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 /* ld80 version of __ieee754_rem_pio2l(x,y) 18 * 19 * return the remainder of x rem pi/2 in y[0]+y[1] 20 * use __kernel_rem_pio2() 21 */ 22 23 #include <float.h> 24 25 #include "math.h" 26 #include "math_private.h" 27 #include "fpmath.h" 28 29 #define BIAS (LDBL_MAX_EXP - 1) 30 31 /* 32 * invpio2: 64 bits of 2/pi 33 * pio2_1: first 39 bits of pi/2 34 * pio2_1t: pi/2 - pio2_1 35 * pio2_2: second 39 bits of pi/2 36 * pio2_2t: pi/2 - (pio2_1+pio2_2) 37 * pio2_3: third 39 bits of pi/2 38 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) 39 */ 40 41 static const double 42 zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 43 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ 44 pio2_1 = 1.57079632679597125389e+00, /* 0x3FF921FB, 0x54444000 */ 45 pio2_2 = -1.07463465549783099519e-12, /* -0x12e7b967674000.0p-92 */ 46 pio2_3 = 6.36831716351370313614e-25; /* 0x18a2e037074000.0p-133 */ 47 48 #if defined(__amd64__) || defined(__i386__) 49 /* Long double constants are slow on these arches, and broken on i386. */ 50 static const volatile double 51 invpio2hi = 6.3661977236758138e-01, /* 0x145f306dc9c883.0p-53 */ 52 invpio2lo = -3.9356538861223811e-17, /* -0x16b00000000000.0p-107 */ 53 pio2_1thi = -1.0746346554971943e-12, /* -0x12e7b9676733af.0p-92 */ 54 pio2_1tlo = 8.8451028997905949e-29, /* 0x1c080000000000.0p-146 */ 55 pio2_2thi = 6.3683171635109499e-25, /* 0x18a2e03707344a.0p-133 */ 56 pio2_2tlo = 2.3183081793789774e-41, /* 0x10280000000000.0p-187 */ 57 pio2_3thi = -2.7529965190440717e-37, /* -0x176b7ed8fbbacc.0p-174 */ 58 pio2_3tlo = -4.2006647512740502e-54; /* -0x19c00000000000.0p-230 */ 59 #define invpio2 ((long double)invpio2hi + invpio2lo) 60 #define pio2_1t ((long double)pio2_1thi + pio2_1tlo) 61 #define pio2_2t ((long double)pio2_2thi + pio2_2tlo) 62 #define pio2_3t ((long double)pio2_3thi + pio2_3tlo) 63 #else 64 static const long double 65 invpio2 = 6.36619772367581343076e-01L, /* 0xa2f9836e4e44152a.0p-64 */ 66 pio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */ 67 pio2_2t = 6.36831716351095013979e-25L, /* 0xc51701b839a25205.0p-144 */ 68 pio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */ 69 #endif 70 71 static inline __always_inline int 72 __ieee754_rem_pio2l(long double x, long double *y) 73 { 74 union IEEEl2bits u,u1; 75 long double z,w,t,r,fn; 76 double tx[3],ty[2]; 77 int e0,ex,i,j,nx,n; 78 int16_t expsign; 79 80 u.e = x; 81 expsign = u.xbits.expsign; 82 ex = expsign & 0x7fff; 83 if (ex < BIAS + 25 || (ex == BIAS + 25 && u.bits.manh < 0xc90fdaa2)) { 84 /* |x| ~< 2^25*(pi/2), medium size */ 85 fn = rnintl(x*invpio2); 86 n = irint(fn); 87 r = x-fn*pio2_1; 88 w = fn*pio2_1t; /* 1st round good to 102 bit */ 89 { 90 union IEEEl2bits u2; 91 int ex1; 92 j = ex; 93 y[0] = r-w; 94 u2.e = y[0]; 95 ex1 = u2.xbits.expsign & 0x7fff; 96 i = j-ex1; 97 if(i>22) { /* 2nd iteration needed, good to 141 */ 98 t = r; 99 w = fn*pio2_2; 100 r = t-w; 101 w = fn*pio2_2t-((t-r)-w); 102 y[0] = r-w; 103 u2.e = y[0]; 104 ex1 = u2.xbits.expsign & 0x7fff; 105 i = j-ex1; 106 if(i>61) { /* 3rd iteration need, 180 bits acc */ 107 t = r; /* will cover all possible cases */ 108 w = fn*pio2_3; 109 r = t-w; 110 w = fn*pio2_3t-((t-r)-w); 111 y[0] = r-w; 112 } 113 } 114 } 115 y[1] = (r-y[0])-w; 116 return n; 117 } 118 /* 119 * all other (large) arguments 120 */ 121 if(ex==0x7fff) { /* x is inf or NaN */ 122 y[0]=y[1]=x-x; return 0; 123 } 124 /* set z = scalbn(|x|,ilogb(x)-23) */ 125 u1.e = x; 126 e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */ 127 u1.xbits.expsign = ex - e0; 128 z = u1.e; 129 for(i=0;i<2;i++) { 130 tx[i] = (double)((int32_t)(z)); 131 z = (z-tx[i])*two24; 132 } 133 tx[2] = z; 134 nx = 3; 135 while(tx[nx-1]==zero) nx--; /* skip zero term */ 136 n = __kernel_rem_pio2(tx,ty,e0,nx,2); 137 r = (long double)ty[0] + ty[1]; 138 w = ty[1] - (r - ty[0]); 139 if(expsign<0) {y[0] = -r; y[1] = -w; return -n;} 140 y[0] = r; y[1] = w; return n; 141 } 142