1 /* From: @(#)e_rem_pio2.c 1.4 95/01/18 */ 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 16 #include <sys/cdefs.h> 17 #if 0 18 __FBSDID("$FreeBSD: head/lib/msun/ld80/e_rem_pio2l.h 336545 2018-07-20 12:42:24Z bde $"); 19 #endif 20 21 /* ld80 version of __ieee754_rem_pio2l(x,y) 22 * 23 * return the remainder of x rem pi/2 in y[0]+y[1] 24 * use __kernel_rem_pio2() 25 */ 26 27 #include <float.h> 28 #include <machine/ieee.h> 29 30 #include "math.h" 31 #include "math_private.h" 32 33 #define BIAS (LDBL_MAX_EXP - 1) 34 35 /* 36 * invpio2: 64 bits of 2/pi 37 * pio2_1: first 39 bits of pi/2 38 * pio2_1t: pi/2 - pio2_1 39 * pio2_2: second 39 bits of pi/2 40 * pio2_2t: pi/2 - (pio2_1+pio2_2) 41 * pio2_3: third 39 bits of pi/2 42 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) 43 */ 44 45 static const double 46 zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 47 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ 48 pio2_1 = 1.57079632679597125389e+00, /* 0x3FF921FB, 0x54444000 */ 49 pio2_2 = -1.07463465549783099519e-12, /* -0x12e7b967674000.0p-92 */ 50 pio2_3 = 6.36831716351370313614e-25; /* 0x18a2e037074000.0p-133 */ 51 52 #if defined(__amd64__) || defined(__i386__) 53 /* Long double constants are slow on these arches, and broken on i386. */ 54 static const volatile double 55 invpio2hi = 6.3661977236758138e-01, /* 0x145f306dc9c883.0p-53 */ 56 invpio2lo = -3.9356538861223811e-17, /* -0x16b00000000000.0p-107 */ 57 pio2_1thi = -1.0746346554971943e-12, /* -0x12e7b9676733af.0p-92 */ 58 pio2_1tlo = 8.8451028997905949e-29, /* 0x1c080000000000.0p-146 */ 59 pio2_2thi = 6.3683171635109499e-25, /* 0x18a2e03707344a.0p-133 */ 60 pio2_2tlo = 2.3183081793789774e-41, /* 0x10280000000000.0p-187 */ 61 pio2_3thi = -2.7529965190440717e-37, /* -0x176b7ed8fbbacc.0p-174 */ 62 pio2_3tlo = -4.2006647512740502e-54; /* -0x19c00000000000.0p-230 */ 63 #define invpio2 ((long double)invpio2hi + invpio2lo) 64 #define pio2_1t ((long double)pio2_1thi + pio2_1tlo) 65 #define pio2_2t ((long double)pio2_2thi + pio2_2tlo) 66 #define pio2_3t ((long double)pio2_3thi + pio2_3tlo) 67 #else 68 static const long double 69 invpio2 = 6.36619772367581343076e-01L, /* 0xa2f9836e4e44152a.0p-64 */ 70 pio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */ 71 pio2_2t = 6.36831716351095013979e-25L, /* 0xc51701b839a25205.0p-144 */ 72 pio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */ 73 #endif 74 75 static inline __always_inline int 76 __ieee754_rem_pio2l(long double x, long double *y) 77 { 78 union ieee_ext_u u, u1; 79 long double z,w,t,r,fn; 80 double tx[3],ty[2]; 81 int e0,ex,i,j,nx,n; 82 int16_t expsign, expsign1; 83 84 u.extu_ld = x; 85 ex = u.extu_exp; 86 expsign = u.extu_exp | (u.extu_sign << EXT_EXPBITS); 87 if (ex < BIAS + 25 || (ex == BIAS + 25 && u.extu_frach < 0xc90fdaa2U)) { 88 /* |x| ~< 2^25*(pi/2), medium size */ 89 fn = rnintl(x*invpio2); 90 n = irint(fn); 91 r = x-fn*pio2_1; 92 w = fn*pio2_1t; /* 1st round good to 102 bit */ 93 { 94 union ieee_ext_u u2; 95 int ex1; 96 j = ex; 97 y[0] = r-w; 98 u2.extu_ld = y[0]; 99 ex1 = u2.extu_exp; 100 i = j-ex1; 101 if(i>22) { /* 2nd iteration needed, good to 141 */ 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 u2.extu_ld = y[0]; 108 ex1 = u2.extu_exp; 109 i = j-ex1; 110 if(i>61) { /* 3rd iteration need, 180 bits acc */ 111 t = r; /* will cover all possible cases */ 112 w = fn*pio2_3; 113 r = t-w; 114 w = fn*pio2_3t-((t-r)-w); 115 y[0] = r-w; 116 } 117 } 118 } 119 y[1] = (r-y[0])-w; 120 return n; 121 } 122 /* 123 * all other (large) arguments 124 */ 125 if(ex==0x7fff) { /* x is inf or NaN */ 126 y[0]=y[1]=x-x; return 0; 127 } 128 /* set z = scalbn(|x|,ilogb(x)-23) */ 129 u1.extu_ld = x; 130 e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */ 131 expsign1 = ex - e0; 132 u1.extu_exp = expsign1; 133 u1.extu_sign = expsign1 >> EXT_EXPBITS; 134 z = u1.extu_ld; 135 for(i=0;i<2;i++) { 136 tx[i] = (double)((int32_t)(z)); 137 z = (z-tx[i])*two24; 138 } 139 tx[2] = z; 140 nx = 3; 141 while(tx[nx-1]==zero) nx--; /* skip zero term */ 142 n = __kernel_rem_pio2(tx,ty,e0,nx,2); 143 r = (long double)ty[0] + ty[1]; 144 w = ty[1] - (r - ty[0]); 145 if(expsign<0) {y[0] = -r; y[1] = -w; return -n;} 146 y[0] = r; y[1] = w; return n; 147 } 148