1 /* 2 * ==================================================== 3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 4 * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans. 5 * 6 * Developed at SunSoft, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this 8 * software is freely granted, provided that this notice 9 * is preserved. 10 * ==================================================== 11 * 12 * Optimized by Bruce D. Evans. 13 */ 14 15 /* ld80 version of __ieee754_rem_pio2l(x,y) 16 * 17 * return the remainder of x rem pi/2 in y[0]+y[1] 18 * use __kernel_rem_pio2() 19 */ 20 21 #include <float.h> 22 23 #include "math.h" 24 #include "math_private.h" 25 #include "fpmath.h" 26 27 #define BIAS (LDBL_MAX_EXP - 1) 28 29 /* 30 * invpio2: 64 bits of 2/pi 31 * pio2_1: first 39 bits of pi/2 32 * pio2_1t: pi/2 - pio2_1 33 * pio2_2: second 39 bits of pi/2 34 * pio2_2t: pi/2 - (pio2_1+pio2_2) 35 * pio2_3: third 39 bits of pi/2 36 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) 37 */ 38 39 static const double 40 zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 41 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ 42 pio2_1 = 1.57079632679597125389e+00, /* 0x3FF921FB, 0x54444000 */ 43 pio2_2 = -1.07463465549783099519e-12, /* -0x12e7b967674000.0p-92 */ 44 pio2_3 = 6.36831716351370313614e-25; /* 0x18a2e037074000.0p-133 */ 45 46 #if defined(__amd64__) || defined(__i386__) 47 /* Long double constants are slow on these arches, and broken on i386. */ 48 static const volatile double 49 invpio2hi = 6.3661977236758138e-01, /* 0x145f306dc9c883.0p-53 */ 50 invpio2lo = -3.9356538861223811e-17, /* -0x16b00000000000.0p-107 */ 51 pio2_1thi = -1.0746346554971943e-12, /* -0x12e7b9676733af.0p-92 */ 52 pio2_1tlo = 8.8451028997905949e-29, /* 0x1c080000000000.0p-146 */ 53 pio2_2thi = 6.3683171635109499e-25, /* 0x18a2e03707344a.0p-133 */ 54 pio2_2tlo = 2.3183081793789774e-41, /* 0x10280000000000.0p-187 */ 55 pio2_3thi = -2.7529965190440717e-37, /* -0x176b7ed8fbbacc.0p-174 */ 56 pio2_3tlo = -4.2006647512740502e-54; /* -0x19c00000000000.0p-230 */ 57 #define invpio2 ((long double)invpio2hi + invpio2lo) 58 #define pio2_1t ((long double)pio2_1thi + pio2_1tlo) 59 #define pio2_2t ((long double)pio2_2thi + pio2_2tlo) 60 #define pio2_3t ((long double)pio2_3thi + pio2_3tlo) 61 #else 62 static const long double 63 invpio2 = 6.36619772367581343076e-01L, /* 0xa2f9836e4e44152a.0p-64 */ 64 pio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */ 65 pio2_2t = 6.36831716351095013979e-25L, /* 0xc51701b839a25205.0p-144 */ 66 pio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */ 67 #endif 68 69 static inline __always_inline int 70 __ieee754_rem_pio2l(long double x, long double *y) 71 { 72 union IEEEl2bits u,u1; 73 long double z,w,t,r,fn; 74 double tx[3],ty[2]; 75 int e0,ex,i,j,nx,n; 76 int16_t expsign; 77 78 u.e = x; 79 expsign = u.xbits.expsign; 80 ex = expsign & 0x7fff; 81 if (ex < BIAS + 25 || (ex == BIAS + 25 && u.bits.manh < 0xc90fdaa2)) { 82 /* |x| ~< 2^25*(pi/2), medium size */ 83 fn = rnintl(x*invpio2); 84 n = irint(fn); 85 r = x-fn*pio2_1; 86 w = fn*pio2_1t; /* 1st round good to 102 bit */ 87 { 88 union IEEEl2bits u2; 89 int ex1; 90 j = ex; 91 y[0] = r-w; 92 u2.e = y[0]; 93 ex1 = u2.xbits.expsign & 0x7fff; 94 i = j-ex1; 95 if(i>22) { /* 2nd iteration needed, good to 141 */ 96 t = r; 97 w = fn*pio2_2; 98 r = t-w; 99 w = fn*pio2_2t-((t-r)-w); 100 y[0] = r-w; 101 u2.e = y[0]; 102 ex1 = u2.xbits.expsign & 0x7fff; 103 i = j-ex1; 104 if(i>61) { /* 3rd iteration need, 180 bits acc */ 105 t = r; /* will cover all possible cases */ 106 w = fn*pio2_3; 107 r = t-w; 108 w = fn*pio2_3t-((t-r)-w); 109 y[0] = r-w; 110 } 111 } 112 } 113 y[1] = (r-y[0])-w; 114 return n; 115 } 116 /* 117 * all other (large) arguments 118 */ 119 if(ex==0x7fff) { /* x is inf or NaN */ 120 y[0]=y[1]=x-x; return 0; 121 } 122 /* set z = scalbn(|x|,ilogb(x)-23) */ 123 u1.e = x; 124 e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */ 125 u1.xbits.expsign = ex - e0; 126 z = u1.e; 127 for(i=0;i<2;i++) { 128 tx[i] = (double)((int32_t)(z)); 129 z = (z-tx[i])*two24; 130 } 131 tx[2] = z; 132 nx = 3; 133 while(tx[nx-1]==zero) nx--; /* skip zero term */ 134 n = __kernel_rem_pio2(tx,ty,e0,nx,2); 135 r = (long double)ty[0] + ty[1]; 136 w = ty[1] - (r - ty[0]); 137 if(expsign<0) {y[0] = -r; y[1] = -w; return -n;} 138 y[0] = r; y[1] = w; return n; 139 } 140