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 two24   = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
  33 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
  34 pio2_1  = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
  35 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
  36 pio2_2  = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
  37 pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
  38 pio2_3  = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
  39 pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
  40
  41 /* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */
  42 int __rem_pio2(double x, double *y)
  43 {
  44         double z,w,t,r,fn;
  45         double tx[3],ty[2];
  46         int32_t e0,i,j,nx,n,ix,hx;
  47         uint32_t low;
  48
  49         GET_HIGH_WORD(hx,x);
  50         ix = hx & 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 (hx > 0) {
  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 (hx > 0) {
  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 (hx > 0) {
  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 (hx > 0) {
  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                 uint32_t high;
 113 medium:
 114                 /* Use a specialized rint() to get fn.  Assume round-to-nearest. */
 115                 STRICT_ASSIGN(double, fn, x*invpio2 + 0x1.8p52);
 116                 fn = fn - 0x1.8p52;
 117 // FIXME
 118 #ifdef HAVE_EFFICIENT_IRINT
 119                 n = irint(fn);
 120 #else
 121                 n = (int32_t)fn;
 122 #endif
 123                 r = x - fn*pio2_1;
 124                 w = fn*pio2_1t;  /* 1st round, good to 85 bits */
 125                 j = ix>>20;
 126                 y[0] = r - w;
 127                 GET_HIGH_WORD(high,y[0]);
 128                 i = j - ((high>>20)&0x7ff);
 129                 if (i > 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                         GET_HIGH_WORD(high,y[0]);
 136                         i = j - ((high>>20)&0x7ff);
 137                         if (i > 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         GET_LOW_WORD(low,x);
 157         e0 = (ix>>20) - 1046;  /* e0 = ilogb(z)-23; */
 158         INSERT_WORDS(z, ix - ((int32_t)(e0<<20)), low);
 159         for (i=0; i<2; i++) {
 160                 tx[i] = (double)((int32_t)(z));
 161                 z = (z-tx[i])*two24;
 162         }
 163         tx[2] = z;
 164         nx = 3;
 165         while (tx[nx-1] == 0.0) nx--;  /* skip zero term */
 166         n = __rem_pio2_large(tx,ty,e0,nx,1);
 167         if (hx < 0) {
 168                 y[0] = -ty[0];
 169                 y[1] = -ty[1];
 170                 return -n;
 171         }
 172         y[0] = ty[0];
 173         y[1] = ty[1];
 174         return n;
 175 }