1 /* derived from /netlib/fdlibm */ 2 3 /* @(#)e_rem_pio2.c 1.4 95/01/18 */ 4 /* 5 * ==================================================== 6 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 7 * 8 * Developed at SunSoft, a Sun Microsystems, Inc. business. 9 * Permission to use, copy, modify, and distribute this 10 * software is freely granted, provided that this notice 11 * is preserved. 12 * ==================================================== 13 * 14 */ 15 16 /* __ieee754_rem_pio2(x,y) 17 * 18 * return the remainder of x rem pi/2 in y[0]+y[1] 19 * use __kernel_rem_pio2() 20 */ 21 22 #include "fdlibm.h" 23 24 /* 25 * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi 26 */ 27 static const int two_over_pi[] = { 28 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, 29 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, 30 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, 31 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41, 32 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8, 33 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF, 34 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, 35 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, 36 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, 37 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, 38 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B, 39 }; 40 41 static const int npio2_hw[] = { 42 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C, 43 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C, 44 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A, 45 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C, 46 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB, 47 0x404858EB, 0x404921FB, 48 }; 49 50 /* 51 * invpio2: 53 bits of 2/pi 52 * pio2_1: first 33 bit of pi/2 53 * pio2_1t: pi/2 - pio2_1 54 * pio2_2: second 33 bit of pi/2 55 * pio2_2t: pi/2 - (pio2_1+pio2_2) 56 * pio2_3: third 33 bit of pi/2 57 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) 58 */ 59 60 static const double 61 zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 62 half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 63 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ 64 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ 65 pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */ 66 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */ 67 pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */ 68 pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */ 69 pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */ 70 pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */ 71 72 int __ieee754_rem_pio2(double x, double *y) 73 { 74 double z,w,t,r,fn; 75 double tx[3]; 76 int e0,i,j,nx,n,ix,hx; 77 78 hx = __HI(x); /* high word of x */ 79 ix = hx&0x7fffffff; 80 if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */ 81 {y[0] = x; y[1] = 0; return 0;} 82 if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */ 83 if(hx>0) { 84 z = x - pio2_1; 85 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ 86 y[0] = z - pio2_1t; 87 y[1] = (z-y[0])-pio2_1t; 88 } else { /* near pi/2, use 33+33+53 bit pi */ 89 z -= pio2_2; 90 y[0] = z - pio2_2t; 91 y[1] = (z-y[0])-pio2_2t; 92 } 93 return 1; 94 } else { /* negative x */ 95 z = x + pio2_1; 96 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ 97 y[0] = z + pio2_1t; 98 y[1] = (z-y[0])+pio2_1t; 99 } else { /* near pi/2, use 33+33+53 bit pi */ 100 z += pio2_2; 101 y[0] = z + pio2_2t; 102 y[1] = (z-y[0])+pio2_2t; 103 } 104 return -1; 105 } 106 } 107 if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */ 108 t = fabs(x); 109 n = (int) (t*invpio2+half); 110 fn = (double)n; 111 r = t-fn*pio2_1; 112 w = fn*pio2_1t; /* 1st round good to 85 bit */ 113 if(n<32&&ix!=npio2_hw[n-1]) { 114 y[0] = r-w; /* quick check no cancellation */ 115 } else { 116 j = ix>>20; 117 y[0] = r-w; 118 i = j-(((__HI(y[0]))>>20)&0x7ff); 119 if(i>16) { /* 2nd iteration needed, good to 118 */ 120 t = r; 121 w = fn*pio2_2; 122 r = t-w; 123 w = fn*pio2_2t-((t-r)-w); 124 y[0] = r-w; 125 i = j-(((__HI(y[0]))>>20)&0x7ff); 126 if(i>49) { /* 3rd iteration need, 151 bits acc */ 127 t = r; /* will cover all possible cases */ 128 w = fn*pio2_3; 129 r = t-w; 130 w = fn*pio2_3t-((t-r)-w); 131 y[0] = r-w; 132 } 133 } 134 } 135 y[1] = (r-y[0])-w; 136 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} 137 else return n; 138 } 139 /* 140 * all other (large) arguments 141 */ 142 if(ix>=0x7ff00000) { /* x is inf or NaN */ 143 y[0]=y[1]=x-x; return 0; 144 } 145 /* set z = scalbn(|x|,ilogb(x)-23) */ 146 __LO(z) = __LO(x); 147 e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */ 148 __HI(z) = ix - (e0<<20); 149 for(i=0;i<2;i++) { 150 tx[i] = (double)((int)(z)); 151 z = (z-tx[i])*two24; 152 } 153 tx[2] = z; 154 nx = 3; 155 while(tx[nx-1]==zero) nx--; /* skip zero term */ 156 n = __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi); 157 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} 158 return n; 159 } 160