lib/libm/src/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  * Optimized by Bruce D. Evans.
  14  * FreeBSD SVN: 223302 (2011-06-19)
  15  */
  16
  17 /* __libm_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  * invpio2:  53 bits of 2/pi
  28  * pio2_1:   first  33 bit of pi/2
  29  * pio2_1t:  pi/2 - pio2_1
  30  * pio2_2:   second 33 bit of pi/2
  31  * pio2_2t:  pi/2 - (pio2_1+pio2_2)
  32  * pio2_3:   third  33 bit of pi/2
  33  * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
  34  */
  35
  36 static const double
  37 zero =  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
  38 two24 =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
  39 invpio2 =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
  40 pio2_1  =  1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
  41 pio2_1t =  6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
  42 pio2_2  =  6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
  43 pio2_2t =  2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
  44 pio2_3  =  2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
  45 pio2_3t =  8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
  46
  47 __inline int
  48 __libm_rem_pio2(double x, double *y)
  49 {
  50         double z,w,t,r,fn;
  51         double tx[3],ty[2];
  52         int32_t e0,i,j,nx,n,ix,hx;
  53         u_int32_t low;
  54
  55         GET_HIGH_WORD(hx,x);            /* high word of x */
  56         ix = hx&0x7fffffff;
  57 #if 0 /* Must be handled in caller. */
  58         if(ix<=0x3fe921fb)   /* |x| ~<= pi/4 , no need for reduction */
  59             {y[0] = x; y[1] = 0; return 0;}
  60 #endif
  61         if (ix <= 0x400f6a7a) {         /* |x| ~<= 5pi/4 */
  62             if ((ix & 0xfffff) == 0x921fb)  /* |x| ~= pi/2 or 2pi/2 */
  63                 goto medium;            /* cancellation -- use medium case */
  64             if (ix <= 0x4002d97c) {     /* |x| ~<= 3pi/4 */
  65                 if (hx > 0) {
  66                     z = x - pio2_1;     /* one round good to 85 bits */
  67                     y[0] = z - pio2_1t;
  68                     y[1] = (z-y[0])-pio2_1t;
  69                     return 1;
  70                 } else {
  71                     z = x + pio2_1;
  72                     y[0] = z + pio2_1t;
  73                     y[1] = (z-y[0])+pio2_1t;
  74                     return -1;
  75                 }
  76             } else {
  77                 if (hx > 0) {
  78                     z = x - 2*pio2_1;
  79                     y[0] = z - 2*pio2_1t;
  80                     y[1] = (z-y[0])-2*pio2_1t;
  81                     return 2;
  82                 } else {
  83                     z = x + 2*pio2_1;
  84                     y[0] = z + 2*pio2_1t;
  85                     y[1] = (z-y[0])+2*pio2_1t;
  86                     return -2;
  87                 }
  88             }
  89         }
  90         if (ix <= 0x401c463b) {         /* |x| ~<= 9pi/4 */
  91             if (ix <= 0x4015fdbc) {     /* |x| ~<= 7pi/4 */
  92                 if (ix == 0x4012d97c)   /* |x| ~= 3pi/2 */
  93                     goto medium;
  94                 if (hx > 0) {
  95                     z = x - 3*pio2_1;
  96                     y[0] = z - 3*pio2_1t;
  97                     y[1] = (z-y[0])-3*pio2_1t;
  98                     return 3;
  99                 } else {
 100                     z = x + 3*pio2_1;
 101                     y[0] = z + 3*pio2_1t;
 102                     y[1] = (z-y[0])+3*pio2_1t;
 103                     return -3;
 104                 }
 105             } else {
 106                 if (ix == 0x401921fb)   /* |x| ~= 4pi/2 */
 107                     goto medium;
 108                 if (hx > 0) {
 109                     z = x - 4*pio2_1;
 110                     y[0] = z - 4*pio2_1t;
 111                     y[1] = (z-y[0])-4*pio2_1t;
 112                     return 4;
 113                 } else {
 114                     z = x + 4*pio2_1;
 115                     y[0] = z + 4*pio2_1t;
 116                     y[1] = (z-y[0])+4*pio2_1t;
 117                     return -4;
 118                 }
 119             }
 120         }
 121         if(ix<0x413921fb) {     /* |x| ~< 2^20*(pi/2), medium size */
 122 medium:
 123             /* Use a specialized rint() to get fn.  Assume round-to-nearest. */
 124             STRICT_ASSIGN(double,fn,x*invpio2+0x1.8p52);
 125             fn = fn-0x1.8p52;
 126 #ifdef HAVE_EFFICIENT_IRINT
 127             n  = irint(fn);
 128 #else
 129             n  = (int32_t)fn;
 130 #endif
 131             r  = x-fn*pio2_1;
 132             w  = fn*pio2_1t;    /* 1st round good to 85 bit */
 133             {
 134                 u_int32_t high;
 135                 j  = ix>>20;
 136                 y[0] = r-w;
 137                 GET_HIGH_WORD(high,y[0]);
 138                 i = j-((high>>20)&0x7ff);
 139                 if(i>16) {  /* 2nd iteration needed, good to 118 */
 140                     t  = r;
 141                     w  = fn*pio2_2;
 142                     r  = t-w;
 143                     w  = fn*pio2_2t-((t-r)-w);
 144                     y[0] = r-w;
 145                     GET_HIGH_WORD(high,y[0]);
 146                     i = j-((high>>20)&0x7ff);
 147                     if(i>49)  { /* 3rd iteration need, 151 bits acc */
 148                         t  = r; /* will cover all possible cases */
 149                         w  = fn*pio2_3;
 150                         r  = t-w;
 151                         w  = fn*pio2_3t-((t-r)-w);
 152                         y[0] = r-w;
 153                     }
 154                 }
 155             }
 156             y[1] = (r-y[0])-w;
 157             return n;
 158         }
 159     /*
 160      * all other (large) arguments
 161      */
 162         if(ix>=0x7ff00000) {            /* x is inf or NaN */
 163             y[0]=y[1]=x-x; return 0;
 164         }
 165     /* set z = scalbn(|x|,ilogb(x)-23) */
 166         GET_LOW_WORD(low,x);
 167         e0      = (ix>>20)-1046;        /* e0 = ilogb(z)-23; */
 168         INSERT_WORDS(z, ix - ((int32_t)(e0<<20)), low);
 169         for(i=0;i<2;i++) {
 170                 tx[i] = (double)((int32_t)(z));
 171                 z     = (z-tx[i])*two24;
 172         }
 173         tx[2] = z;
 174         nx = 3;
 175         while(tx[nx-1]==zero) nx--;     /* skip zero term */
 176         n  =  __kernel_rem_pio2(tx,ty,e0,nx,1);
 177         if(hx<0) {y[0] = -ty[0]; y[1] = -ty[1]; return -n;}
 178         y[0] = ty[0]; y[1] = ty[1]; return n;
 179 }