1 /* @(#)e_rem_pio2.c 5.1 93/09/24 */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunPro, 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 13 #ifndef lint 14 static char rcsid[] = "$FreeBSD$"; 15 #endif 16 17 /* __ieee754_rem_pio2(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 "math.h" 24 #include "math_private.h" 25 26 /* 27 * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi 28 */ 29 static const int32_t two_over_pi[] = { 30 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, 31 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, 32 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, 33 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41, 34 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8, 35 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF, 36 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, 37 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, 38 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, 39 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, 40 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B, 41 }; 42 43 static const int32_t npio2_hw[] = { 44 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C, 45 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C, 46 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A, 47 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C, 48 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB, 49 0x404858EB, 0x404921FB, 50 }; 51 52 /* 53 * invpio2: 53 bits of 2/pi 54 * pio2_1: first 33 bit of pi/2 55 * pio2_1t: pi/2 - pio2_1 56 * pio2_2: second 33 bit of pi/2 57 * pio2_2t: pi/2 - (pio2_1+pio2_2) 58 * pio2_3: third 33 bit of pi/2 59 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) 60 */ 61 62 static const double 63 zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 64 half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 65 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ 66 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ 67 pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */ 68 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */ 69 pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */ 70 pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */ 71 pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */ 72 pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */ 73 74 int32_t __ieee754_rem_pio2(double x, double *y) 75 { 76 double z,w,t,r,fn; 77 double tx[3]; 78 int32_t e0,i,j,nx,n,ix,hx; 79 u_int32_t low; 80 81 GET_HIGH_WORD(hx,x); /* high word of x */ 82 ix = hx&0x7fffffff; 83 if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */ 84 {y[0] = x; y[1] = 0; return 0;} 85 if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */ 86 if(hx>0) { 87 z = x - pio2_1; 88 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ 89 y[0] = z - pio2_1t; 90 y[1] = (z-y[0])-pio2_1t; 91 } else { /* near pi/2, use 33+33+53 bit pi */ 92 z -= pio2_2; 93 y[0] = z - pio2_2t; 94 y[1] = (z-y[0])-pio2_2t; 95 } 96 return 1; 97 } else { /* negative x */ 98 z = x + pio2_1; 99 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ 100 y[0] = z + pio2_1t; 101 y[1] = (z-y[0])+pio2_1t; 102 } else { /* near pi/2, use 33+33+53 bit pi */ 103 z += pio2_2; 104 y[0] = z + pio2_2t; 105 y[1] = (z-y[0])+pio2_2t; 106 } 107 return -1; 108 } 109 } 110 if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */ 111 t = fabs(x); 112 n = (int32_t) (t*invpio2+half); 113 fn = (double)n; 114 r = t-fn*pio2_1; 115 w = fn*pio2_1t; /* 1st round good to 85 bit */ 116 if(n<32&&ix!=npio2_hw[n-1]) { 117 y[0] = r-w; /* quick check no cancellation */ 118 } else { 119 u_int32_t high; 120 j = ix>>20; 121 y[0] = r-w; 122 GET_HIGH_WORD(high,y[0]); 123 i = j-((high>>20)&0x7ff); 124 if(i>16) { /* 2nd iteration needed, good to 118 */ 125 t = r; 126 w = fn*pio2_2; 127 r = t-w; 128 w = fn*pio2_2t-((t-r)-w); 129 y[0] = r-w; 130 GET_HIGH_WORD(high,y[0]); 131 i = j-((high>>20)&0x7ff); 132 if(i>49) { /* 3rd iteration need, 151 bits acc */ 133 t = r; /* will cover all possible cases */ 134 w = fn*pio2_3; 135 r = t-w; 136 w = fn*pio2_3t-((t-r)-w); 137 y[0] = r-w; 138 } 139 } 140 } 141 y[1] = (r-y[0])-w; 142 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} 143 else return n; 144 } 145 /* 146 * all other (large) arguments 147 */ 148 if(ix>=0x7ff00000) { /* x is inf or NaN */ 149 y[0]=y[1]=x-x; return 0; 150 } 151 /* set z = scalbn(|x|,ilogb(x)-23) */ 152 GET_LOW_WORD(low,x); 153 SET_LOW_WORD(z,low); 154 e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */ 155 SET_HIGH_WORD(z, ix - ((int32_t)(e0<<20))); 156 for(i=0;i<2;i++) { 157 tx[i] = (double)((int32_t)(z)); 158 z = (z-tx[i])*two24; 159 } 160 tx[2] = z; 161 nx = 3; 162 while(tx[nx-1]==zero) nx--; /* skip zero term */ 163 n = __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi); 164 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} 165 return n; 166 } 167