nsz Git - musl/blob - src/math/__rem_pio2.c

   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 /*
  23  * invpio2:  53 bits of 2/pi
  24  * pio2_1:   first  33 bit of pi/2
  25  * pio2_1t:  pi/2 - pio2_1
  26  * pio2_2:   second 33 bit of pi/2
  27  * pio2_2t:  pi/2 - (pio2_1+pio2_2)
  28  * pio2_3:   third  33 bit of pi/2
  29  * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
  30  */
  31 static const double
  32 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
  33 pio2_1  = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
  34 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
  35 pio2_2  = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
  36 pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
  37 pio2_3  = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
  38 pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
  39
  40 /* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */
  41 int __rem_pio2(double x, double *y)
  42 {
  43         union {double f; uint64_t i;} u = {x};
  44         double_t z,w,t,r;
  45         double tx[3],ty[2],fn;
  46         uint32_t ix;
  47         int sign, n, ex, ey, i;
  48
  49         sign = u.i>>63;
  50         ix = u.i>>32 & 0x7fffffff;
  51         if (ix <= 0x400f6a7a) {  /* |x| ~<= 5pi/4 */
  52                 if ((ix & 0xfffff) == 0x921fb)  /* |x| ~= pi/2 or 2pi/2 */
  53                         goto medium;  /* cancellation -- use medium case */
  54                 if (ix <= 0x4002d97c) {  /* |x| ~<= 3pi/4 */
  55                         if (!sign) {
  56                                 z = x - pio2_1;  /* one round good to 85 bits */
  57                                 y[0] = z - pio2_1t;
  58                                 y[1] = (z-y[0]) - pio2_1t;
  59                                 return 1;
  60                         } else {
  61                                 z = x + pio2_1;
  62                                 y[0] = z + pio2_1t;
  63                                 y[1] = (z-y[0]) + pio2_1t;
  64                                 return -1;
  65                         }
  66                 } else {
  67                         if (!sign) {
  68                                 z = x - 2*pio2_1;
  69                                 y[0] = z - 2*pio2_1t;
  70                                 y[1] = (z-y[0]) - 2*pio2_1t;
  71                                 return 2;
  72                         } else {
  73                                 z = x + 2*pio2_1;
  74                                 y[0] = z + 2*pio2_1t;
  75                                 y[1] = (z-y[0]) + 2*pio2_1t;
  76                                 return -2;
  77                         }
  78                 }
  79         }
  80         if (ix <= 0x401c463b) {  /* |x| ~<= 9pi/4 */
  81                 if (ix <= 0x4015fdbc) {  /* |x| ~<= 7pi/4 */
  82                         if (ix == 0x4012d97c)  /* |x| ~= 3pi/2 */
  83                                 goto medium;
  84                         if (!sign) {
  85                                 z = x - 3*pio2_1;
  86                                 y[0] = z - 3*pio2_1t;
  87                                 y[1] = (z-y[0]) - 3*pio2_1t;
  88                                 return 3;
  89                         } else {
  90                                 z = x + 3*pio2_1;
  91                                 y[0] = z + 3*pio2_1t;
  92                                 y[1] = (z-y[0]) + 3*pio2_1t;
  93                                 return -3;
  94                         }
  95                 } else {
  96                         if (ix == 0x401921fb)  /* |x| ~= 4pi/2 */
  97                                 goto medium;
  98                         if (!sign) {
  99                                 z = x - 4*pio2_1;
 100                                 y[0] = z - 4*pio2_1t;
 101                                 y[1] = (z-y[0]) - 4*pio2_1t;
 102                                 return 4;
 103                         } else {
 104                                 z = x + 4*pio2_1;
 105                                 y[0] = z + 4*pio2_1t;
 106                                 y[1] = (z-y[0]) + 4*pio2_1t;
 107                                 return -4;
 108                         }
 109                 }
 110         }
 111         if (ix < 0x413921fb) {  /* |x| ~< 2^20*(pi/2), medium size */
 112 medium:
 113                 /* rint(x/(pi/2)), Assume round-to-nearest. */
 114                 fn = x*invpio2 + 0x1.8p52;
 115                 fn = fn - 0x1.8p52;
 116                 n = (int32_t)fn;
 117                 r = x - fn*pio2_1;
 118                 w = fn*pio2_1t;  /* 1st round, good to 85 bits */
 119                 y[0] = r - w;
 120                 u.f = y[0];
 121                 ey = u.i>>52 & 0x7ff;
 122                 ex = ix>>20;
 123                 if (ex - ey > 16) { /* 2nd round, good to 118 bits */
 124                         t = r;
 125                         w = fn*pio2_2;
 126                         r = t - w;
 127                         w = fn*pio2_2t - ((t-r)-w);
 128                         y[0] = r - w;
 129                         u.f = y[0];
 130                         ey = u.i>>52 & 0x7ff;
 131                         if (ex - ey > 49) {  /* 3rd round, good to 151 bits, covers all cases */
 132                                 t = r;
 133                                 w = fn*pio2_3;
 134                                 r = t - w;
 135                                 w = fn*pio2_3t - ((t-r)-w);
 136                                 y[0] = r - w;
 137                         }
 138                 }
 139                 y[1] = (r - y[0]) - w;
 140                 return n;
 141         }
 142         /*
 143          * all other (large) arguments
 144          */
 145         if (ix >= 0x7ff00000) {  /* x is inf or NaN */
 146                 y[0] = y[1] = x - x;
 147                 return 0;
 148         }
 149         /* set z = scalbn(|x|,-ilogb(x)+23) */
 150         u.f = x;
 151         u.i &= (uint64_t)-1>>12;
 152         u.i |= (uint64_t)(0x3ff + 23)<<52;
 153         z = u.f;
 154         for (i=0; i < 2; i++) {
 155                 tx[i] = (double)(int32_t)z;
 156                 z     = (z-tx[i])*0x1p24;
 157         }
 158         tx[i] = z;
 159         /* skip zero terms, first term is non-zero */
 160         while (tx[i] == 0.0)
 161                 i--;
 162         n = __rem_pio2_large(tx,ty,(int)(ix>>20)-(0x3ff+23),i+1,1);
 163         if (sign) {
 164                 y[0] = -ty[0];
 165                 y[1] = -ty[1];
 166                 return -n;
 167         }
 168         y[0] = ty[0];
 169         y[1] = ty[1];
 170         return n;
 171 }