libm/e_rem_pio2.c

   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 #if defined(LIBM_SCCS) && !defined(lint)
  14 static char rcsid[] = "$NetBSD: e_rem_pio2.c,v 1.8 1995/05/10 20:46:02 jtc Exp $";
  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 /*
  28  * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
  29  */
  30 #ifdef __STDC__
  31 static const int32_t two_over_pi[] = {
  32 #else
  33 static int32_t two_over_pi[] = {
  34 #endif
  35 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
  36 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
  37 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
  38 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
  39 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
  40 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
  41 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
  42 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
  43 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
  44 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
  45 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
  46 };
  47
  48 #ifdef __STDC__
  49 static const int32_t npio2_hw[] = {
  50 #else
  51 static int32_t npio2_hw[] = {
  52 #endif
  53 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
  54 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
  55 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
  56 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
  57 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
  58 0x404858EB, 0x404921FB,
  59 };
  60
  61 /*
  62  * invpio2:  53 bits of 2/pi
  63  * pio2_1:   first  33 bit of pi/2
  64  * pio2_1t:  pi/2 - pio2_1
  65  * pio2_2:   second 33 bit of pi/2
  66  * pio2_2t:  pi/2 - (pio2_1+pio2_2)
  67  * pio2_3:   third  33 bit of pi/2
  68  * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
  69  */
  70
  71 #ifdef __STDC__
  72 static const double
  73 #else
  74 static double
  75 #endif
  76 zero =  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
  77 half =  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
  78 two24 =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
  79 invpio2 =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
  80 pio2_1  =  1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
  81 pio2_1t =  6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
  82 pio2_2  =  6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
  83 pio2_2t =  2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
  84 pio2_3  =  2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
  85 pio2_3t =  8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
  86
  87 #ifdef __STDC__
  88         int32_t attribute_hidden __ieee754_rem_pio2(double x, double *y)
  89 #else
  90         int32_t attribute_hidden __ieee754_rem_pio2(x,y)
  91         double x,y[];
  92 #endif
  93 {
  94         double z=0.0,w,t,r,fn;
  95         double tx[3];
  96         int32_t e0,i,j,nx,n,ix,hx;
  97         u_int32_t low;
  98
  99         GET_HIGH_WORD(hx,x);            /* high word of x */
 100         ix = hx&0x7fffffff;
 101         if(ix<=0x3fe921fb)   /* |x| ~<= pi/4 , no need for reduction */
 102             {y[0] = x; y[1] = 0; return 0;}
 103         if(ix<0x4002d97c) {  /* |x| < 3pi/4, special case with n=+-1 */
 104             if(hx>0) {
 105                 z = x - pio2_1;
 106                 if(ix!=0x3ff921fb) {    /* 33+53 bit pi is good enough */
 107                     y[0] = z - pio2_1t;
 108                     y[1] = (z-y[0])-pio2_1t;
 109                 } else {                /* near pi/2, use 33+33+53 bit pi */
 110                     z -= pio2_2;
 111                     y[0] = z - pio2_2t;
 112                     y[1] = (z-y[0])-pio2_2t;
 113                 }
 114                 return 1;
 115             } else {    /* negative x */
 116                 z = x + pio2_1;
 117                 if(ix!=0x3ff921fb) {    /* 33+53 bit pi is good enough */
 118                     y[0] = z + pio2_1t;
 119                     y[1] = (z-y[0])+pio2_1t;
 120                 } else {                /* near pi/2, use 33+33+53 bit pi */
 121                     z += pio2_2;
 122                     y[0] = z + pio2_2t;
 123                     y[1] = (z-y[0])+pio2_2t;
 124                 }
 125                 return -1;
 126             }
 127         }
 128         if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
 129             t  = fabs(x);
 130             n  = (int32_t) (t*invpio2+half);
 131             fn = (double)n;
 132             r  = t-fn*pio2_1;
 133             w  = fn*pio2_1t;    /* 1st round good to 85 bit */
 134             if(n<32&&ix!=npio2_hw[n-1]) {
 135                 y[0] = r-w;     /* quick check no cancellation */
 136             } else {
 137                 u_int32_t high;
 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      = (ix>>20)-1046;        /* e0 = ilogb(z)-23; */
 173         SET_HIGH_WORD(z, ix - ((int32_t)(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 }