1/* origin: FreeBSD /usr/src/lib/msun/ld80/e_rem_pio2.c */ 2/* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans. 6 * 7 * Developed at SunSoft, a Sun Microsystems, Inc. business. 8 * Permission to use, copy, modify, and distribute this 9 * software is freely granted, provided that this notice 10 * is preserved. 11 * ==================================================== 12 * 13 * Optimized by Bruce D. Evans. 14 */ 15#include "libm.h" 16#if (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384 17/* ld80 and ld128 version of __rem_pio2(x,y) 18 * 19 * return the remainder of x rem pi/2 in y[0]+y[1] 20 * use __rem_pio2_large() for large x 21 */ 22 23static const long double toint = 1.5/LDBL_EPSILON; 24 25#if LDBL_MANT_DIG == 64 26/* u ~< 0x1p25*pi/2 */ 27#define SMALL(u) (((u.i.se & 0x7fffU)<<16 | u.i.m>>48) < ((0x3fff + 25)<<16 | 0x921f>>1 | 0x8000)) 28#define QUOBITS(x) ((uint32_t)(int32_t)x & 0x7fffffff) 29#define ROUND1 22 30#define ROUND2 61 31#define NX 3 32#define NY 2 33/* 34 * invpio2: 64 bits of 2/pi 35 * pio2_1: first 39 bits of pi/2 36 * pio2_1t: pi/2 - pio2_1 37 * pio2_2: second 39 bits of pi/2 38 * pio2_2t: pi/2 - (pio2_1+pio2_2) 39 * pio2_3: third 39 bits of pi/2 40 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) 41 */ 42static const double 43pio2_1 = 1.57079632679597125389e+00, /* 0x3FF921FB, 0x54444000 */ 44pio2_2 = -1.07463465549783099519e-12, /* -0x12e7b967674000.0p-92 */ 45pio2_3 = 6.36831716351370313614e-25; /* 0x18a2e037074000.0p-133 */ 46static const long double 47pio4 = 0x1.921fb54442d1846ap-1L, 48invpio2 = 6.36619772367581343076e-01L, /* 0xa2f9836e4e44152a.0p-64 */ 49pio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */ 50pio2_2t = 6.36831716351095013979e-25L, /* 0xc51701b839a25205.0p-144 */ 51pio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */ 52#elif LDBL_MANT_DIG == 113 53/* u ~< 0x1p45*pi/2 */ 54#define SMALL(u) (((u.i.se & 0x7fffU)<<16 | u.i.top) < ((0x3fff + 45)<<16 | 0x921f)) 55#define QUOBITS(x) ((uint32_t)(int64_t)x & 0x7fffffff) 56#define ROUND1 51 57#define ROUND2 119 58#define NX 5 59#define NY 3 60static const long double 61pio4 = 0x1.921fb54442d18469898cc51701b8p-1L, 62invpio2 = 6.3661977236758134307553505349005747e-01L, /* 0x145f306dc9c882a53f84eafa3ea6a.0p-113 */ 63pio2_1 = 1.5707963267948966192292994253909555e+00L, /* 0x1921fb54442d18469800000000000.0p-112 */ 64pio2_1t = 2.0222662487959507323996846200947577e-21L, /* 0x13198a2e03707344a4093822299f3.0p-181 */ 65pio2_2 = 2.0222662487959507323994779168837751e-21L, /* 0x13198a2e03707344a400000000000.0p-181 */ 66pio2_2t = 2.0670321098263988236496903051604844e-43L, /* 0x127044533e63a0105df531d89cd91.0p-254 */ 67pio2_3 = 2.0670321098263988236499468110329591e-43L, /* 0x127044533e63a0105e00000000000.0p-254 */ 68pio2_3t = -2.5650587247459238361625433492959285e-65L; /* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */ 69#endif 70 71int __rem_pio2l(long double x, long double *y) 72{ 73 union ldshape u,uz; 74 long double z,w,t,r,fn; 75 double tx[NX],ty[NY]; 76 int ex,ey,n,i; 77 78 u.f = x; 79 ex = u.i.se & 0x7fff; 80 if (SMALL(u)) { 81 /* rint(x/(pi/2)) */ 82 fn = x*invpio2 + toint - toint; 83 n = QUOBITS(fn); 84 r = x-fn*pio2_1; 85 w = fn*pio2_1t; /* 1st round good to 102/180 bits (ld80/ld128) */ 86 /* Matters with directed rounding. */ 87 if (predict_false(r - w < -pio4)) { 88 n--; 89 fn--; 90 r = x - fn*pio2_1; 91 w = fn*pio2_1t; 92 } else if (predict_false(r - w > pio4)) { 93 n++; 94 fn++; 95 r = x - fn*pio2_1; 96 w = fn*pio2_1t; 97 } 98 y[0] = r-w; 99 u.f = y[0]; 100 ey = u.i.se & 0x7fff; 101 if (ex - ey > ROUND1) { /* 2nd iteration needed, good to 141/248 (ld80/ld128) */ 102 t = r; 103 w = fn*pio2_2; 104 r = t-w; 105 w = fn*pio2_2t-((t-r)-w); 106 y[0] = r-w; 107 u.f = y[0]; 108 ey = u.i.se & 0x7fff; 109 if (ex - ey > ROUND2) { /* 3rd iteration, good to 180/316 bits */ 110 t = r; /* will cover all possible cases (not verified for ld128) */ 111 w = fn*pio2_3; 112 r = t-w; 113 w = fn*pio2_3t-((t-r)-w); 114 y[0] = r-w; 115 } 116 } 117 y[1] = (r - y[0]) - w; 118 return n; 119 } 120 /* 121 * all other (large) arguments 122 */ 123 if (ex == 0x7fff) { /* x is inf or NaN */ 124 y[0] = y[1] = x - x; 125 return 0; 126 } 127 /* set z = scalbn(|x|,-ilogb(x)+23) */ 128 uz.f = x; 129 uz.i.se = 0x3fff + 23; 130 z = uz.f; 131 for (i=0; i < NX - 1; i++) { 132 tx[i] = (double)(int32_t)z; 133 z = (z-tx[i])*0x1p24; 134 } 135 tx[i] = z; 136 while (tx[i] == 0) 137 i--; 138 n = __rem_pio2_large(tx, ty, ex-0x3fff-23, i+1, NY); 139 w = ty[1]; 140 if (NY == 3) 141 w += ty[2]; 142 r = ty[0] + w; 143 /* TODO: for ld128 this does not follow the recommendation of the 144 comments of __rem_pio2_large which seem wrong if |ty[0]| > |ty[1]+ty[2]| */ 145 w -= r - ty[0]; 146 if (u.i.se >> 15) { 147 y[0] = -r; 148 y[1] = -w; 149 return -n; 150 } 151 y[0] = r; 152 y[1] = w; 153 return n; 154} 155#endif 156