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 23 static 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 */ 42 static const double 43 pio2_1 = 1.57079632679597125389e+00, /* 0x3FF921FB, 0x54444000 */ 44 pio2_2 = -1.07463465549783099519e-12, /* -0x12e7b967674000.0p-92 */ 45 pio2_3 = 6.36831716351370313614e-25; /* 0x18a2e037074000.0p-133 */ 46 static const long double 47 invpio2 = 6.36619772367581343076e-01L, /* 0xa2f9836e4e44152a.0p-64 */ 48 pio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */ 49 pio2_2t = 6.36831716351095013979e-25L, /* 0xc51701b839a25205.0p-144 */ 50 pio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */ 51 #elif LDBL_MANT_DIG == 113 52 /* u ~< 0x1p45*pi/2 */ 53 #define SMALL(u) (((u.i.se & 0x7fffU)<<16 | u.i.top) < ((0x3fff + 45)<<16 | 0x921f)) 54 #define QUOBITS(x) ((uint32_t)(int64_t)x & 0x7fffffff) 55 #define ROUND1 51 56 #define ROUND2 119 57 #define NX 5 58 #define NY 3 59 static const long double 60 invpio2 = 6.3661977236758134307553505349005747e-01L, /* 0x145f306dc9c882a53f84eafa3ea6a.0p-113 */ 61 pio2_1 = 1.5707963267948966192292994253909555e+00L, /* 0x1921fb54442d18469800000000000.0p-112 */ 62 pio2_1t = 2.0222662487959507323996846200947577e-21L, /* 0x13198a2e03707344a4093822299f3.0p-181 */ 63 pio2_2 = 2.0222662487959507323994779168837751e-21L, /* 0x13198a2e03707344a400000000000.0p-181 */ 64 pio2_2t = 2.0670321098263988236496903051604844e-43L, /* 0x127044533e63a0105df531d89cd91.0p-254 */ 65 pio2_3 = 2.0670321098263988236499468110329591e-43L, /* 0x127044533e63a0105e00000000000.0p-254 */ 66 pio2_3t = -2.5650587247459238361625433492959285e-65L; /* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */ 67 #endif 68 69 int __rem_pio2l(long double x, long double *y) 70 { 71 union ldshape u,uz; 72 long double z,w,t,r,fn; 73 double tx[NX],ty[NY]; 74 int ex,ey,n,i; 75 76 u.f = x; 77 ex = u.i.se & 0x7fff; 78 if (SMALL(u)) { 79 /* rint(x/(pi/2)), Assume round-to-nearest. */ 80 fn = x*invpio2 + toint - toint; 81 n = QUOBITS(fn); 82 r = x-fn*pio2_1; 83 w = fn*pio2_1t; /* 1st round good to 102/180 bits (ld80/ld128) */ 84 y[0] = r-w; 85 u.f = y[0]; 86 ey = u.i.se & 0x7fff; 87 if (ex - ey > ROUND1) { /* 2nd iteration needed, good to 141/248 (ld80/ld128) */ 88 t = r; 89 w = fn*pio2_2; 90 r = t-w; 91 w = fn*pio2_2t-((t-r)-w); 92 y[0] = r-w; 93 u.f = y[0]; 94 ey = u.i.se & 0x7fff; 95 if (ex - ey > ROUND2) { /* 3rd iteration, good to 180/316 bits */ 96 t = r; /* will cover all possible cases (not verified for ld128) */ 97 w = fn*pio2_3; 98 r = t-w; 99 w = fn*pio2_3t-((t-r)-w); 100 y[0] = r-w; 101 } 102 } 103 y[1] = (r - y[0]) - w; 104 return n; 105 } 106 /* 107 * all other (large) arguments 108 */ 109 if (ex == 0x7fff) { /* x is inf or NaN */ 110 y[0] = y[1] = x - x; 111 return 0; 112 } 113 /* set z = scalbn(|x|,-ilogb(x)+23) */ 114 uz.f = x; 115 uz.i.se = 0x3fff + 23; 116 z = uz.f; 117 for (i=0; i < NX - 1; i++) { 118 tx[i] = (double)(int32_t)z; 119 z = (z-tx[i])*0x1p24; 120 } 121 tx[i] = z; 122 while (tx[i] == 0) 123 i--; 124 n = __rem_pio2_large(tx, ty, ex-0x3fff-23, i+1, NY); 125 w = ty[1]; 126 if (NY == 3) 127 w += ty[2]; 128 r = ty[0] + w; 129 /* TODO: for ld128 this does not follow the recommendation of the 130 comments of __rem_pio2_large which seem wrong if |ty[0]| > |ty[1]+ty[2]| */ 131 w -= r - ty[0]; 132 if (u.i.se >> 15) { 133 y[0] = -r; 134 y[1] = -w; 135 return -n; 136 } 137 y[0] = r; 138 y[1] = w; 139 return n; 140 } 141 #endif 142