - ((x * (s1 + s2) - atanlo[id]) - x);
1252 632505 : return (hx < 0) ? -z : z;
1253 : }
1254 : }
1255 :
1256 : /* atan2(y,x)
1257 : * Method :
1258 : * 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
1259 : * 2. Reduce x to positive by (if x and y are unexceptional):
1260 : * ARG (x+iy) = arctan(y/x) ... if x > 0,
1261 : * ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0,
1262 : *
1263 : * Special cases:
1264 : *
1265 : * ATAN2((anything), NaN ) is NaN;
1266 : * ATAN2(NAN , (anything) ) is NaN;
1267 : * ATAN2(+-0, +(anything but NaN)) is +-0 ;
1268 : * ATAN2(+-0, -(anything but NaN)) is +-pi ;
1269 : * ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2;
1270 : * ATAN2(+-(anything but INF and NaN), +INF) is +-0 ;
1271 : * ATAN2(+-(anything but INF and NaN), -INF) is +-pi;
1272 : * ATAN2(+-INF,+INF ) is +-pi/4 ;
1273 : * ATAN2(+-INF,-INF ) is +-3pi/4;
1274 : * ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2;
1275 : *
1276 : * Constants:
1277 : * The hexadecimal values are the intended ones for the following
1278 : * constants. The decimal values may be used, provided that the
1279 : * compiler will convert from decimal to binary accurately enough
1280 : * to produce the hexadecimal values shown.
1281 : */
1282 1435367 : double atan2(double y, double x) {
1283 : static volatile double tiny = 1.0e-300;
1284 : static const double
1285 : zero = 0.0,
1286 : pi_o_4 = 7.8539816339744827900E-01, /* 0x3FE921FB, 0x54442D18 */
1287 : pi_o_2 = 1.5707963267948965580E+00, /* 0x3FF921FB, 0x54442D18 */
1288 : pi = 3.1415926535897931160E+00; /* 0x400921FB, 0x54442D18 */
1289 : static volatile double pi_lo =
1290 : 1.2246467991473531772E-16; /* 0x3CA1A626, 0x33145C07 */
1291 :
1292 : double z;
1293 : int32_t k, m, hx, hy, ix, iy;
1294 : uint32_t lx, ly;
1295 :
1296 1435367 : EXTRACT_WORDS(hx, lx, x);
1297 1435367 : ix = hx & 0x7fffffff;
1298 1435367 : EXTRACT_WORDS(hy, ly, y);
1299 1435367 : iy = hy & 0x7fffffff;
1300 2851230 : if (((ix | ((lx | -static_cast<int32_t>(lx)) >> 31)) > 0x7ff00000) ||
1301 1415863 : ((iy | ((ly | -static_cast<int32_t>(ly)) >> 31)) > 0x7ff00000)) {
1302 38702 : return x + y; /* x or y is NaN */
1303 : }
1304 1396665 : if (((hx - 0x3ff00000) | lx) == 0) return atan(y); /* x=1.0 */
1305 1382601 : m = ((hy >> 31) & 1) | ((hx >> 30) & 2); /* 2*sign(x)+sign(y) */
1306 :
1307 : /* when y = 0 */
1308 1382601 : if ((iy | ly) == 0) {
1309 37144 : switch (m) {
1310 : case 0:
1311 : case 1:
1312 : return y; /* atan(+-0,+anything)=+-0 */
1313 : case 2:
1314 5834 : return pi + tiny; /* atan(+0,-anything) = pi */
1315 : case 3:
1316 5834 : return -pi - tiny; /* atan(-0,-anything) =-pi */
1317 : }
1318 : }
1319 : /* when x = 0 */
1320 1345457 : if ((ix | lx) == 0) return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny;
1321 :
1322 : /* when x is INF */
1323 1308945 : if (ix == 0x7ff00000) {
1324 36590 : if (iy == 0x7ff00000) {
1325 1080 : switch (m) {
1326 : case 0:
1327 270 : return pi_o_4 + tiny; /* atan(+INF,+INF) */
1328 : case 1:
1329 270 : return -pi_o_4 - tiny; /* atan(-INF,+INF) */
1330 : case 2:
1331 270 : return 3.0 * pi_o_4 + tiny; /*atan(+INF,-INF)*/
1332 : case 3:
1333 270 : return -3.0 * pi_o_4 - tiny; /*atan(-INF,-INF)*/
1334 : }
1335 : } else {
1336 35510 : switch (m) {
1337 : case 0:
1338 : return zero; /* atan(+...,+INF) */
1339 : case 1:
1340 7080 : return -zero; /* atan(-...,+INF) */
1341 : case 2:
1342 10682 : return pi + tiny; /* atan(+...,-INF) */
1343 : case 3:
1344 7066 : return -pi - tiny; /* atan(-...,-INF) */
1345 : }
1346 : }
1347 : }
1348 : /* when y is INF */
1349 1272355 : if (iy == 0x7ff00000) return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny;
1350 :
1351 : /* compute y/x */
1352 1237243 : k = (iy - ix) >> 20;
1353 1237243 : if (k > 60) { /* |y/x| > 2**60 */
1354 35162 : z = pi_o_2 + 0.5 * pi_lo;
1355 35162 : m &= 1;
1356 1202081 : } else if (hx < 0 && k < -60) {
1357 : z = 0.0; /* 0 > |y|/x > -2**-60 */
1358 : } else {
1359 1191365 : z = atan(fabs(y / x)); /* safe to do y/x */
1360 : }
1361 1237243 : switch (m) {
1362 : case 0:
1363 : return z; /* atan(+,+) */
1364 : case 1:
1365 345412 : return -z; /* atan(-,+) */
1366 : case 2:
1367 222426 : return pi - (z - pi_lo); /* atan(+,-) */
1368 : default: /* case 3 */
1369 147536 : return (z - pi_lo) - pi; /* atan(-,-) */
1370 : }
1371 : }
1372 :
1373 : /* cos(x)
1374 : * Return cosine function of x.
1375 : *
1376 : * kernel function:
1377 : * __kernel_sin ... sine function on [-pi/4,pi/4]
1378 : * __kernel_cos ... cosine function on [-pi/4,pi/4]
1379 : * __ieee754_rem_pio2 ... argument reduction routine
1380 : *
1381 : * Method.
1382 : * Let S,C and T denote the sin, cos and tan respectively on
1383 : * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
1384 : * in [-pi/4 , +pi/4], and let n = k mod 4.
1385 : * We have
1386 : *
1387 : * n sin(x) cos(x) tan(x)
1388 : * ----------------------------------------------------------
1389 : * 0 S C T
1390 : * 1 C -S -1/T
1391 : * 2 -S -C T
1392 : * 3 -C S -1/T
1393 : * ----------------------------------------------------------
1394 : *
1395 : * Special cases:
1396 : * Let trig be any of sin, cos, or tan.
1397 : * trig(+-INF) is NaN, with signals;
1398 : * trig(NaN) is that NaN;
1399 : *
1400 : * Accuracy:
1401 : * TRIG(x) returns trig(x) nearly rounded
1402 : */
1403 15908749 : double cos(double x) {
1404 : double y[2], z = 0.0;
1405 : int32_t n, ix;
1406 :
1407 : /* High word of x. */
1408 15908749 : GET_HIGH_WORD(ix, x);
1409 :
1410 : /* |x| ~< pi/4 */
1411 15908749 : ix &= 0x7fffffff;
1412 15908749 : if (ix <= 0x3fe921fb) {
1413 15879289 : return __kernel_cos(x, z);
1414 29460 : } else if (ix >= 0x7ff00000) {
1415 : /* cos(Inf or NaN) is NaN */
1416 2480 : return x - x;
1417 : } else {
1418 : /* argument reduction needed */
1419 26980 : n = __ieee754_rem_pio2(x, y);
1420 26980 : switch (n & 3) {
1421 : case 0:
1422 19956 : return __kernel_cos(y[0], y[1]);
1423 : case 1:
1424 13558 : return -__kernel_sin(y[0], y[1], 1);
1425 : case 2:
1426 7604 : return -__kernel_cos(y[0], y[1]);
1427 : default:
1428 12842 : return __kernel_sin(y[0], y[1], 1);
1429 : }
1430 : }
1431 : }
1432 :
1433 : /* exp(x)
1434 : * Returns the exponential of x.
1435 : *
1436 : * Method
1437 : * 1. Argument reduction:
1438 : * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
1439 : * Given x, find r and integer k such that
1440 : *
1441 : * x = k*ln2 + r, |r| <= 0.5*ln2.
1442 : *
1443 : * Here r will be represented as r = hi-lo for better
1444 : * accuracy.
1445 : *
1446 : * 2. Approximation of exp(r) by a special rational function on
1447 : * the interval [0,0.34658]:
1448 : * Write
1449 : * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
1450 : * We use a special Remes algorithm on [0,0.34658] to generate
1451 : * a polynomial of degree 5 to approximate R. The maximum error
1452 : * of this polynomial approximation is bounded by 2**-59. In
1453 : * other words,
1454 : * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
1455 : * (where z=r*r, and the values of P1 to P5 are listed below)
1456 : * and
1457 : * | 5 | -59
1458 : * | 2.0+P1*z+...+P5*z - R(z) | <= 2
1459 : * | |
1460 : * The computation of exp(r) thus becomes
1461 : * 2*r
1462 : * exp(r) = 1 + -------
1463 : * R - r
1464 : * r*R1(r)
1465 : * = 1 + r + ----------- (for better accuracy)
1466 : * 2 - R1(r)
1467 : * where
1468 : * 2 4 10
1469 : * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
1470 : *
1471 : * 3. Scale back to obtain exp(x):
1472 : * From step 1, we have
1473 : * exp(x) = 2^k * exp(r)
1474 : *
1475 : * Special cases:
1476 : * exp(INF) is INF, exp(NaN) is NaN;
1477 : * exp(-INF) is 0, and
1478 : * for finite argument, only exp(0)=1 is exact.
1479 : *
1480 : * Accuracy:
1481 : * according to an error analysis, the error is always less than
1482 : * 1 ulp (unit in the last place).
1483 : *
1484 : * Misc. info.
1485 : * For IEEE double
1486 : * if x > 7.09782712893383973096e+02 then exp(x) overflow
1487 : * if x < -7.45133219101941108420e+02 then exp(x) underflow
1488 : *
1489 : * Constants:
1490 : * The hexadecimal values are the intended ones for the following
1491 : * constants. The decimal values may be used, provided that the
1492 : * compiler will convert from decimal to binary accurately enough
1493 : * to produce the hexadecimal values shown.
1494 : */
1495 329321 : double exp(double x) {
1496 : static const double
1497 : one = 1.0,
1498 : halF[2] = {0.5, -0.5},
1499 : o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
1500 : u_threshold = -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
1501 : ln2HI[2] = {6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
1502 : -6.93147180369123816490e-01}, /* 0xbfe62e42, 0xfee00000 */
1503 : ln2LO[2] = {1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
1504 : -1.90821492927058770002e-10}, /* 0xbdea39ef, 0x35793c76 */
1505 : invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
1506 : P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
1507 : P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
1508 : P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
1509 : P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
1510 : P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
1511 : E = 2.718281828459045; /* 0x4005bf0a, 0x8b145769 */
1512 :
1513 : static volatile double
1514 : huge = 1.0e+300,
1515 : twom1000 = 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
1516 : two1023 = 8.988465674311579539e307; /* 0x1p1023 */
1517 :
1518 : double y, hi = 0.0, lo = 0.0, c, t, twopk;
1519 : int32_t k = 0, xsb;
1520 : uint32_t hx;
1521 :
1522 329321 : GET_HIGH_WORD(hx, x);
1523 329321 : xsb = (hx >> 31) & 1; /* sign bit of x */
1524 329321 : hx &= 0x7fffffff; /* high word of |x| */
1525 :
1526 : /* filter out non-finite argument */
1527 329321 : if (hx >= 0x40862E42) { /* if |x|>=709.78... */
1528 14490 : if (hx >= 0x7ff00000) {
1529 : uint32_t lx;
1530 2530 : GET_LOW_WORD(lx, x);
1531 2530 : if (((hx & 0xfffff) | lx) != 0)
1532 866 : return x + x; /* NaN */
1533 : else
1534 1664 : return (xsb == 0) ? x : 0.0; /* exp(+-inf)={inf,0} */
1535 : }
1536 11960 : if (x > o_threshold) return huge * huge; /* overflow */
1537 4679 : if (x < u_threshold) return twom1000 * twom1000; /* underflow */
1538 : }
1539 :
1540 : /* argument reduction */
1541 314834 : if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
1542 307877 : if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
1543 : /* TODO(rtoy): We special case exp(1) here to return the correct
1544 : * value of E, as the computation below would get the last bit
1545 : * wrong. We should probably fix the algorithm instead.
1546 : */
1547 3765 : if (x == 1.0) return E;
1548 2909 : hi = x - ln2HI[xsb];
1549 2909 : lo = ln2LO[xsb];
1550 2909 : k = 1 - xsb - xsb;
1551 : } else {
1552 304112 : k = static_cast<int>(invln2 * x + halF[xsb]);
1553 304112 : t = k;
1554 304112 : hi = x - t * ln2HI[0]; /* t*ln2HI is exact here */
1555 304112 : lo = t * ln2LO[0];
1556 : }
1557 307021 : STRICT_ASSIGN(double, x, hi - lo);
1558 6957 : } else if (hx < 0x3e300000) { /* when |x|<2**-28 */
1559 2668 : if (huge + x > one) return one + x; /* trigger inexact */
1560 : } else {
1561 : k = 0;
1562 : }
1563 :
1564 : /* x is now in primary range */
1565 311310 : t = x * x;
1566 311310 : if (k >= -1021) {
1567 311308 : INSERT_WORDS(twopk, 0x3ff00000 + (k << 20), 0);
1568 : } else {
1569 2 : INSERT_WORDS(twopk, 0x3ff00000 + ((k + 1000) << 20), 0);
1570 : }
1571 311310 : c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
1572 311310 : if (k == 0) {
1573 4289 : return one - ((x * c) / (c - 2.0) - x);
1574 : } else {
1575 307021 : y = one - ((lo - (x * c) / (2.0 - c)) - hi);
1576 : }
1577 307021 : if (k >= -1021) {
1578 307019 : if (k == 1024) return y * 2.0 * two1023;
1579 306961 : return y * twopk;
1580 : } else {
1581 2 : return y * twopk * twom1000;
1582 : }
1583 : }
1584 :
1585 : /*
1586 : * Method :
1587 : * 1.Reduced x to positive by atanh(-x) = -atanh(x)
1588 : * 2.For x>=0.5
1589 : * 1 2x x
1590 : * atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------)
1591 : * 2 1 - x 1 - x
1592 : *
1593 : * For x<0.5
1594 : * atanh(x) = 0.5*log1p(2x+2x*x/(1-x))
1595 : *
1596 : * Special cases:
1597 : * atanh(x) is NaN if |x| > 1 with signal;
1598 : * atanh(NaN) is that NaN with no signal;
1599 : * atanh(+-1) is +-INF with signal.
1600 : *
1601 : */
1602 1586 : double atanh(double x) {
1603 : static const double one = 1.0, huge = 1e300;
1604 : static const double zero = 0.0;
1605 :
1606 : double t;
1607 : int32_t hx, ix;
1608 : uint32_t lx;
1609 1586 : EXTRACT_WORDS(hx, lx, x);
1610 1586 : ix = hx & 0x7fffffff;
1611 1586 : if ((ix | ((lx | -static_cast<int32_t>(lx)) >> 31)) > 0x3ff00000) /* |x|>1 */
1612 903 : return (x - x) / (x - x);
1613 683 : if (ix == 0x3ff00000) return x / zero;
1614 617 : if (ix < 0x3e300000 && (huge + x) > zero) return x; /* x<2**-28 */
1615 319 : SET_HIGH_WORD(x, ix);
1616 319 : if (ix < 0x3fe00000) { /* x < 0.5 */
1617 126 : t = x + x;
1618 126 : t = 0.5 * log1p(t + t * x / (one - x));
1619 : } else {
1620 193 : t = 0.5 * log1p((x + x) / (one - x));
1621 : }
1622 319 : if (hx >= 0)
1623 : return t;
1624 : else
1625 128 : return -t;
1626 : }
1627 :
1628 : /* log(x)
1629 : * Return the logrithm of x
1630 : *
1631 : * Method :
1632 : * 1. Argument Reduction: find k and f such that
1633 : * x = 2^k * (1+f),
1634 : * where sqrt(2)/2 < 1+f < sqrt(2) .
1635 : *
1636 : * 2. Approximation of log(1+f).
1637 : * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
1638 : * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
1639 : * = 2s + s*R
1640 : * We use a special Reme algorithm on [0,0.1716] to generate
1641 : * a polynomial of degree 14 to approximate R The maximum error
1642 : * of this polynomial approximation is bounded by 2**-58.45. In
1643 : * other words,
1644 : * 2 4 6 8 10 12 14
1645 : * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
1646 : * (the values of Lg1 to Lg7 are listed in the program)
1647 : * and
1648 : * | 2 14 | -58.45
1649 : * | Lg1*s +...+Lg7*s - R(z) | <= 2
1650 : * | |
1651 : * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
1652 : * In order to guarantee error in log below 1ulp, we compute log
1653 : * by
1654 : * log(1+f) = f - s*(f - R) (if f is not too large)
1655 : * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
1656 : *
1657 : * 3. Finally, log(x) = k*ln2 + log(1+f).
1658 : * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
1659 : * Here ln2 is split into two floating point number:
1660 : * ln2_hi + ln2_lo,
1661 : * where n*ln2_hi is always exact for |n| < 2000.
1662 : *
1663 : * Special cases:
1664 : * log(x) is NaN with signal if x < 0 (including -INF) ;
1665 : * log(+INF) is +INF; log(0) is -INF with signal;
1666 : * log(NaN) is that NaN with no signal.
1667 : *
1668 : * Accuracy:
1669 : * according to an error analysis, the error is always less than
1670 : * 1 ulp (unit in the last place).
1671 : *
1672 : * Constants:
1673 : * The hexadecimal values are the intended ones for the following
1674 : * constants. The decimal values may be used, provided that the
1675 : * compiler will convert from decimal to binary accurately enough
1676 : * to produce the hexadecimal values shown.
1677 : */
1678 1265373 : double log(double x) {
1679 : static const double /* -- */
1680 : ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
1681 : ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
1682 : two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
1683 : Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
1684 : Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
1685 : Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
1686 : Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
1687 : Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
1688 : Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
1689 : Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
1690 :
1691 : static const double zero = 0.0;
1692 : static volatile double vzero = 0.0;
1693 :
1694 : double hfsq, f, s, z, R, w, t1, t2, dk;
1695 : int32_t k, hx, i, j;
1696 : uint32_t lx;
1697 :
1698 1265373 : EXTRACT_WORDS(hx, lx, x);
1699 :
1700 : k = 0;
1701 1265373 : if (hx < 0x00100000) { /* x < 2**-1022 */
1702 43843 : if (((hx & 0x7fffffff) | lx) == 0)
1703 18197 : return -two54 / vzero; /* log(+-0)=-inf */
1704 25646 : if (hx < 0) return (x - x) / zero; /* log(-#) = NaN */
1705 : k -= 54;
1706 2 : x *= two54; /* subnormal number, scale up x */
1707 2 : GET_HIGH_WORD(hx, x);
1708 : }
1709 1221532 : if (hx >= 0x7ff00000) return x + x;
1710 1204861 : k += (hx >> 20) - 1023;
1711 1204861 : hx &= 0x000fffff;
1712 1204861 : i = (hx + 0x95f64) & 0x100000;
1713 1204861 : SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */
1714 1204861 : k += (i >> 20);
1715 1204861 : f = x - 1.0;
1716 1204861 : if ((0x000fffff & (2 + hx)) < 3) { /* -2**-20 <= f < 2**-20 */
1717 8921 : if (f == zero) {
1718 7426 : if (k == 0) {
1719 : return zero;
1720 : } else {
1721 6442 : dk = static_cast<double>(k);
1722 6442 : return dk * ln2_hi + dk * ln2_lo;
1723 : }
1724 : }
1725 1495 : R = f * f * (0.5 - 0.33333333333333333 * f);
1726 1495 : if (k == 0) {
1727 74 : return f - R;
1728 : } else {
1729 1421 : dk = static_cast<double>(k);
1730 1421 : return dk * ln2_hi - ((R - dk * ln2_lo) - f);
1731 : }
1732 : }
1733 1195940 : s = f / (2.0 + f);
1734 1195940 : dk = static_cast<double>(k);
1735 1195940 : z = s * s;
1736 1195940 : i = hx - 0x6147a;
1737 1195940 : w = z * z;
1738 1195940 : j = 0x6b851 - hx;
1739 1195940 : t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
1740 1195940 : t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
1741 1195940 : i |= j;
1742 1195940 : R = t2 + t1;
1743 1195940 : if (i > 0) {
1744 48268 : hfsq = 0.5 * f * f;
1745 48268 : if (k == 0)
1746 904 : return f - (hfsq - s * (hfsq + R));
1747 : else
1748 47364 : return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f);
1749 : } else {
1750 1147672 : if (k == 0)
1751 9360 : return f - s * (f - R);
1752 : else
1753 1138312 : return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
1754 : }
1755 : }
1756 :
1757 : /* double log1p(double x)
1758 : *
1759 : * Method :
1760 : * 1. Argument Reduction: find k and f such that
1761 : * 1+x = 2^k * (1+f),
1762 : * where sqrt(2)/2 < 1+f < sqrt(2) .
1763 : *
1764 : * Note. If k=0, then f=x is exact. However, if k!=0, then f
1765 : * may not be representable exactly. In that case, a correction
1766 : * term is need. Let u=1+x rounded. Let c = (1+x)-u, then
1767 : * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
1768 : * and add back the correction term c/u.
1769 : * (Note: when x > 2**53, one can simply return log(x))
1770 : *
1771 : * 2. Approximation of log1p(f).
1772 : * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
1773 : * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
1774 : * = 2s + s*R
1775 : * We use a special Reme algorithm on [0,0.1716] to generate
1776 : * a polynomial of degree 14 to approximate R The maximum error
1777 : * of this polynomial approximation is bounded by 2**-58.45. In
1778 : * other words,
1779 : * 2 4 6 8 10 12 14
1780 : * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
1781 : * (the values of Lp1 to Lp7 are listed in the program)
1782 : * and
1783 : * | 2 14 | -58.45
1784 : * | Lp1*s +...+Lp7*s - R(z) | <= 2
1785 : * | |
1786 : * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
1787 : * In order to guarantee error in log below 1ulp, we compute log
1788 : * by
1789 : * log1p(f) = f - (hfsq - s*(hfsq+R)).
1790 : *
1791 : * 3. Finally, log1p(x) = k*ln2 + log1p(f).
1792 : * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
1793 : * Here ln2 is split into two floating point number:
1794 : * ln2_hi + ln2_lo,
1795 : * where n*ln2_hi is always exact for |n| < 2000.
1796 : *
1797 : * Special cases:
1798 : * log1p(x) is NaN with signal if x < -1 (including -INF) ;
1799 : * log1p(+INF) is +INF; log1p(-1) is -INF with signal;
1800 : * log1p(NaN) is that NaN with no signal.
1801 : *
1802 : * Accuracy:
1803 : * according to an error analysis, the error is always less than
1804 : * 1 ulp (unit in the last place).
1805 : *
1806 : * Constants:
1807 : * The hexadecimal values are the intended ones for the following
1808 : * constants. The decimal values may be used, provided that the
1809 : * compiler will convert from decimal to binary accurately enough
1810 : * to produce the hexadecimal values shown.
1811 : *
1812 : * Note: Assuming log() return accurate answer, the following
1813 : * algorithm can be used to compute log1p(x) to within a few ULP:
1814 : *
1815 : * u = 1+x;
1816 : * if(u==1.0) return x ; else
1817 : * return log(u)*(x/(u-1.0));
1818 : *
1819 : * See HP-15C Advanced Functions Handbook, p.193.
1820 : */
1821 86257 : double log1p(double x) {
1822 : static const double /* -- */
1823 : ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
1824 : ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
1825 : two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
1826 : Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
1827 : Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
1828 : Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
1829 : Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
1830 : Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
1831 : Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
1832 : Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
1833 :
1834 : static const double zero = 0.0;
1835 : static volatile double vzero = 0.0;
1836 :
1837 : double hfsq, f, c, s, z, R, u;
1838 : int32_t k, hx, hu, ax;
1839 :
1840 86257 : GET_HIGH_WORD(hx, x);
1841 86257 : ax = hx & 0x7fffffff;
1842 :
1843 : k = 1;
1844 86257 : if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */
1845 44025 : if (ax >= 0x3ff00000) { /* x <= -1.0 */
1846 302 : if (x == -1.0)
1847 33 : return -two54 / vzero; /* log1p(-1)=+inf */
1848 : else
1849 269 : return (x - x) / (x - x); /* log1p(x<-1)=NaN */
1850 : }
1851 43723 : if (ax < 0x3e200000) { /* |x| < 2**-29 */
1852 42302 : if (two54 + x > zero /* raise inexact */
1853 42302 : && ax < 0x3c900000) /* |x| < 2**-54 */
1854 : return x;
1855 : else
1856 1123 : return x - x * x * 0.5;
1857 : }
1858 1421 : if (hx > 0 || hx <= static_cast<int32_t>(0xbfd2bec4)) {
1859 : k = 0;
1860 : f = x;
1861 : hu = 1;
1862 : } /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
1863 : }
1864 43653 : if (hx >= 0x7ff00000) return x + x;
1865 43536 : if (k != 0) {
1866 42127 : if (hx < 0x43400000) {
1867 919 : STRICT_ASSIGN(double, u, 1.0 + x);
1868 919 : GET_HIGH_WORD(hu, u);
1869 919 : k = (hu >> 20) - 1023;
1870 919 : c = (k > 0) ? 1.0 - (u - x) : x - (u - 1.0); /* correction term */
1871 919 : c /= u;
1872 : } else {
1873 : u = x;
1874 41208 : GET_HIGH_WORD(hu, u);
1875 41208 : k = (hu >> 20) - 1023;
1876 : c = 0;
1877 : }
1878 42127 : hu &= 0x000fffff;
1879 : /*
1880 : * The approximation to sqrt(2) used in thresholds is not
1881 : * critical. However, the ones used above must give less
1882 : * strict bounds than the one here so that the k==0 case is
1883 : * never reached from here, since here we have committed to
1884 : * using the correction term but don't use it if k==0.
1885 : */
1886 42127 : if (hu < 0x6a09e) { /* u ~< sqrt(2) */
1887 21047 : SET_HIGH_WORD(u, hu | 0x3ff00000); /* normalize u */
1888 : } else {
1889 21080 : k += 1;
1890 21080 : SET_HIGH_WORD(u, hu | 0x3fe00000); /* normalize u/2 */
1891 21080 : hu = (0x00100000 - hu) >> 2;
1892 : }
1893 42127 : f = u - 1.0;
1894 : }
1895 43536 : hfsq = 0.5 * f * f;
1896 43536 : if (hu == 0) { /* |f| < 2**-20 */
1897 243 : if (f == zero) {
1898 87 : if (k == 0) {
1899 : return zero;
1900 : } else {
1901 87 : c += k * ln2_lo;
1902 87 : return k * ln2_hi + c;
1903 : }
1904 : }
1905 156 : R = hfsq * (1.0 - 0.66666666666666666 * f);
1906 156 : if (k == 0)
1907 0 : return f - R;
1908 : else
1909 156 : return k * ln2_hi - ((R - (k * ln2_lo + c)) - f);
1910 : }
1911 43293 : s = f / (2.0 + f);
1912 43293 : z = s * s;
1913 43293 : R = z * (Lp1 +
1914 86586 : z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 + z * Lp7))))));
1915 43293 : if (k == 0)
1916 1409 : return f - (hfsq - s * (hfsq + R));
1917 : else
1918 41884 : return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f);
1919 : }
1920 :
1921 : /*
1922 : * k_log1p(f):
1923 : * Return log(1+f) - f for 1+f in ~[sqrt(2)/2, sqrt(2)].
1924 : *
1925 : * The following describes the overall strategy for computing
1926 : * logarithms in base e. The argument reduction and adding the final
1927 : * term of the polynomial are done by the caller for increased accuracy
1928 : * when different bases are used.
1929 : *
1930 : * Method :
1931 : * 1. Argument Reduction: find k and f such that
1932 : * x = 2^k * (1+f),
1933 : * where sqrt(2)/2 < 1+f < sqrt(2) .
1934 : *
1935 : * 2. Approximation of log(1+f).
1936 : * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
1937 : * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
1938 : * = 2s + s*R
1939 : * We use a special Reme algorithm on [0,0.1716] to generate
1940 : * a polynomial of degree 14 to approximate R The maximum error
1941 : * of this polynomial approximation is bounded by 2**-58.45. In
1942 : * other words,
1943 : * 2 4 6 8 10 12 14
1944 : * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
1945 : * (the values of Lg1 to Lg7 are listed in the program)
1946 : * and
1947 : * | 2 14 | -58.45
1948 : * | Lg1*s +...+Lg7*s - R(z) | <= 2
1949 : * | |
1950 : * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
1951 : * In order to guarantee error in log below 1ulp, we compute log
1952 : * by
1953 : * log(1+f) = f - s*(f - R) (if f is not too large)
1954 : * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
1955 : *
1956 : * 3. Finally, log(x) = k*ln2 + log(1+f).
1957 : * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
1958 : * Here ln2 is split into two floating point number:
1959 : * ln2_hi + ln2_lo,
1960 : * where n*ln2_hi is always exact for |n| < 2000.
1961 : *
1962 : * Special cases:
1963 : * log(x) is NaN with signal if x < 0 (including -INF) ;
1964 : * log(+INF) is +INF; log(0) is -INF with signal;
1965 : * log(NaN) is that NaN with no signal.
1966 : *
1967 : * Accuracy:
1968 : * according to an error analysis, the error is always less than
1969 : * 1 ulp (unit in the last place).
1970 : *
1971 : * Constants:
1972 : * The hexadecimal values are the intended ones for the following
1973 : * constants. The decimal values may be used, provided that the
1974 : * compiler will convert from decimal to binary accurately enough
1975 : * to produce the hexadecimal values shown.
1976 : */
1977 :
1978 : static const double Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
1979 : Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
1980 : Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
1981 : Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
1982 : Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
1983 : Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
1984 : Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
1985 :
1986 : /*
1987 : * We always inline k_log1p(), since doing so produces a
1988 : * substantial performance improvement (~40% on amd64).
1989 : */
1990 89096 : static inline double k_log1p(double f) {
1991 : double hfsq, s, z, R, w, t1, t2;
1992 :
1993 89096 : s = f / (2.0 + f);
1994 89096 : z = s * s;
1995 89096 : w = z * z;
1996 89096 : t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
1997 89096 : t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
1998 89096 : R = t2 + t1;
1999 89096 : hfsq = 0.5 * f * f;
2000 89096 : return s * (hfsq + R);
2001 : }
2002 :
2003 : /*
2004 : * Return the base 2 logarithm of x. See e_log.c and k_log.h for most
2005 : * comments.
2006 : *
2007 : * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel,
2008 : * then does the combining and scaling steps
2009 : * log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k
2010 : * in not-quite-routine extra precision.
2011 : */
2012 89637 : double log2(double x) {
2013 : static const double
2014 : two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
2015 : ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */
2016 : ivln2lo = 1.67517131648865118353e-10; /* 0x3de705fc, 0x2eefa200 */
2017 :
2018 : static const double zero = 0.0;
2019 : static volatile double vzero = 0.0;
2020 :
2021 : double f, hfsq, hi, lo, r, val_hi, val_lo, w, y;
2022 : int32_t i, k, hx;
2023 : uint32_t lx;
2024 :
2025 89637 : EXTRACT_WORDS(hx, lx, x);
2026 :
2027 : k = 0;
2028 89637 : if (hx < 0x00100000) { /* x < 2**-1022 */
2029 1066 : if (((hx & 0x7fffffff) | lx) == 0)
2030 66 : return -two54 / vzero; /* log(+-0)=-inf */
2031 1000 : if (hx < 0) return (x - x) / zero; /* log(-#) = NaN */
2032 : k -= 54;
2033 728 : x *= two54; /* subnormal number, scale up x */
2034 728 : GET_HIGH_WORD(hx, x);
2035 : }
2036 89299 : if (hx >= 0x7ff00000) return x + x;
2037 89184 : if (hx == 0x3ff00000 && lx == 0) return zero; /* log(1) = +0 */
2038 89096 : k += (hx >> 20) - 1023;
2039 89096 : hx &= 0x000fffff;
2040 89096 : i = (hx + 0x95f64) & 0x100000;
2041 89096 : SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */
2042 89096 : k += (i >> 20);
2043 89096 : y = static_cast<double>(k);
2044 89096 : f = x - 1.0;
2045 89096 : hfsq = 0.5 * f * f;
2046 89096 : r = k_log1p(f);
2047 :
2048 : /*
2049 : * f-hfsq must (for args near 1) be evaluated in extra precision
2050 : * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2).
2051 : * This is fairly efficient since f-hfsq only depends on f, so can
2052 : * be evaluated in parallel with R. Not combining hfsq with R also
2053 : * keeps R small (though not as small as a true `lo' term would be),
2054 : * so that extra precision is not needed for terms involving R.
2055 : *
2056 : * Compiler bugs involving extra precision used to break Dekker's
2057 : * theorem for spitting f-hfsq as hi+lo, unless double_t was used
2058 : * or the multi-precision calculations were avoided when double_t
2059 : * has extra precision. These problems are now automatically
2060 : * avoided as a side effect of the optimization of combining the
2061 : * Dekker splitting step with the clear-low-bits step.
2062 : *
2063 : * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra
2064 : * precision to avoid a very large cancellation when x is very near
2065 : * these values. Unlike the above cancellations, this problem is
2066 : * specific to base 2. It is strange that adding +-1 is so much
2067 : * harder than adding +-ln2 or +-log10_2.
2068 : *
2069 : * This uses Dekker's theorem to normalize y+val_hi, so the
2070 : * compiler bugs are back in some configurations, sigh. And I
2071 : * don't want to used double_t to avoid them, since that gives a
2072 : * pessimization and the support for avoiding the pessimization
2073 : * is not yet available.
2074 : *
2075 : * The multi-precision calculations for the multiplications are
2076 : * routine.
2077 : */
2078 89096 : hi = f - hfsq;
2079 89096 : SET_LOW_WORD(hi, 0);
2080 89096 : lo = (f - hi) - hfsq + r;
2081 89096 : val_hi = hi * ivln2hi;
2082 89096 : val_lo = (lo + hi) * ivln2lo + lo * ivln2hi;
2083 :
2084 : /* spadd(val_hi, val_lo, y), except for not using double_t: */
2085 89096 : w = y + val_hi;
2086 89096 : val_lo += (y - w) + val_hi;
2087 : val_hi = w;
2088 :
2089 89096 : return val_lo + val_hi;
2090 : }
2091 :
2092 : /*
2093 : * Return the base 10 logarithm of x
2094 : *
2095 : * Method :
2096 : * Let log10_2hi = leading 40 bits of log10(2) and
2097 : * log10_2lo = log10(2) - log10_2hi,
2098 : * ivln10 = 1/log(10) rounded.
2099 : * Then
2100 : * n = ilogb(x),
2101 : * if(n<0) n = n+1;
2102 : * x = scalbn(x,-n);
2103 : * log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x))
2104 : *
2105 : * Note 1:
2106 : * To guarantee log10(10**n)=n, where 10**n is normal, the rounding
2107 : * mode must set to Round-to-Nearest.
2108 : * Note 2:
2109 : * [1/log(10)] rounded to 53 bits has error .198 ulps;
2110 : * log10 is monotonic at all binary break points.
2111 : *
2112 : * Special cases:
2113 : * log10(x) is NaN if x < 0;
2114 : * log10(+INF) is +INF; log10(0) is -INF;
2115 : * log10(NaN) is that NaN;
2116 : * log10(10**N) = N for N=0,1,...,22.
2117 : */
2118 18265 : double log10(double x) {
2119 : static const double
2120 : two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
2121 : ivln10 = 4.34294481903251816668e-01,
2122 : log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */
2123 : log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */
2124 :
2125 : static const double zero = 0.0;
2126 : static volatile double vzero = 0.0;
2127 :
2128 : double y;
2129 : int32_t i, k, hx;
2130 : uint32_t lx;
2131 :
2132 18265 : EXTRACT_WORDS(hx, lx, x);
2133 :
2134 : k = 0;
2135 18265 : if (hx < 0x00100000) { /* x < 2**-1022 */
2136 422 : if (((hx & 0x7fffffff) | lx) == 0)
2137 66 : return -two54 / vzero; /* log(+-0)=-inf */
2138 356 : if (hx < 0) return (x - x) / zero; /* log(-#) = NaN */
2139 : k -= 54;
2140 84 : x *= two54; /* subnormal number, scale up x */
2141 84 : GET_HIGH_WORD(hx, x);
2142 84 : GET_LOW_WORD(lx, x);
2143 : }
2144 17927 : if (hx >= 0x7ff00000) return x + x;
2145 17812 : if (hx == 0x3ff00000 && lx == 0) return zero; /* log(1) = +0 */
2146 17772 : k += (hx >> 20) - 1023;
2147 :
2148 17772 : i = (k & 0x80000000) >> 31;
2149 17772 : hx = (hx & 0x000fffff) | ((0x3ff - i) << 20);
2150 17772 : y = k + i;
2151 17772 : SET_HIGH_WORD(x, hx);
2152 17772 : SET_LOW_WORD(x, lx);
2153 :
2154 17772 : double z = y * log10_2lo + ivln10 * log(x);
2155 17772 : return z + y * log10_2hi;
2156 : }
2157 :
2158 : /* expm1(x)
2159 : * Returns exp(x)-1, the exponential of x minus 1.
2160 : *
2161 : * Method
2162 : * 1. Argument reduction:
2163 : * Given x, find r and integer k such that
2164 : *
2165 : * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
2166 : *
2167 : * Here a correction term c will be computed to compensate
2168 : * the error in r when rounded to a floating-point number.
2169 : *
2170 : * 2. Approximating expm1(r) by a special rational function on
2171 : * the interval [0,0.34658]:
2172 : * Since
2173 : * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
2174 : * we define R1(r*r) by
2175 : * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
2176 : * That is,
2177 : * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
2178 : * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
2179 : * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
2180 : * We use a special Reme algorithm on [0,0.347] to generate
2181 : * a polynomial of degree 5 in r*r to approximate R1. The
2182 : * maximum error of this polynomial approximation is bounded
2183 : * by 2**-61. In other words,
2184 : * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
2185 : * where Q1 = -1.6666666666666567384E-2,
2186 : * Q2 = 3.9682539681370365873E-4,
2187 : * Q3 = -9.9206344733435987357E-6,
2188 : * Q4 = 2.5051361420808517002E-7,
2189 : * Q5 = -6.2843505682382617102E-9;
2190 : * z = r*r,
2191 : * with error bounded by
2192 : * | 5 | -61
2193 : * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
2194 : * | |
2195 : *
2196 : * expm1(r) = exp(r)-1 is then computed by the following
2197 : * specific way which minimize the accumulation rounding error:
2198 : * 2 3
2199 : * r r [ 3 - (R1 + R1*r/2) ]
2200 : * expm1(r) = r + --- + --- * [--------------------]
2201 : * 2 2 [ 6 - r*(3 - R1*r/2) ]
2202 : *
2203 : * To compensate the error in the argument reduction, we use
2204 : * expm1(r+c) = expm1(r) + c + expm1(r)*c
2205 : * ~ expm1(r) + c + r*c
2206 : * Thus c+r*c will be added in as the correction terms for
2207 : * expm1(r+c). Now rearrange the term to avoid optimization
2208 : * screw up:
2209 : * ( 2 2 )
2210 : * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
2211 : * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
2212 : * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
2213 : * ( )
2214 : *
2215 : * = r - E
2216 : * 3. Scale back to obtain expm1(x):
2217 : * From step 1, we have
2218 : * expm1(x) = either 2^k*[expm1(r)+1] - 1
2219 : * = or 2^k*[expm1(r) + (1-2^-k)]
2220 : * 4. Implementation notes:
2221 : * (A). To save one multiplication, we scale the coefficient Qi
2222 : * to Qi*2^i, and replace z by (x^2)/2.
2223 : * (B). To achieve maximum accuracy, we compute expm1(x) by
2224 : * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
2225 : * (ii) if k=0, return r-E
2226 : * (iii) if k=-1, return 0.5*(r-E)-0.5
2227 : * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
2228 : * else return 1.0+2.0*(r-E);
2229 : * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
2230 : * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
2231 : * (vii) return 2^k(1-((E+2^-k)-r))
2232 : *
2233 : * Special cases:
2234 : * expm1(INF) is INF, expm1(NaN) is NaN;
2235 : * expm1(-INF) is -1, and
2236 : * for finite argument, only expm1(0)=0 is exact.
2237 : *
2238 : * Accuracy:
2239 : * according to an error analysis, the error is always less than
2240 : * 1 ulp (unit in the last place).
2241 : *
2242 : * Misc. info.
2243 : * For IEEE double
2244 : * if x > 7.09782712893383973096e+02 then expm1(x) overflow
2245 : *
2246 : * Constants:
2247 : * The hexadecimal values are the intended ones for the following
2248 : * constants. The decimal values may be used, provided that the
2249 : * compiler will convert from decimal to binary accurately enough
2250 : * to produce the hexadecimal values shown.
2251 : */
2252 124001 : double expm1(double x) {
2253 : static const double
2254 : one = 1.0,
2255 : tiny = 1.0e-300,
2256 : o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
2257 : ln2_hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
2258 : ln2_lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
2259 : invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
2260 : /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs =
2261 : x*x/2: */
2262 : Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
2263 : Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
2264 : Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
2265 : Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
2266 : Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
2267 :
2268 : static volatile double huge = 1.0e+300;
2269 :
2270 : double y, hi, lo, c, t, e, hxs, hfx, r1, twopk;
2271 : int32_t k, xsb;
2272 : uint32_t hx;
2273 :
2274 124001 : GET_HIGH_WORD(hx, x);
2275 124001 : xsb = hx & 0x80000000; /* sign bit of x */
2276 124001 : hx &= 0x7fffffff; /* high word of |x| */
2277 :
2278 : /* filter out huge and non-finite argument */
2279 124001 : if (hx >= 0x4043687A) { /* if |x|>=56*ln2 */
2280 74553 : if (hx >= 0x40862E42) { /* if |x|>=709.78... */
2281 477 : if (hx >= 0x7ff00000) {
2282 : uint32_t low;
2283 160 : GET_LOW_WORD(low, x);
2284 160 : if (((hx & 0xfffff) | low) != 0)
2285 94 : return x + x; /* NaN */
2286 : else
2287 66 : return (xsb == 0) ? x : -1.0; /* exp(+-inf)={inf,-1} */
2288 : }
2289 317 : if (x > o_threshold) return huge * huge; /* overflow */
2290 : }
2291 74222 : if (xsb != 0) { /* x < -56*ln2, return -1.0 with inexact */
2292 37190 : if (x + tiny < 0.0) /* raise inexact */
2293 : return tiny - one; /* return -1 */
2294 : }
2295 : }
2296 :
2297 : /* argument reduction */
2298 86480 : if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
2299 42643 : if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
2300 460 : if (xsb == 0) {
2301 262 : hi = x - ln2_hi;
2302 : lo = ln2_lo;
2303 : k = 1;
2304 : } else {
2305 198 : hi = x + ln2_hi;
2306 : lo = -ln2_lo;
2307 : k = -1;
2308 : }
2309 : } else {
2310 42183 : k = invln2 * x + ((xsb == 0) ? 0.5 : -0.5);
2311 42183 : t = k;
2312 42183 : hi = x - t * ln2_hi; /* t*ln2_hi is exact here */
2313 42183 : lo = t * ln2_lo;
2314 : }
2315 42643 : STRICT_ASSIGN(double, x, hi - lo);
2316 42643 : c = (hi - x) - lo;
2317 43837 : } else if (hx < 0x3c900000) { /* when |x|<2**-54, return x */
2318 41341 : t = huge + x; /* return x with inexact flags when x!=0 */
2319 41341 : return x - (t - (huge + x));
2320 : } else {
2321 : k = 0;
2322 : }
2323 :
2324 : /* x is now in primary range */
2325 45139 : hfx = 0.5 * x;
2326 45139 : hxs = x * hfx;
2327 45139 : r1 = one + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5))));
2328 45139 : t = 3.0 - r1 * hfx;
2329 45139 : e = hxs * ((r1 - t) / (6.0 - x * t));
2330 45139 : if (k == 0) {
2331 2496 : return x - (x * e - hxs); /* c is 0 */
2332 : } else {
2333 42643 : INSERT_WORDS(twopk, 0x3ff00000 + (k << 20), 0); /* 2^k */
2334 42643 : e = (x * (e - c) - c);
2335 42643 : e -= hxs;
2336 42643 : if (k == -1) return 0.5 * (x - e) - 0.5;
2337 42445 : if (k == 1) {
2338 262 : if (x < -0.25)
2339 24 : return -2.0 * (e - (x + 0.5));
2340 : else
2341 238 : return one + 2.0 * (x - e);
2342 : }
2343 42183 : if (k <= -2 || k > 56) { /* suffice to return exp(x)-1 */
2344 39198 : y = one - (e - x);
2345 : // TODO(mvstanton): is this replacement for the hex float
2346 : // sufficient?
2347 : // if (k == 1024) y = y*2.0*0x1p1023;
2348 39198 : if (k == 1024)
2349 0 : y = y * 2.0 * 8.98846567431158e+307;
2350 : else
2351 39198 : y = y * twopk;
2352 39198 : return y - one;
2353 : }
2354 : t = one;
2355 2985 : if (k < 20) {
2356 1543 : SET_HIGH_WORD(t, 0x3ff00000 - (0x200000 >> k)); /* t=1-2^-k */
2357 1543 : y = t - (e - x);
2358 1543 : y = y * twopk;
2359 : } else {
2360 1442 : SET_HIGH_WORD(t, ((0x3ff - k) << 20)); /* 2^-k */
2361 1442 : y = x - (e + t);
2362 1442 : y += one;
2363 1442 : y = y * twopk;
2364 : }
2365 : }
2366 2985 : return y;
2367 : }
2368 :
2369 152827 : double cbrt(double x) {
2370 : static const uint32_t
2371 : B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
2372 : B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */
2373 :
2374 : /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
2375 : static const double P0 = 1.87595182427177009643, /* 0x3ffe03e6, 0x0f61e692 */
2376 : P1 = -1.88497979543377169875, /* 0xbffe28e0, 0x92f02420 */
2377 : P2 = 1.621429720105354466140, /* 0x3ff9f160, 0x4a49d6c2 */
2378 : P3 = -0.758397934778766047437, /* 0xbfe844cb, 0xbee751d9 */
2379 : P4 = 0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */
2380 :
2381 : int32_t hx;
2382 : union {
2383 : double value;
2384 : uint64_t bits;
2385 : } u;
2386 : double r, s, t = 0.0, w;
2387 : uint32_t sign;
2388 : uint32_t high, low;
2389 :
2390 152827 : EXTRACT_WORDS(hx, low, x);
2391 152827 : sign = hx & 0x80000000; /* sign= sign(x) */
2392 152827 : hx ^= sign;
2393 152827 : if (hx >= 0x7ff00000) return (x + x); /* cbrt(NaN,INF) is itself */
2394 :
2395 : /*
2396 : * Rough cbrt to 5 bits:
2397 : * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
2398 : * where e is integral and >= 0, m is real and in [0, 1), and "/" and
2399 : * "%" are integer division and modulus with rounding towards minus
2400 : * infinity. The RHS is always >= the LHS and has a maximum relative
2401 : * error of about 1 in 16. Adding a bias of -0.03306235651 to the
2402 : * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
2403 : * floating point representation, for finite positive normal values,
2404 : * ordinary integer divison of the value in bits magically gives
2405 : * almost exactly the RHS of the above provided we first subtract the
2406 : * exponent bias (1023 for doubles) and later add it back. We do the
2407 : * subtraction virtually to keep e >= 0 so that ordinary integer
2408 : * division rounds towards minus infinity; this is also efficient.
2409 : */
2410 152667 : if (hx < 0x00100000) { /* zero or subnormal? */
2411 52 : if ((hx | low) == 0) return (x); /* cbrt(0) is itself */
2412 0 : SET_HIGH_WORD(t, 0x43500000); /* set t= 2**54 */
2413 0 : t *= x;
2414 0 : GET_HIGH_WORD(high, t);
2415 0 : INSERT_WORDS(t, sign | ((high & 0x7fffffff) / 3 + B2), 0);
2416 : } else {
2417 152615 : INSERT_WORDS(t, sign | (hx / 3 + B1), 0);
2418 : }
2419 :
2420 : /*
2421 : * New cbrt to 23 bits:
2422 : * cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
2423 : * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
2424 : * to within 2**-23.5 when |r - 1| < 1/10. The rough approximation
2425 : * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
2426 : * gives us bounds for r = t**3/x.
2427 : *
2428 : * Try to optimize for parallel evaluation as in k_tanf.c.
2429 : */
2430 152615 : r = (t * t) * (t / x);
2431 152615 : t = t * ((P0 + r * (P1 + r * P2)) + ((r * r) * r) * (P3 + r * P4));
2432 :
2433 : /*
2434 : * Round t away from zero to 23 bits (sloppily except for ensuring that
2435 : * the result is larger in magnitude than cbrt(x) but not much more than
2436 : * 2 23-bit ulps larger). With rounding towards zero, the error bound
2437 : * would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps
2438 : * in the rounded t, the infinite-precision error in the Newton
2439 : * approximation barely affects third digit in the final error
2440 : * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
2441 : * before the final error is larger than 0.667 ulps.
2442 : */
2443 : u.value = t;
2444 152615 : u.bits = (u.bits + 0x80000000) & 0xffffffffc0000000ULL;
2445 : t = u.value;
2446 :
2447 : /* one step Newton iteration to 53 bits with error < 0.667 ulps */
2448 152615 : s = t * t; /* t*t is exact */
2449 152615 : r = x / s; /* error <= 0.5 ulps; |r| < |t| */
2450 152615 : w = t + t; /* t+t is exact */
2451 152615 : r = (r - t) / (w + r); /* r-t is exact; w+r ~= 3*t */
2452 152615 : t = t + t * r; /* error <= 0.5 + 0.5/3 + epsilon */
2453 :
2454 152615 : return (t);
2455 : }
2456 :
2457 : /* sin(x)
2458 : * Return sine function of x.
2459 : *
2460 : * kernel function:
2461 : * __kernel_sin ... sine function on [-pi/4,pi/4]
2462 : * __kernel_cos ... cose function on [-pi/4,pi/4]
2463 : * __ieee754_rem_pio2 ... argument reduction routine
2464 : *
2465 : * Method.
2466 : * Let S,C and T denote the sin, cos and tan respectively on
2467 : * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
2468 : * in [-pi/4 , +pi/4], and let n = k mod 4.
2469 : * We have
2470 : *
2471 : * n sin(x) cos(x) tan(x)
2472 : * ----------------------------------------------------------
2473 : * 0 S C T
2474 : * 1 C -S -1/T
2475 : * 2 -S -C T
2476 : * 3 -C S -1/T
2477 : * ----------------------------------------------------------
2478 : *
2479 : * Special cases:
2480 : * Let trig be any of sin, cos, or tan.
2481 : * trig(+-INF) is NaN, with signals;
2482 : * trig(NaN) is that NaN;
2483 : *
2484 : * Accuracy:
2485 : * TRIG(x) returns trig(x) nearly rounded
2486 : */
2487 45632414 : double sin(double x) {
2488 : double y[2], z = 0.0;
2489 : int32_t n, ix;
2490 :
2491 : /* High word of x. */
2492 45632414 : GET_HIGH_WORD(ix, x);
2493 :
2494 : /* |x| ~< pi/4 */
2495 45632414 : ix &= 0x7fffffff;
2496 45632414 : if (ix <= 0x3fe921fb) {
2497 37842753 : return __kernel_sin(x, z, 0);
2498 7789661 : } else if (ix >= 0x7ff00000) {
2499 : /* sin(Inf or NaN) is NaN */
2500 2465 : return x - x;
2501 : } else {
2502 : /* argument reduction needed */
2503 7787196 : n = __ieee754_rem_pio2(x, y);
2504 7787196 : switch (n & 3) {
2505 : case 0:
2506 900480 : return __kernel_sin(y[0], y[1], 1);
2507 : case 1:
2508 12872038 : return __kernel_cos(y[0], y[1]);
2509 : case 2:
2510 899870 : return -__kernel_sin(y[0], y[1], 1);
2511 : default:
2512 902004 : return -__kernel_cos(y[0], y[1]);
2513 : }
2514 : }
2515 : }
2516 :
2517 : /* tan(x)
2518 : * Return tangent function of x.
2519 : *
2520 : * kernel function:
2521 : * __kernel_tan ... tangent function on [-pi/4,pi/4]
2522 : * __ieee754_rem_pio2 ... argument reduction routine
2523 : *
2524 : * Method.
2525 : * Let S,C and T denote the sin, cos and tan respectively on
2526 : * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
2527 : * in [-pi/4 , +pi/4], and let n = k mod 4.
2528 : * We have
2529 : *
2530 : * n sin(x) cos(x) tan(x)
2531 : * ----------------------------------------------------------
2532 : * 0 S C T
2533 : * 1 C -S -1/T
2534 : * 2 -S -C T
2535 : * 3 -C S -1/T
2536 : * ----------------------------------------------------------
2537 : *
2538 : * Special cases:
2539 : * Let trig be any of sin, cos, or tan.
2540 : * trig(+-INF) is NaN, with signals;
2541 : * trig(NaN) is that NaN;
2542 : *
2543 : * Accuracy:
2544 : * TRIG(x) returns trig(x) nearly rounded
2545 : */
2546 1529307 : double tan(double x) {
2547 : double y[2], z = 0.0;
2548 : int32_t n, ix;
2549 :
2550 : /* High word of x. */
2551 1529307 : GET_HIGH_WORD(ix, x);
2552 :
2553 : /* |x| ~< pi/4 */
2554 1529307 : ix &= 0x7fffffff;
2555 1529307 : if (ix <= 0x3fe921fb) {
2556 7350 : return __kernel_tan(x, z, 1);
2557 1521957 : } else if (ix >= 0x7ff00000) {
2558 : /* tan(Inf or NaN) is NaN */
2559 2450 : return x - x; /* NaN */
2560 : } else {
2561 : /* argument reduction needed */
2562 1519507 : n = __ieee754_rem_pio2(x, y);
2563 : /* 1 -> n even, -1 -> n odd */
2564 1519507 : return __kernel_tan(y[0], y[1], 1 - ((n & 1) << 1));
2565 : }
2566 : }
2567 :
2568 : /*
2569 : * ES6 draft 09-27-13, section 20.2.2.12.
2570 : * Math.cosh
2571 : * Method :
2572 : * mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2
2573 : * 1. Replace x by |x| (cosh(x) = cosh(-x)).
2574 : * 2.
2575 : * [ exp(x) - 1 ]^2
2576 : * 0 <= x <= ln2/2 : cosh(x) := 1 + -------------------
2577 : * 2*exp(x)
2578 : *
2579 : * exp(x) + 1/exp(x)
2580 : * ln2/2 <= x <= 22 : cosh(x) := -------------------
2581 : * 2
2582 : * 22 <= x <= lnovft : cosh(x) := exp(x)/2
2583 : * lnovft <= x <= ln2ovft: cosh(x) := exp(x/2)/2 * exp(x/2)
2584 : * ln2ovft < x : cosh(x) := huge*huge (overflow)
2585 : *
2586 : * Special cases:
2587 : * cosh(x) is |x| if x is +INF, -INF, or NaN.
2588 : * only cosh(0)=1 is exact for finite x.
2589 : */
2590 1453 : double cosh(double x) {
2591 : static const double KCOSH_OVERFLOW = 710.4758600739439;
2592 : static const double one = 1.0, half = 0.5;
2593 : static volatile double huge = 1.0e+300;
2594 :
2595 : int32_t ix;
2596 :
2597 : /* High word of |x|. */
2598 1453 : GET_HIGH_WORD(ix, x);
2599 1453 : ix &= 0x7fffffff;
2600 :
2601 : // |x| in [0,0.5*log2], return 1+expm1(|x|)^2/(2*exp(|x|))
2602 1453 : if (ix < 0x3fd62e43) {
2603 333 : double t = expm1(fabs(x));
2604 333 : double w = one + t;
2605 : // For |x| < 2^-55, cosh(x) = 1
2606 333 : if (ix < 0x3c800000) return w;
2607 102 : return one + (t * t) / (w + w);
2608 : }
2609 :
2610 : // |x| in [0.5*log2, 22], return (exp(|x|)+1/exp(|x|)/2
2611 1120 : if (ix < 0x40360000) {
2612 439 : double t = exp(fabs(x));
2613 439 : return half * t + half / t;
2614 : }
2615 :
2616 : // |x| in [22, log(maxdouble)], return half*exp(|x|)
2617 681 : if (ix < 0x40862e42) return half * exp(fabs(x));
2618 :
2619 : // |x| in [log(maxdouble), overflowthreshold]
2620 641 : if (fabs(x) <= KCOSH_OVERFLOW) {
2621 28 : double w = exp(half * fabs(x));
2622 28 : double t = half * w;
2623 28 : return t * w;
2624 : }
2625 :
2626 : /* x is INF or NaN */
2627 613 : if (ix >= 0x7ff00000) return x * x;
2628 :
2629 : // |x| > overflowthreshold.
2630 418 : return huge * huge;
2631 : }
2632 :
2633 : /*
2634 : * ES6 draft 09-27-13, section 20.2.2.30.
2635 : * Math.sinh
2636 : * Method :
2637 : * mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2
2638 : * 1. Replace x by |x| (sinh(-x) = -sinh(x)).
2639 : * 2.
2640 : * E + E/(E+1)
2641 : * 0 <= x <= 22 : sinh(x) := --------------, E=expm1(x)
2642 : * 2
2643 : *
2644 : * 22 <= x <= lnovft : sinh(x) := exp(x)/2
2645 : * lnovft <= x <= ln2ovft: sinh(x) := exp(x/2)/2 * exp(x/2)
2646 : * ln2ovft < x : sinh(x) := x*shuge (overflow)
2647 : *
2648 : * Special cases:
2649 : * sinh(x) is |x| if x is +Infinity, -Infinity, or NaN.
2650 : * only sinh(0)=0 is exact for finite x.
2651 : */
2652 1607 : double sinh(double x) {
2653 : static const double KSINH_OVERFLOW = 710.4758600739439,
2654 : TWO_M28 =
2655 : 3.725290298461914e-9, // 2^-28, empty lower half
2656 : LOG_MAXD = 709.7822265625; // 0x40862e42 00000000, empty lower half
2657 : static const double shuge = 1.0e307;
2658 :
2659 1607 : double h = (x < 0) ? -0.5 : 0.5;
2660 : // |x| in [0, 22]. return sign(x)*0.5*(E+E/(E+1))
2661 1607 : double ax = fabs(x);
2662 1607 : if (ax < 22) {
2663 : // For |x| < 2^-28, sinh(x) = x
2664 870 : if (ax < TWO_M28) return x;
2665 595 : double t = expm1(ax);
2666 595 : if (ax < 1) {
2667 178 : return h * (2 * t - t * t / (t + 1));
2668 : }
2669 417 : return h * (t + t / (t + 1));
2670 : }
2671 : // |x| in [22, log(maxdouble)], return 0.5 * exp(|x|)
2672 737 : if (ax < LOG_MAXD) return h * exp(ax);
2673 : // |x| in [log(maxdouble), overflowthreshold]
2674 : // overflowthreshold = 710.4758600739426
2675 669 : if (ax <= KSINH_OVERFLOW) {
2676 28 : double w = exp(0.5 * ax);
2677 28 : double t = h * w;
2678 28 : return t * w;
2679 : }
2680 : // |x| > overflowthreshold or is NaN.
2681 : // Return Infinity of the appropriate sign or NaN.
2682 641 : return x * shuge;
2683 : }
2684 :
2685 : /* Tanh(x)
2686 : * Return the Hyperbolic Tangent of x
2687 : *
2688 : * Method :
2689 : * x -x
2690 : * e - e
2691 : * 0. tanh(x) is defined to be -----------
2692 : * x -x
2693 : * e + e
2694 : * 1. reduce x to non-negative by tanh(-x) = -tanh(x).
2695 : * 2. 0 <= x < 2**-28 : tanh(x) := x with inexact if x != 0
2696 : * -t
2697 : * 2**-28 <= x < 1 : tanh(x) := -----; t = expm1(-2x)
2698 : * t + 2
2699 : * 2
2700 : * 1 <= x < 22 : tanh(x) := 1 - -----; t = expm1(2x)
2701 : * t + 2
2702 : * 22 <= x <= INF : tanh(x) := 1.
2703 : *
2704 : * Special cases:
2705 : * tanh(NaN) is NaN;
2706 : * only tanh(0)=0 is exact for finite argument.
2707 : */
2708 1467 : double tanh(double x) {
2709 : static const volatile double tiny = 1.0e-300;
2710 : static const double one = 1.0, two = 2.0, huge = 1.0e300;
2711 : double t, z;
2712 : int32_t jx, ix;
2713 :
2714 1467 : GET_HIGH_WORD(jx, x);
2715 1467 : ix = jx & 0x7fffffff;
2716 :
2717 : /* x is INF or NaN */
2718 1467 : if (ix >= 0x7ff00000) {
2719 167 : if (jx >= 0)
2720 126 : return one / x + one; /* tanh(+-inf)=+-1 */
2721 : else
2722 41 : return one / x - one; /* tanh(NaN) = NaN */
2723 : }
2724 :
2725 : /* |x| < 22 */
2726 1300 : if (ix < 0x40360000) { /* |x|<22 */
2727 842 : if (ix < 0x3e300000) { /* |x|<2**-28 */
2728 247 : if (huge + x > one) return x; /* tanh(tiny) = tiny with inexact */
2729 : }
2730 595 : if (ix >= 0x3ff00000) { /* |x|>=1 */
2731 389 : t = expm1(two * fabs(x));
2732 389 : z = one - two / (t + two);
2733 : } else {
2734 206 : t = expm1(-two * fabs(x));
2735 206 : z = -t / (t + two);
2736 : }
2737 : /* |x| >= 22, return +-1 */
2738 : } else {
2739 458 : z = one - tiny; /* raise inexact flag */
2740 : }
2741 1053 : return (jx >= 0) ? z : -z;
2742 : }
2743 :
2744 : } // namespace ieee754
2745 : } // namespace base
2746 : } // namespace v8
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