lib/libm/src/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 /* __ieee754_rem_pio2(x,y)
  14  *
  15  * return the remainder of x rem pi/2 in y[0]+y[1]
  16  * use __kernel_rem_pio2()
  17  */
  18
  19 #include "math.h"
  20 #include "math_private.h"
  21
  22 static const int32_t npio2_hw[] = {
  23 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
  24 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
  25 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
  26 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
  27 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
  28 0x404858EB, 0x404921FB,
  29 };
  30
  31 /*
  32  * invpio2:  53 bits of 2/pi
  33  * pio2_1:   first  33 bit of pi/2
  34  * pio2_1t:  pi/2 - pio2_1
  35  * pio2_2:   second 33 bit of pi/2
  36  * pio2_2t:  pi/2 - (pio2_1+pio2_2)
  37  * pio2_3:   third  33 bit of pi/2
  38  * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
  39  */
  40
  41 static const double
  42 zero =  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
  43 half =  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
  44 two24 =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
  45 invpio2 =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
  46 pio2_1  =  1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
  47 pio2_1t =  6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
  48 pio2_2  =  6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
  49 pio2_2t =  2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
  50 pio2_3  =  2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
  51 pio2_3t =  8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
  52
  53 int32_t
  54 __ieee754_rem_pio2(double x, double *y)
  55 {
  56         double z,w,t,r,fn;
  57         double tx[3];
  58         int32_t e0,i,j,nx,n,ix,hx;
  59         u_int32_t low;
  60
  61         GET_HIGH_WORD(hx,x);            /* high word of x */
  62         ix = hx&0x7fffffff;
  63         if(ix<=0x3fe921fb)   /* |x| ~<= pi/4 , no need for reduction */
  64             {y[0] = x; y[1] = 0; return 0;}
  65         if(ix<0x4002d97c) {  /* |x| < 3pi/4, special case with n=+-1 */
  66             if(hx>0) {
  67                 z = x - pio2_1;
  68                 if(ix!=0x3ff921fb) {    /* 33+53 bit pi is good enough */
  69                     y[0] = z - pio2_1t;
  70                     y[1] = (z-y[0])-pio2_1t;
  71                 } else {                /* near pi/2, use 33+33+53 bit pi */
  72                     z -= pio2_2;
  73                     y[0] = z - pio2_2t;
  74                     y[1] = (z-y[0])-pio2_2t;
  75                 }
  76                 return 1;
  77             } else {    /* negative x */
  78                 z = x + pio2_1;
  79                 if(ix!=0x3ff921fb) {    /* 33+53 bit pi is good enough */
  80                     y[0] = z + pio2_1t;
  81                     y[1] = (z-y[0])+pio2_1t;
  82                 } else {                /* near pi/2, use 33+33+53 bit pi */
  83                     z += pio2_2;
  84                     y[0] = z + pio2_2t;
  85                     y[1] = (z-y[0])+pio2_2t;
  86                 }
  87                 return -1;
  88             }
  89         }
  90         if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
  91             t  = fabs(x);
  92             n  = (int32_t) (t*invpio2+half);
  93             fn = (double)n;
  94             r  = t-fn*pio2_1;
  95             w  = fn*pio2_1t;    /* 1st round good to 85 bit */
  96             if(n<32&&ix!=npio2_hw[n-1]) {
  97                 y[0] = r-w;     /* quick check no cancellation */
  98             } else {
  99                 u_int32_t high;
 100                 j  = ix>>20;
 101                 y[0] = r-w;
 102                 GET_HIGH_WORD(high,y[0]);
 103                 i = j-((high>>20)&0x7ff);
 104                 if(i>16) {  /* 2nd iteration needed, good to 118 */
 105                     t  = r;
 106                     w  = fn*pio2_2;
 107                     r  = t-w;
 108                     w  = fn*pio2_2t-((t-r)-w);
 109                     y[0] = r-w;
 110                     GET_HIGH_WORD(high,y[0]);
 111                     i = j-((high>>20)&0x7ff);
 112                     if(i>49)  { /* 3rd iteration need, 151 bits acc */
 113                         t  = r; /* will cover all possible cases */
 114                         w  = fn*pio2_3;
 115                         r  = t-w;
 116                         w  = fn*pio2_3t-((t-r)-w);
 117                         y[0] = r-w;
 118                     }
 119                 }
 120             }
 121             y[1] = (r-y[0])-w;
 122             if(hx<0)    {y[0] = -y[0]; y[1] = -y[1]; return -n;}
 123             else         return n;
 124         }
 125     /*
 126      * all other (large) arguments
 127      */
 128         if(ix>=0x7ff00000) {            /* x is inf or NaN */
 129             y[0]=y[1]=x-x; return 0;
 130         }
 131     /* set z = scalbn(|x|,ilogb(x)-23) */
 132         GET_LOW_WORD(low,x);
 133         SET_LOW_WORD(z,low);
 134         e0      = (ix>>20)-1046;        /* e0 = ilogb(z)-23; */
 135         SET_HIGH_WORD(z, ix - ((int32_t)(e0<<20)));
 136         for(i=0;i<2;i++) {
 137                 tx[i] = (double)((int32_t)(z));
 138                 z     = (z-tx[i])*two24;
 139         }
 140         tx[2] = z;
 141         nx = 3;
 142         while(tx[nx-1]==zero) nx--;     /* skip zero term */
 143         n  =  __kernel_rem_pio2(tx,y,e0,nx,2);
 144         if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
 145         return n;
 146 }