libm/e_rem_pio2.c

   1 /*
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  *
   5  * Developed at SunPro, a Sun Microsystems, Inc. business.
   6  * Permission to use, copy, modify, and distribute this
   7  * software is freely granted, provided that this notice
   8  * is preserved.
   9  * ====================================================
  10  */
  11
  12 /* __ieee754_rem_pio2(x,y)
  13  *
  14  * return the remainder of x rem pi/2 in y[0]+y[1]
  15  * use __kernel_rem_pio2()
  16  */
  17
  18 #include "math.h"
  19 #include "math_private.h"
  20
  21 /*
  22  * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
  23  */
  24 static const int32_t two_over_pi[] = {
  25 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
  26 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
  27 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
  28 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
  29 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
  30 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
  31 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
  32 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
  33 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
  34 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
  35 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
  36 };
  37
  38 static const int32_t npio2_hw[] = {
  39 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
  40 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
  41 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
  42 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
  43 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
  44 0x404858EB, 0x404921FB,
  45 };
  46
  47 /*
  48  * invpio2:  53 bits of 2/pi
  49  * pio2_1:   first  33 bit of pi/2
  50  * pio2_1t:  pi/2 - pio2_1
  51  * pio2_2:   second 33 bit of pi/2
  52  * pio2_2t:  pi/2 - (pio2_1+pio2_2)
  53  * pio2_3:   third  33 bit of pi/2
  54  * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
  55  */
  56
  57 static const double
  58 zero =  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
  59 half =  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
  60 two24 =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
  61 invpio2 =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
  62 pio2_1  =  1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
  63 pio2_1t =  6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
  64 pio2_2  =  6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
  65 pio2_2t =  2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
  66 pio2_3  =  2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
  67 pio2_3t =  8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
  68
  69 int32_t __ieee754_rem_pio2(double x, double *y)
  70 {
  71         double z=0.0,w,t,r,fn;
  72         double tx[3];
  73         int32_t e0,i,j,nx,n,ix,hx;
  74         u_int32_t low;
  75
  76         GET_HIGH_WORD(hx,x);            /* high word of x */
  77         ix = hx&0x7fffffff;
  78         if(ix<=0x3fe921fb)   /* |x| ~<= pi/4 , no need for reduction */
  79             {y[0] = x; y[1] = 0; return 0;}
  80         if(ix<0x4002d97c) {  /* |x| < 3pi/4, special case with n=+-1 */
  81             if(hx>0) {
  82                 z = x - pio2_1;
  83                 if(ix!=0x3ff921fb) {    /* 33+53 bit pi is good enough */
  84                     y[0] = z - pio2_1t;
  85                     y[1] = (z-y[0])-pio2_1t;
  86                 } else {                /* near pi/2, use 33+33+53 bit pi */
  87                     z -= pio2_2;
  88                     y[0] = z - pio2_2t;
  89                     y[1] = (z-y[0])-pio2_2t;
  90                 }
  91                 return 1;
  92             } else {    /* negative x */
  93                 z = x + pio2_1;
  94                 if(ix!=0x3ff921fb) {    /* 33+53 bit pi is good enough */
  95                     y[0] = z + pio2_1t;
  96                     y[1] = (z-y[0])+pio2_1t;
  97                 } else {                /* near pi/2, use 33+33+53 bit pi */
  98                     z += pio2_2;
  99                     y[0] = z + pio2_2t;
 100                     y[1] = (z-y[0])+pio2_2t;
 101                 }
 102                 return -1;
 103             }
 104         }
 105         if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
 106             t  = fabs(x);
 107             n  = (int32_t) (t*invpio2+half);
 108             fn = (double)n;
 109             r  = t-fn*pio2_1;
 110             w  = fn*pio2_1t;    /* 1st round good to 85 bit */
 111             if(n<32&&ix!=npio2_hw[n-1]) {
 112                 y[0] = r-w;     /* quick check no cancellation */
 113             } else {
 114                 u_int32_t high;
 115                 j  = ix>>20;
 116                 y[0] = r-w;
 117                 GET_HIGH_WORD(high,y[0]);
 118                 i = j-((high>>20)&0x7ff);
 119                 if(i>16) {  /* 2nd iteration needed, good to 118 */
 120                     t  = r;
 121                     w  = fn*pio2_2;
 122                     r  = t-w;
 123                     w  = fn*pio2_2t-((t-r)-w);
 124                     y[0] = r-w;
 125                     GET_HIGH_WORD(high,y[0]);
 126                     i = j-((high>>20)&0x7ff);
 127                     if(i>49)  { /* 3rd iteration need, 151 bits acc */
 128                         t  = r; /* will cover all possible cases */
 129                         w  = fn*pio2_3;
 130                         r  = t-w;
 131                         w  = fn*pio2_3t-((t-r)-w);
 132                         y[0] = r-w;
 133                     }
 134                 }
 135             }
 136             y[1] = (r-y[0])-w;
 137             if(hx<0)    {y[0] = -y[0]; y[1] = -y[1]; return -n;}
 138             else         return n;
 139         }
 140     /*
 141      * all other (large) arguments
 142      */
 143         if(ix>=0x7ff00000) {            /* x is inf or NaN */
 144             y[0]=y[1]=x-x; return 0;
 145         }
 146     /* set z = scalbn(|x|,ilogb(x)-23) */
 147         GET_LOW_WORD(low,x);
 148         SET_LOW_WORD(z,low);
 149         e0      = (ix>>20)-1046;        /* e0 = ilogb(z)-23; */
 150         SET_HIGH_WORD(z, ix - ((int32_t)(e0<<20)));
 151         for(i=0;i<2;i++) {
 152                 tx[i] = (double)((int32_t)(z));
 153                 z     = (z-tx[i])*two24;
 154         }
 155         tx[2] = z;
 156         nx = 3;
 157         while(tx[nx-1]==zero) nx--;     /* skip zero term */
 158         n  =  __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi);
 159         if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
 160         return n;
 161 }