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 __FBSDID("$FreeBSD$"); 18 19 /* ld80 version of __ieee754_rem_pio2l(x,y) 20 * 21 * return the remainder of x rem pi/2 in y[0]+y[1] 22 * use __kernel_rem_pio2() 23 */ 24 25 #include <float.h> 26 27 #include "math.h" 28 #include "math_private.h" 29 #include "fpmath.h" 30 31 #define BIAS (LDBL_MAX_EXP - 1) 32 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 43 static const double 44 zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 45 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ 46 pio2_1 = 1.57079632679597125389e+00, /* 0x3FF921FB, 0x54444000 */ 47 pio2_2 = -1.07463465549783099519e-12, /* -0x12e7b967674000.0p-92 */ 48 pio2_3 = 6.36831716351370313614e-25; /* 0x18a2e037074000.0p-133 */ 49 50 #if defined(__amd64__) || defined(__i386__) 51 /* Long double constants are slow on these arches, and broken on i386. */ 52 static const volatile double 53 invpio2hi = 6.3661977236758138e-01, /* 0x145f306dc9c883.0p-53 */ 54 invpio2lo = -3.9356538861223811e-17, /* -0x16b00000000000.0p-107 */ 55 pio2_1thi = -1.0746346554971943e-12, /* -0x12e7b9676733af.0p-92 */ 56 pio2_1tlo = 8.8451028997905949e-29, /* 0x1c080000000000.0p-146 */ 57 pio2_2thi = 6.3683171635109499e-25, /* 0x18a2e03707344a.0p-133 */ 58 pio2_2tlo = 2.3183081793789774e-41, /* 0x10280000000000.0p-187 */ 59 pio2_3thi = -2.7529965190440717e-37, /* -0x176b7ed8fbbacc.0p-174 */ 60 pio2_3tlo = -4.2006647512740502e-54; /* -0x19c00000000000.0p-230 */ 61 #define invpio2 ((long double)invpio2hi + invpio2lo) 62 #define pio2_1t ((long double)pio2_1thi + pio2_1tlo) 63 #define pio2_2t ((long double)pio2_2thi + pio2_2tlo) 64 #define pio2_3t ((long double)pio2_3thi + pio2_3tlo) 65 #else 66 static const long double 67 invpio2 = 6.36619772367581343076e-01L, /* 0xa2f9836e4e44152a.0p-64 */ 68 pio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */ 69 pio2_2t = 6.36831716351095013979e-25L, /* 0xc51701b839a25205.0p-144 */ 70 pio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */ 71 #endif 72 73 static inline __always_inline int 74 __ieee754_rem_pio2l(long double x, long double *y) 75 { 76 union IEEEl2bits u,u1; 77 long double z,w,t,r,fn; 78 double tx[3],ty[2]; 79 int e0,ex,i,j,nx,n; 80 int16_t expsign; 81 82 u.e = x; 83 expsign = u.xbits.expsign; 84 ex = expsign & 0x7fff; 85 if (ex < BIAS + 25 || (ex == BIAS + 25 && u.bits.manh < 0xc90fdaa2)) { 86 /* |x| ~< 2^25*(pi/2), medium size */ 87 /* Use a specialized rint() to get fn. Assume round-to-nearest. */ 88 fn = x*invpio2+0x1.8p63; 89 fn = fn-0x1.8p63; 90 #ifdef HAVE_EFFICIENT_IRINT 91 n = irint(fn); 92 #else 93 n = fn; 94 #endif 95 r = x-fn*pio2_1; 96 w = fn*pio2_1t; /* 1st round good to 102 bit */ 97 { 98 union IEEEl2bits u2; 99 int ex1; 100 j = ex; 101 y[0] = r-w; 102 u2.e = y[0]; 103 ex1 = u2.xbits.expsign & 0x7fff; 104 i = j-ex1; 105 if(i>22) { /* 2nd iteration needed, good to 141 */ 106 t = r; 107 w = fn*pio2_2; 108 r = t-w; 109 w = fn*pio2_2t-((t-r)-w); 110 y[0] = r-w; 111 u2.e = y[0]; 112 ex1 = u2.xbits.expsign & 0x7fff; 113 i = j-ex1; 114 if(i>61) { /* 3rd iteration need, 180 bits acc */ 115 t = r; /* will cover all possible cases */ 116 w = fn*pio2_3; 117 r = t-w; 118 w = fn*pio2_3t-((t-r)-w); 119 y[0] = r-w; 120 } 121 } 122 } 123 y[1] = (r-y[0])-w; 124 return n; 125 } 126 /* 127 * all other (large) arguments 128 */ 129 if(ex==0x7fff) { /* x is inf or NaN */ 130 y[0]=y[1]=x-x; return 0; 131 } 132 /* set z = scalbn(|x|,ilogb(x)-23) */ 133 u1.e = x; 134 e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */ 135 u1.xbits.expsign = ex - e0; 136 z = u1.e; 137 for(i=0;i<2;i++) { 138 tx[i] = (double)((int32_t)(z)); 139 z = (z-tx[i])*two24; 140 } 141 tx[2] = z; 142 nx = 3; 143 while(tx[nx-1]==zero) nx--; /* skip zero term */ 144 n = __kernel_rem_pio2(tx,ty,e0,nx,2); 145 r = (long double)ty[0] + ty[1]; 146 w = ty[1] - (r - ty[0]); 147 if(expsign<0) {y[0] = -r; y[1] = -w; return -n;} 148 y[0] = r; y[1] = w; return n; 149 } 150