1 /* origin: FreeBSD /usr/src/lib/msun/src/e_rem_pio2.c */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 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 /* __rem_pio2(x,y) 15 * 16 * return the remainder of x rem pi/2 in y[0]+y[1] 17 * use __rem_pio2_large() for large x 18 */ 19 20 #include "libm.h" 21 22 #if FLT_EVAL_METHOD==0 || FLT_EVAL_METHOD==1 23 #define EPS DBL_EPSILON 24 #elif FLT_EVAL_METHOD==2 25 #define EPS LDBL_EPSILON 26 #endif 27 28 /* 29 * invpio2: 53 bits of 2/pi 30 * pio2_1: first 33 bit of pi/2 31 * pio2_1t: pi/2 - pio2_1 32 * pio2_2: second 33 bit of pi/2 33 * pio2_2t: pi/2 - (pio2_1+pio2_2) 34 * pio2_3: third 33 bit of pi/2 35 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) 36 */ 37 static const double 38 toint = 1.5/EPS, 39 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ 40 pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */ 41 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */ 42 pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */ 43 pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */ 44 pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */ 45 pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */ 46 47 /* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */ 48 int __rem_pio2(double x, double *y) 49 { 50 union {double f; uint64_t i;} u = {x}; 51 double_t z,w,t,r,fn; 52 double tx[3],ty[2]; 53 uint32_t ix; 54 int sign, n, ex, ey, i; 55 56 sign = u.i>>63; 57 ix = u.i>>32 & 0x7fffffff; 58 if (ix <= 0x400f6a7a) { /* |x| ~<= 5pi/4 */ 59 if ((ix & 0xfffff) == 0x921fb) /* |x| ~= pi/2 or 2pi/2 */ 60 goto medium; /* cancellation -- use medium case */ 61 if (ix <= 0x4002d97c) { /* |x| ~<= 3pi/4 */ 62 if (!sign) { 63 z = x - pio2_1; /* one round good to 85 bits */ 64 y[0] = z - pio2_1t; 65 y[1] = (z-y[0]) - pio2_1t; 66 return 1; 67 } else { 68 z = x + pio2_1; 69 y[0] = z + pio2_1t; 70 y[1] = (z-y[0]) + pio2_1t; 71 return -1; 72 } 73 } else { 74 if (!sign) { 75 z = x - 2*pio2_1; 76 y[0] = z - 2*pio2_1t; 77 y[1] = (z-y[0]) - 2*pio2_1t; 78 return 2; 79 } else { 80 z = x + 2*pio2_1; 81 y[0] = z + 2*pio2_1t; 82 y[1] = (z-y[0]) + 2*pio2_1t; 83 return -2; 84 } 85 } 86 } 87 if (ix <= 0x401c463b) { /* |x| ~<= 9pi/4 */ 88 if (ix <= 0x4015fdbc) { /* |x| ~<= 7pi/4 */ 89 if (ix == 0x4012d97c) /* |x| ~= 3pi/2 */ 90 goto medium; 91 if (!sign) { 92 z = x - 3*pio2_1; 93 y[0] = z - 3*pio2_1t; 94 y[1] = (z-y[0]) - 3*pio2_1t; 95 return 3; 96 } else { 97 z = x + 3*pio2_1; 98 y[0] = z + 3*pio2_1t; 99 y[1] = (z-y[0]) + 3*pio2_1t; 100 return -3; 101 } 102 } else { 103 if (ix == 0x401921fb) /* |x| ~= 4pi/2 */ 104 goto medium; 105 if (!sign) { 106 z = x - 4*pio2_1; 107 y[0] = z - 4*pio2_1t; 108 y[1] = (z-y[0]) - 4*pio2_1t; 109 return 4; 110 } else { 111 z = x + 4*pio2_1; 112 y[0] = z + 4*pio2_1t; 113 y[1] = (z-y[0]) + 4*pio2_1t; 114 return -4; 115 } 116 } 117 } 118 if (ix < 0x413921fb) { /* |x| ~< 2^20*(pi/2), medium size */ 119 medium: 120 /* rint(x/(pi/2)), Assume round-to-nearest. */ 121 fn = (double_t)x*invpio2 + toint - toint; 122 n = (int32_t)fn; 123 r = x - fn*pio2_1; 124 w = fn*pio2_1t; /* 1st round, good to 85 bits */ 125 y[0] = r - w; 126 u.f = y[0]; 127 ey = u.i>>52 & 0x7ff; 128 ex = ix>>20; 129 if (ex - ey > 16) { /* 2nd round, good to 118 bits */ 130 t = r; 131 w = fn*pio2_2; 132 r = t - w; 133 w = fn*pio2_2t - ((t-r)-w); 134 y[0] = r - w; 135 u.f = y[0]; 136 ey = u.i>>52 & 0x7ff; 137 if (ex - ey > 49) { /* 3rd round, good to 151 bits, covers all cases */ 138 t = r; 139 w = fn*pio2_3; 140 r = t - w; 141 w = fn*pio2_3t - ((t-r)-w); 142 y[0] = r - w; 143 } 144 } 145 y[1] = (r - y[0]) - w; 146 return n; 147 } 148 /* 149 * all other (large) arguments 150 */ 151 if (ix >= 0x7ff00000) { /* x is inf or NaN */ 152 y[0] = y[1] = x - x; 153 return 0; 154 } 155 /* set z = scalbn(|x|,-ilogb(x)+23) */ 156 u.f = x; 157 u.i &= (uint64_t)-1>>12; 158 u.i |= (uint64_t)(0x3ff + 23)<<52; 159 z = u.f; 160 for (i=0; i < 2; i++) { 161 tx[i] = (double)(int32_t)z; 162 z = (z-tx[i])*0x1p24; 163 } 164 tx[i] = z; 165 /* skip zero terms, first term is non-zero */ 166 while (tx[i] == 0.0) 167 i--; 168 n = __rem_pio2_large(tx,ty,(int)(ix>>20)-(0x3ff+23),i+1,1); 169 if (sign) { 170 y[0] = -ty[0]; 171 y[1] = -ty[1]; 172 return -n; 173 } 174 y[0] = ty[0]; 175 y[1] = ty[1]; 176 return n; 177 } 178