src/fdlibm/e_rem_pio2.c

   1
   2 /* @(#)e_rem_pio2.c 1.4 95/01/18 */
   3 /*
   4  * ====================================================
   5  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   6  *
   7  * Developed at SunSoft, a Sun Microsystems, Inc. business.
   8  * Permission to use, copy, modify, and distribute this
   9  * software is freely granted, provided that this notice
  10  * is preserved.
  11  * ====================================================
  12  *
  13  */
  14
  15 /* __ieee754_rem_pio2(x,y)
  16  *
  17  * return the remainder of x rem pi/2 in y[0]+y[1]
  18  * use __kernel_rem_pio2()
  19  */
  20
  21 #include "fdlibm.h"
  22
  23 #ifndef _DOUBLE_IS_32BITS
  24
  25 /*
  26  * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
  27  */
  28 #ifdef __STDC__
  29 static const int32_t two_over_pi[] = {
  30 #else
  31 static int32_t two_over_pi[] = {
  32 #endif
  33 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
  34 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
  35 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
  36 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
  37 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
  38 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
  39 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
  40 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
  41 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
  42 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
  43 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
  44 };
  45
  46 #ifdef __STDC__
  47 static const int32_t npio2_hw[] = {
  48 #else
  49 static int32_t npio2_hw[] = {
  50 #endif
  51 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
  52 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
  53 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
  54 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
  55 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
  56 0x404858EB, 0x404921FB,
  57 };
  58
  59 /*
  60  * invpio2:  53 bits of 2/pi
  61  * pio2_1:   first  33 bit of pi/2
  62  * pio2_1t:  pi/2 - pio2_1
  63  * pio2_2:   second 33 bit of pi/2
  64  * pio2_2t:  pi/2 - (pio2_1+pio2_2)
  65  * pio2_3:   third  33 bit of pi/2
  66  * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
  67  */
  68
  69 #ifdef __STDC__
  70 static const double
  71 #else
  72 static double
  73 #endif
  74 zero =  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
  75 half =  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
  76 two24 =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
  77 invpio2 =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
  78 pio2_1  =  1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
  79 pio2_1t =  6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
  80 pio2_2  =  6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
  81 pio2_2t =  2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
  82 pio2_3  =  2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
  83 pio2_3t =  8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
  84
  85 #ifdef __STDC__
  86         int32_t __ieee754_rem_pio2(double x, double *y)
  87 #else
  88         int32_t __ieee754_rem_pio2(x,y)
  89         double x,y[];
  90 #endif
  91 {
  92         double z = 0.,w,t,r,fn;
  93         double tx[3];
  94         int32_t i,j,n,ix,hx;
  95         int e0,nx;
  96         uint32_t low;
  97
  98         GET_HIGH_WORD(hx,x);            /* high word of x */
  99         ix = hx&0x7fffffff;
 100         if(ix<=0x3fe921fb)   /* |x| ~<= pi/4 , no need for reduction */
 101             {y[0] = x; y[1] = 0; return 0;}
 102         if(ix<0x4002d97c) {  /* |x| < 3pi/4, special case with n=+-1 */
 103             if(hx>0) {
 104                 z = x - pio2_1;
 105                 if(ix!=0x3ff921fb) {    /* 33+53 bit pi is good enough */
 106                     y[0] = z - pio2_1t;
 107                     y[1] = (z-y[0])-pio2_1t;
 108                 } else {                /* near pi/2, use 33+33+53 bit pi */
 109                     z -= pio2_2;
 110                     y[0] = z - pio2_2t;
 111                     y[1] = (z-y[0])-pio2_2t;
 112                 }
 113                 return 1;
 114             } else {    /* negative x */
 115                 z = x + pio2_1;
 116                 if(ix!=0x3ff921fb) {    /* 33+53 bit pi is good enough */
 117                     y[0] = z + pio2_1t;
 118                     y[1] = (z-y[0])+pio2_1t;
 119                 } else {                /* near pi/2, use 33+33+53 bit pi */
 120                     z += pio2_2;
 121                     y[0] = z + pio2_2t;
 122                     y[1] = (z-y[0])+pio2_2t;
 123                 }
 124                 return -1;
 125             }
 126         }
 127         if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
 128             t  = fabs(x);
 129             n  = (int32_t) (t*invpio2+half);
 130             fn = (double)n;
 131             r  = t-fn*pio2_1;
 132             w  = fn*pio2_1t;    /* 1st round good to 85 bit */
 133             if(n<32&&ix!=npio2_hw[n-1]) {
 134                 y[0] = r-w;     /* quick check no cancellation */
 135             } else {
 136                 uint32_t high;
 137
 138                 j  = ix>>20;
 139                 y[0] = r-w;
 140                 GET_HIGH_WORD(high, y[0]);
 141                 i = j-((high>>20)&0x7ff);
 142                 if(i>16) {  /* 2nd iteration needed, good to 118 */
 143                     t  = r;
 144                     w  = fn*pio2_2;
 145                     r  = t-w;
 146                     w  = fn*pio2_2t-((t-r)-w);
 147                     y[0] = r-w;
 148                     GET_HIGH_WORD(high,y[0]);
 149                     i = j-((high>>20)&0x7ff);
 150                     if(i>49)  { /* 3rd iteration need, 151 bits acc */
 151                         t  = r; /* will cover all possible cases */
 152                         w  = fn*pio2_3;
 153                         r  = t-w;
 154                         w  = fn*pio2_3t-((t-r)-w);
 155                         y[0] = r-w;
 156                     }
 157                 }
 158             }
 159             y[1] = (r-y[0])-w;
 160             if(hx<0)    {y[0] = -y[0]; y[1] = -y[1]; return -n;}
 161             else         return n;
 162         }
 163     /*
 164      * all other (large) arguments
 165      */
 166         if(ix>=0x7ff00000) {            /* x is inf or NaN */
 167             y[0]=y[1]=x-x; return 0;
 168         }
 169     /* set z = scalbn(|x|,ilogb(x)-23) */
 170         GET_LOW_WORD(low,x);
 171         SET_LOW_WORD(z,low);
 172         e0      = (int32_t)(ix>>20)-1046;       /* e0 = ilogb(z)-23; */
 173         SET_HIGH_WORD(z,ix - (e0<<20));
 174         for(i=0;i<2;i++) {
 175                 tx[i] = (double)((int32_t)(z));
 176                 z     = (z-tx[i])*two24;
 177         }
 178         tx[2] = z;
 179         nx = 3;
 180         while(tx[nx-1]==zero) nx--;     /* skip zero term */
 181         n  =  __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi);
 182         if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
 183         return n;
 184 }
 185 #endif /* defined(_DOUBLE_IS_32BITS) */