/* Copyright 2016 Samsung Electronics Co., Ltd. * Copyright 2016 University of Szeged * * Licensed under the Apache License, Version 2.0 (the "License"); * you may not use this file except in compliance with the License. * You may obtain a copy of the License at * * http://www.apache.org/licenses/LICENSE-2.0 * * Unless required by applicable law or agreed to in writing, software * distributed under the License is distributed on an "AS IS" BASIS * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. * See the License for the specific language governing permissions and * limitations under the License. * * This file is based on work under the following copyright and permission * notice: * * Copyright (C) 1993, 2004 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * * @(#)k_rem_pio2.c 1.3 95/01/18 * @(#)e_rem_pio2.c 1.4 95/01/18 * @(#)k_sin.c 1.3 95/01/18 * @(#)k_cos.c 1.3 95/01/18 * @(#)k_tan.c 1.5 04/04/22 * @(#)s_sin.c 1.3 95/01/18 * @(#)s_cos.c 1.3 95/01/18 * @(#)s_tan.c 1.3 95/01/18 */ #include "jerry-libm-internal.h" #define zero 0.00000000000000000000e+00 /* 0x00000000, 0x00000000 */ #define half 5.00000000000000000000e-01 /* 0x3FE00000, 0x00000000 */ #define one 1.00000000000000000000e+00 /* 0x3FF00000, 0x00000000 */ #define two24 1.67772160000000000000e+07 /* 0x41700000, 0x00000000 */ #define twon24 5.96046447753906250000e-08 /* 0x3E700000, 0x00000000 */ /* __kernel_rem_pio2(x,y,e0,nx,prec) * double x[],y[]; int e0,nx,prec; * * __kernel_rem_pio2 return the last three digits of N with * y = x - N*pi/2 * so that |y| < pi/2. * * The method is to compute the integer (mod 8) and fraction parts of * (2/pi)*x without doing the full multiplication. In general we * skip the part of the product that are known to be a huge integer ( * more accurately, = 0 mod 8 ). Thus the number of operations are * independent of the exponent of the input. * * (2/pi) is represented by an array of 24-bit integers in ipio2[]. * * Input parameters: * x[] The input value (must be positive) is broken into nx * pieces of 24-bit integers in double precision format. * x[i] will be the i-th 24 bit of x. The scaled exponent * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 * match x's up to 24 bits. * * Example of breaking a double positive z into x[0]+x[1]+x[2]: * e0 = ilogb(z)-23 * z = scalbn(z,-e0) * for i = 0,1,2 * x[i] = floor(z) * z = (z-x[i])*2**24 * * y[] ouput result in an array of double precision numbers. * The dimension of y[] is: * 24-bit precision 1 * 53-bit precision 2 * 64-bit precision 2 * 113-bit precision 3 * The actual value is the sum of them. Thus for 113-bit * precison, one may have to do something like: * * long double t,w,r_head, r_tail; * t = (long double)y[2] + (long double)y[1]; * w = (long double)y[0]; * r_head = t+w; * r_tail = w - (r_head - t); * * e0 The exponent of x[0] * * nx dimension of x[] * * prec an integer indicating the precision: * 0 24 bits (single) * 1 53 bits (double) * 2 64 bits (extended) * 3 113 bits (quad) * * External function: * double scalbn(), floor(); * * Here is the description of some local variables: * * ipio2[] integer array, contains the (24*i)-th to (24*i+23)-th * bit of 2/pi after binary point. The corresponding * floating value is * * ipio2[i] * 2^(-24(i+1)). * * jk jk+1 is the initial number of terms of ipio2[] needed * in the computation. The recommended value is 2,3,4, * 6 for single, double, extended,and quad. * * jz local integer variable indicating the number of * terms of ipio2[] used. * * jx nx - 1 * * jv index for pointing to the suitable ipio2[] for the * computation. In general, we want * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 * is an integer. Thus * e0-3-24*jv >= 0 or (e0-3)/24 >= jv * Hence jv = max(0,(e0-3)/24). * * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. * * q[] double array with integral value, representing the * 24-bits chunk of the product of x and 2/pi. * * q0 the corresponding exponent of q[0]. Note that the * exponent for q[i] would be q0-24*i. * * PIo2[] double precision array, obtained by cutting pi/2 * into 24 bits chunks. * * f[] ipio2[] in floating point * * iq[] integer array by breaking up q[] in 24-bits chunk. * * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] * * ih integer. If >0 it indicates q[] is >= 0.5, hence * it also indicates the *sign* of the result. */ /* * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ /* initial value for jk */ static const int init_jk[] = { 2, 3, 4, 6 }; static const double PIo2[] = { 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ }; /* * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi */ static const int ipio2[] = { 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41, 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8, 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF, 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B, }; static int __kernel_rem_pio2 (double *x, double *y, int e0, int nx, int prec) { int jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih; double z, fw, f[20], fq[20], q[20]; /* initialize jk */ jk = init_jk[prec]; jp = jk; /* determine jx, jv, q0, note that 3 > q0 */ jx = nx - 1; jv = (e0 - 3) / 24; if (jv < 0) { jv = 0; } q0 = e0 - 24 * (jv + 1); /* set up f[0] to f[jx + jk] where f[jx + jk] = ipio2[jv + jk] */ j = jv - jx; m = jx + jk; for (i = 0; i <= m; i++, j++) { f[i] = (j < 0) ? zero : (double) ipio2[j]; } /* compute q[0], q[1], ... q[jk] */ for (i = 0; i <= jk; i++) { for (j = 0, fw = 0.0; j <= jx; j++) { fw += x[j] * f[jx + i - j]; } q[i] = fw; } jz = jk; recompute: /* distill q[] into iq[] reversingly */ for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) { fw = (double) ((int) (twon24 * z)); iq[i] = (int) (z - two24 * fw); z = q[j - 1] + fw; } /* compute n */ z = scalbn (z, q0); /* actual value of z */ z -= 8.0 * floor (z * 0.125); /* trim off integer >= 8 */ n = (int) z; z -= (double) n; ih = 0; if (q0 > 0) /* need iq[jz - 1] to determine n */ { i = (iq[jz - 1] >> (24 - q0)); n += i; iq[jz - 1] -= i << (24 - q0); ih = iq[jz - 1] >> (23 - q0); } else if (q0 == 0) { ih = iq[jz - 1] >> 23; } else if (z >= 0.5) { ih = 2; } if (ih > 0) /* q > 0.5 */ { n += 1; carry = 0; for (i = 0; i < jz; i++) /* compute 1 - q */ { j = iq[i]; if (carry == 0) { if (j != 0) { carry = 1; iq[i] = 0x1000000 - j; } } else { iq[i] = 0xffffff - j; } } if (q0 > 0) /* rare case: chance is 1 in 12 */ { switch (q0) { case 1: { iq[jz - 1] &= 0x7fffff; break; } case 2: { iq[jz - 1] &= 0x3fffff; break; } } } if (ih == 2) { z = one - z; if (carry != 0) { z -= scalbn (one, q0); } } } /* check if recomputation is needed */ if (z == zero) { j = 0; for (i = jz - 1; i >= jk; i--) { j |= iq[i]; } if (j == 0) /* need recomputation */ { for (k = 1; iq[jk - k] == 0; k++) /* k = no. of terms needed */ { } for (i = jz + 1; i <= jz + k; i++) /* add q[jz + 1] to q[jz + k] */ { f[jx + i] = (double) ipio2[jv + i]; for (j = 0, fw = 0.0; j <= jx; j++) { fw += x[j] * f[jx + i - j]; } q[i] = fw; } jz += k; goto recompute; } } /* chop off zero terms */ if (z == 0.0) { jz -= 1; q0 -= 24; while (iq[jz] == 0) { jz--; q0 -= 24; } } else { /* break z into 24-bit if necessary */ z = scalbn (z, -q0); if (z >= two24) { fw = (double) ((int) (twon24 * z)); iq[jz] = (int) (z - two24 * fw); jz += 1; q0 += 24; iq[jz] = (int) fw; } else { iq[jz] = (int) z; } } /* convert integer "bit" chunk to floating-point value */ fw = scalbn (one, q0); for (i = jz; i >= 0; i--) { q[i] = fw * (double) iq[i]; fw *= twon24; } /* compute PIo2[0, ..., jp] * q[jz, ..., 0] */ for (i = jz; i >= 0; i--) { for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++) { fw += PIo2[k] * q[i + k]; } fq[jz - i] = fw; } /* compress fq[] into y[] */ switch (prec) { case 0: { fw = 0.0; for (i = jz; i >= 0; i--) { fw += fq[i]; } y[0] = (ih == 0) ? fw : -fw; break; } case 1: case 2: { fw = 0.0; for (i = jz; i >= 0; i--) { fw += fq[i]; } y[0] = (ih == 0) ? fw : -fw; fw = fq[0] - fw; for (i = 1; i <= jz; i++) { fw += fq[i]; } y[1] = (ih == 0) ? fw : -fw; break; } case 3: /* painful */ { for (i = jz; i > 0; i--) { fw = fq[i - 1] + fq[i]; fq[i] += fq[i - 1] - fw; fq[i - 1] = fw; } for (i = jz; i > 1; i--) { fw = fq[i - 1] + fq[i]; fq[i] += fq[i - 1] - fw; fq[i - 1] = fw; } for (fw = 0.0, i = jz; i >= 2; i--) { fw += fq[i]; } if (ih == 0) { y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; } else { y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; } } } return n & 7; } /* __kernel_rem_pio2 */ /* __ieee754_rem_pio2(x,y) * return the remainder of x rem pi/2 in y[0]+y[1] * use __kernel_rem_pio2() */ static const int npio2_hw[] = { 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C, 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C, 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A, 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C, 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB, 0x404858EB, 0x404921FB, }; /* * invpio2: 53 bits of 2/pi * pio2_1: first 33 bit of pi/2 * pio2_1t: pi/2 - pio2_1 * pio2_2: second 33 bit of pi/2 * pio2_2t: pi/2 - (pio2_1 + pio2_2) * pio2_3: third 33 bit of pi/2 * pio2_3t: pi/2 - (pio2_1 + pio2_2 + pio2_3) */ #define invpio2 6.36619772367581382433e-01 /* 0x3FE45F30, 0x6DC9C883 */ #define pio2_1 1.57079632673412561417e+00 /* 0x3FF921FB, 0x54400000 */ #define pio2_1t 6.07710050650619224932e-11 /* 0x3DD0B461, 0x1A626331 */ #define pio2_2 6.07710050630396597660e-11 /* 0x3DD0B461, 0x1A600000 */ #define pio2_2t 2.02226624879595063154e-21 /* 0x3BA3198A, 0x2E037073 */ #define pio2_3 2.02226624871116645580e-21 /* 0x3BA3198A, 0x2E000000 */ #define pio2_3t 8.47842766036889956997e-32 /* 0x397B839A, 0x252049C1 */ static int __ieee754_rem_pio2 (double x, double *y) { double z, w, t, r, fn; double tx[3]; int e0, i, j, nx, n, ix, hx; hx = __HI (x); /* high word of x */ ix = hx & 0x7fffffff; if (ix <= 0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */ { y[0] = x; y[1] = 0; return 0; } if (ix < 0x4002d97c) /* |x| < 3pi/4, special case with n = +-1 */ { if (hx > 0) { z = x - pio2_1; if (ix != 0x3ff921fb) /* 33 + 53 bit pi is good enough */ { y[0] = z - pio2_1t; y[1] = (z - y[0]) - pio2_1t; } else /* near pi/2, use 33 + 33 + 53 bit pi */ { z -= pio2_2; y[0] = z - pio2_2t; y[1] = (z - y[0]) - pio2_2t; } return 1; } else /* negative x */ { z = x + pio2_1; if (ix != 0x3ff921fb) /* 33 + 53 bit pi is good enough */ { y[0] = z + pio2_1t; y[1] = (z - y[0]) + pio2_1t; } else /* near pi/2, use 33 + 33 + 53 bit pi */ { z += pio2_2; y[0] = z + pio2_2t; y[1] = (z - y[0]) + pio2_2t; } return -1; } } if (ix <= 0x413921fb) /* |x| ~<= 2^19 * (pi/2), medium size */ { t = fabs (x); n = (int) (t * invpio2 + half); fn = (double) n; r = t - fn * pio2_1; w = fn * pio2_1t; /* 1st round good to 85 bit */ if (n < 32 && ix != npio2_hw[n - 1]) { y[0] = r - w; /* quick check no cancellation */ } else { j = ix >> 20; y[0] = r - w; i = j - (((__HI (y[0])) >> 20) & 0x7ff); if (i > 16) /* 2nd iteration needed, good to 118 */ { t = r; w = fn * pio2_2; r = t - w; w = fn * pio2_2t - ((t - r) - w); y[0] = r - w; i = j - (((__HI (y[0])) >> 20) & 0x7ff); if (i > 49) /* 3rd iteration need, 151 bits acc, will cover all possible cases */ { t = r; w = fn * pio2_3; r = t - w; w = fn * pio2_3t - ((t - r) - w); y[0] = r - w; } } } y[1] = (r - y[0]) - w; if (hx < 0) { y[0] = -y[0]; y[1] = -y[1]; return -n; } else { return n; } } /* * all other (large) arguments */ if (ix >= 0x7ff00000) /* x is inf or NaN */ { y[0] = y[1] = x - x; return 0; } /* set z = scalbn(|x|, ilogb(x) - 23) */ __LO (z) = __LO (x); e0 = (ix >> 20) - 1046; /* e0 = ilogb(z) - 23; */ __HI (z) = ix - (e0 << 20); for (i = 0; i < 2; i++) { tx[i] = (double) ((int) (z)); z = (z - tx[i]) * two24; } tx[2] = z; nx = 3; while (tx[nx - 1] == zero) /* skip zero term */ { nx--; } n = __kernel_rem_pio2 (tx, y, e0, nx, 2); if (hx < 0) { y[0] = -y[0]; y[1] = -y[1]; return -n; } return n; } /* __ieee754_rem_pio2 */ /* __kernel_sin( x, y, iy) * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 * Input x is assumed to be bounded by ~pi/4 in magnitude. * Input y is the tail of x. * Input iy indicates whether y is 0. (if iy=0, y assume to be 0). * * Algorithm * 1. Since sin(-x) = -sin(x), we need only to consider positive x. * 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0. * 3. sin(x) is approximated by a polynomial of degree 13 on * [0,pi/4] * 3 13 * sin(x) ~ x + S1*x + ... + S6*x * where * * |sin(x) 2 4 6 8 10 12 | -58 * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 * | x | * * 4. sin(x+y) = sin(x) + sin'(x')*y * ~ sin(x) + (1-x*x/2)*y * For better accuracy, let * 3 2 2 2 2 * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6)))) * then 3 2 * sin(x) = x + (S1*x + (x *(r-y/2)+y)) */ #define S1 -1.66666666666666324348e-01 /* 0xBFC55555, 0x55555549 */ #define S2 8.33333333332248946124e-03 /* 0x3F811111, 0x1110F8A6 */ #define S3 -1.98412698298579493134e-04 /* 0xBF2A01A0, 0x19C161D5 */ #define S4 2.75573137070700676789e-06 /* 0x3EC71DE3, 0x57B1FE7D */ #define S5 -2.50507602534068634195e-08 /* 0xBE5AE5E6, 0x8A2B9CEB */ #define S6 1.58969099521155010221e-10 /* 0x3DE5D93A, 0x5ACFD57C */ static double __kernel_sin (double x, double y, int iy) { double z, r, v; int ix; ix = __HI (x) & 0x7fffffff; /* high word of x */ if (ix < 0x3e400000) /* |x| < 2**-27 */ { if ((int) x == 0) { return x; /* generate inexact */ } } z = x * x; v = z * x; r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6))); if (iy == 0) { return x + v * (S1 + z * r); } else { return x - ((z * (half * y - v * r) - y) - v * S1); } } /* __kernel_sin */ /* * __kernel_cos( x, y ) * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 * Input x is assumed to be bounded by ~pi/4 in magnitude. * Input y is the tail of x. * * Algorithm * 1. Since cos(-x) = cos(x), we need only to consider positive x. * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. * 3. cos(x) is approximated by a polynomial of degree 14 on * [0,pi/4] * 4 14 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x * where the remez error is * * | 2 4 6 8 10 12 14 | -58 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 * | | * * 4 6 8 10 12 14 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then * cos(x) = 1 - x*x/2 + r * since cos(x+y) ~ cos(x) - sin(x)*y * ~ cos(x) - x*y, * a correction term is necessary in cos(x) and hence * cos(x+y) = 1 - (x*x/2 - (r - x*y)) * For better accuracy when x > 0.3, let qx = |x|/4 with * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. * Then * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). * Note that 1-qx and (x*x/2-qx) is EXACT here, and the * magnitude of the latter is at least a quarter of x*x/2, * thus, reducing the rounding error in the subtraction. */ #define C1 4.16666666666666019037e-02 /* 0x3FA55555, 0x5555554C */ #define C2 -1.38888888888741095749e-03 /* 0xBF56C16C, 0x16C15177 */ #define C3 2.48015872894767294178e-05 /* 0x3EFA01A0, 0x19CB1590 */ #define C4 -2.75573143513906633035e-07 /* 0xBE927E4F, 0x809C52AD */ #define C5 2.08757232129817482790e-09 /* 0x3E21EE9E, 0xBDB4B1C4 */ #define C6 -1.13596475577881948265e-11 /* 0xBDA8FAE9, 0xBE8838D4 */ static double __kernel_cos (double x, double y) { double a, hz, z, r, qx; int ix; ix = __HI (x) & 0x7fffffff; /* ix = |x|'s high word */ if (ix < 0x3e400000) /* if x < 2**27 */ { if (((int) x) == 0) { return one; /* generate inexact */ } } z = x * x; r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6))))); if (ix < 0x3FD33333) /* if |x| < 0.3 */ { return one - (0.5 * z - (z * r - x * y)); } else { if (ix > 0x3fe90000) /* x > 0.78125 */ { qx = 0.28125; } else { __HI (qx) = ix - 0x00200000; /* x / 4 */ __LO (qx) = 0; } hz = 0.5 * z - qx; a = one - qx; return a - (hz - (z * r - x * y)); } } /* __kernel_cos */ /* __kernel_tan( x, y, k ) * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 * Input x is assumed to be bounded by ~pi/4 in magnitude. * Input y is the tail of x. * Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned. * * Algorithm * 1. Since tan(-x) = -tan(x), we need only to consider positive x. * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. * 3. tan(x) is approximated by a odd polynomial of degree 27 on * [0,0.67434] * 3 27 * tan(x) ~ x + T1*x + ... + T13*x * where * * |tan(x) 2 4 26 | -59.2 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 * | x | * * Note: tan(x+y) = tan(x) + tan'(x)*y * ~ tan(x) + (1+x*x)*y * Therefore, for better accuracy in computing tan(x+y), let * 3 2 2 2 2 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) * then * 3 2 * tan(x+y) = x + (T1*x + (x *(r+y)+y)) * * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) */ #define T0 3.33333333333334091986e-01 /* 3FD55555, 55555563 */ #define T1 1.33333333333201242699e-01 /* 3FC11111, 1110FE7A */ #define T2 5.39682539762260521377e-02 /* 3FABA1BA, 1BB341FE */ #define T3 2.18694882948595424599e-02 /* 3F9664F4, 8406D637 */ #define T4 8.86323982359930005737e-03 /* 3F8226E3, E96E8493 */ #define T5 3.59207910759131235356e-03 /* 3F6D6D22, C9560328 */ #define T6 1.45620945432529025516e-03 /* 3F57DBC8, FEE08315 */ #define T7 5.88041240820264096874e-04 /* 3F4344D8, F2F26501 */ #define T8 2.46463134818469906812e-04 /* 3F3026F7, 1A8D1068 */ #define T9 7.81794442939557092300e-05 /* 3F147E88, A03792A6 */ #define T10 7.14072491382608190305e-05 /* 3F12B80F, 32F0A7E9 */ #define T11 -1.85586374855275456654e-05 /* BEF375CB, DB605373 */ #define T12 2.59073051863633712884e-05 /* 3EFB2A70, 74BF7AD4 */ #define pio4 7.85398163397448278999e-01 /* 3FE921FB, 54442D18 */ #define pio4lo 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ static double __kernel_tan (double x, double y, int iy) { double z, r, v, w, s; int ix, hx; hx = __HI (x); /* high word of x */ ix = hx & 0x7fffffff; /* high word of |x| */ if (ix < 0x3e300000) /* x < 2**-28 */ { if ((int) x == 0) /* generate inexact */ { if (((ix | __LO (x)) | (iy + 1)) == 0) { return one / fabs (x); } else { if (iy == 1) { return x; } else /* compute -1 / (x + y) carefully */ { double a, t; z = w = x + y; __LO (z) = 0; v = y - (z - x); t = a = -one / w; __LO (t) = 0; s = one + t * z; return t + a * (s + t * v); } } } } if (ix >= 0x3FE59428) /* |x| >= 0.6744 */ { if (hx < 0) { x = -x; y = -y; } z = pio4 - x; w = pio4lo - y; x = z + w; y = 0.0; } z = x * x; w = z * z; /* * Break x^5 * (T[1] + x^2 * T[2] + ...) into * x^5 (T[1] + x^4 * T[3] + ... + x^20 * T[11]) + * x^5 (x^2 * (T[2] + x^4 * T[4] + ... + x^22 * [T12])) */ r = T1 + w * (T3 + w * (T5 + w * (T7 + w * (T9 + w * T11)))); v = z * (T2 + w * (T4 + w * (T6 + w * (T8 + w * (T10 + w * T12))))); s = z * x; r = y + z * (s * (r + v) + y); r += T0 * s; w = x + r; if (ix >= 0x3FE59428) { v = (double) iy; return (double) (1 - ((hx >> 30) & 2)) * (v - 2.0 * (x - (w * w / (w + v) - r))); } if (iy == 1) { return w; } else { /* * if allow error up to 2 ulp, simply return * -1.0 / (x + r) here */ /* compute -1.0 / (x + r) accurately */ double a, t; z = w; __LO (z) = 0; v = r - (z - x); /* z + v = r + x */ t = a = -1.0 / w; /* a = -1.0 / w */ __LO (t) = 0; s = 1.0 + t * z; return t + a * (s + t * v); } } /* __kernel_tan */ /* Method: * Let S,C and T denote the sin, cos and tan respectively on * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 * in [-pi/4 , +pi/4], and let n = k mod 4. * We have * * n sin(x) cos(x) tan(x) * ---------------------------------------------------------- * 0 S C T * 1 C -S -1/T * 2 -S -C T * 3 -C S -1/T * ---------------------------------------------------------- * * Special cases: * Let trig be any of sin, cos, or tan. * trig(+-INF) is NaN, with signals; * trig(NaN) is that NaN; * * Accuracy: * TRIG(x) returns trig(x) nearly rounded */ /* sin(x) * Return sine function of x. * * kernel function: * __kernel_sin ... sine function on [-pi/4,pi/4] * __kernel_cos ... cose function on [-pi/4,pi/4] * __ieee754_rem_pio2 ... argument reduction routine */ double sin (double x) { double y[2], z = 0.0; int n, ix; /* High word of x. */ ix = __HI (x); /* |x| ~< pi/4 */ ix &= 0x7fffffff; if (ix <= 0x3fe921fb) { return __kernel_sin (x, z, 0); } /* sin(Inf or NaN) is NaN */ else if (ix >= 0x7ff00000) { return x - x; } /* argument reduction needed */ else { n = __ieee754_rem_pio2 (x, y); switch (n & 3) { case 0: { return __kernel_sin (y[0], y[1], 1); } case 1: { return __kernel_cos (y[0], y[1]); } case 2: { return -__kernel_sin (y[0], y[1], 1); } default: { return -__kernel_cos (y[0], y[1]); } } } } /* sin */ /* cos(x) * Return cosine function of x. * * kernel function: * __kernel_sin ... sine function on [-pi/4,pi/4] * __kernel_cos ... cosine function on [-pi/4,pi/4] * __ieee754_rem_pio2 ... argument reduction routine */ double cos (double x) { double y[2], z = 0.0; int n, ix; /* High word of x. */ ix = __HI (x); /* |x| ~< pi/4 */ ix &= 0x7fffffff; if (ix <= 0x3fe921fb) { return __kernel_cos (x, z); } /* cos(Inf or NaN) is NaN */ else if (ix >= 0x7ff00000) { return x - x; } /* argument reduction needed */ else { n = __ieee754_rem_pio2 (x, y); switch (n & 3) { case 0: { return __kernel_cos (y[0], y[1]); } case 1: { return -__kernel_sin (y[0], y[1], 1); } case 2: { return -__kernel_cos (y[0], y[1]); } default: { return __kernel_sin (y[0], y[1], 1); } } } } /* cos */ /* tan(x) * Return tangent function of x. * * kernel function: * __kernel_tan ... tangent function on [-pi/4,pi/4] * __ieee754_rem_pio2 ... argument reduction routine */ double tan (double x) { double y[2], z = 0.0; int n, ix; /* High word of x. */ ix = __HI (x); /* |x| ~< pi/4 */ ix &= 0x7fffffff; if (ix <= 0x3fe921fb) { return __kernel_tan (x, z, 1); } /* tan(Inf or NaN) is NaN */ else if (ix >= 0x7ff00000) { return x - x; /* NaN */ } /* argument reduction needed */ else { n = __ieee754_rem_pio2 (x, y); return __kernel_tan (y[0], y[1], 1 - ((n & 1) << 1)); /* 1 -- n even, -1 -- n odd */ } } /* tan */ #undef zero #undef half #undef one #undef two24 #undef twon24 #undef invpio2 #undef pio2_1 #undef pio2_1t #undef pio2_2 #undef pio2_2t #undef pio2_3 #undef pio2_3t #undef S1 #undef S2 #undef S3 #undef S4 #undef S5 #undef S6 #undef C1 #undef C2 #undef C3 #undef C4 #undef C5 #undef C6 #undef T0 #undef T1 #undef T2 #undef T3 #undef T4 #undef T5 #undef T6 #undef T7 #undef T8 #undef T9 #undef T10 #undef T11 #undef T12 #undef pio4 #undef pio4lo