lib/libm/src/e_rem_pio2.c

   1
   2 /* @(#)e_rem_pio2.c 1.4 95/01/18 */
   3 /* $FreeBSD: head/lib/msun/src/e_rem_pio2.c 239195 2012-08-11 15:47:22Z dim $ */
   4 /*
   5  * ====================================================
   6  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   7  *
   8  * Developed at SunSoft, a Sun Microsystems, Inc. business.
   9  * Permission to use, copy, modify, and distribute this
  10  * software is freely granted, provided that this notice
  11  * is preserved.
  12  * ====================================================
  13  *
  14  * Optimized by Bruce D. Evans.
  15  */
  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 <float.h>
  24
  25 #include "math.h"
  26 #include "math_private.h"
  27
  28 /*
  29  * invpio2:  53 bits of 2/pi
  30  * pio2_1:   first  33 bit of pi/2
  31  * pio2_1t:  pi/2 - pio2_1
  32  * pio2_2:   second 33 bit of pi/2
  33  * pio2_2t:  pi/2 - (pio2_1+pio2_2)
  34  * pio2_3:   third  33 bit of pi/2
  35  * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
  36  */
  37
  38 static const double
  39 zero =  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
  40 two24 =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
  41 invpio2 =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
  42 pio2_1  =  1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
  43 pio2_1t =  6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
  44 pio2_2  =  6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
  45 pio2_2t =  2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
  46 pio2_3  =  2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
  47 pio2_3t =  8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
  48
  49 #ifdef INLINE_REM_PIO2
  50 static __inline __always_inline
  51 #endif
  52 int
  53 __ieee754_rem_pio2(double x, double *y)
  54 {
  55         double z,w,t,r,fn;
  56         double tx[3],ty[2];
  57         int32_t e0,i,j,nx,n,ix,hx;
  58         u_int32_t low;
  59
  60         GET_HIGH_WORD(hx,x);            /* high word of x */
  61         ix = hx&0x7fffffff;
  62 #if 0 /* Must be handled in caller. */
  63         if(ix<=0x3fe921fb)   /* |x| ~<= pi/4 , no need for reduction */
  64             {y[0] = x; y[1] = 0; return 0;}
  65 #endif
  66         if (ix <= 0x400f6a7a) {         /* |x| ~<= 5pi/4 */
  67             if ((ix & 0xfffff) == 0x921fb)  /* |x| ~= pi/2 or 2pi/2 */
  68                 goto medium;            /* cancellation -- use medium case */
  69             if (ix <= 0x4002d97c) {     /* |x| ~<= 3pi/4 */
  70                 if (hx > 0) {
  71                     z = x - pio2_1;     /* one round good to 85 bits */
  72                     y[0] = z - pio2_1t;
  73                     y[1] = (z-y[0])-pio2_1t;
  74                     return 1;
  75                 } else {
  76                     z = x + pio2_1;
  77                     y[0] = z + pio2_1t;
  78                     y[1] = (z-y[0])+pio2_1t;
  79                     return -1;
  80                 }
  81             } else {
  82                 if (hx > 0) {
  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                 } else {
  88                     z = x + 2*pio2_1;
  89                     y[0] = z + 2*pio2_1t;
  90                     y[1] = (z-y[0])+2*pio2_1t;
  91                     return -2;
  92                 }
  93             }
  94         }
  95         if (ix <= 0x401c463b) {         /* |x| ~<= 9pi/4 */
  96             if (ix <= 0x4015fdbc) {     /* |x| ~<= 7pi/4 */
  97                 if (ix == 0x4012d97c)   /* |x| ~= 3pi/2 */
  98                     goto medium;
  99                 if (hx > 0) {
 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                 } else {
 105                     z = x + 3*pio2_1;
 106                     y[0] = z + 3*pio2_1t;
 107                     y[1] = (z-y[0])+3*pio2_1t;
 108                     return -3;
 109                 }
 110             } else {
 111                 if (ix == 0x401921fb)   /* |x| ~= 4pi/2 */
 112                     goto medium;
 113                 if (hx > 0) {
 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                 } else {
 119                     z = x + 4*pio2_1;
 120                     y[0] = z + 4*pio2_1t;
 121                     y[1] = (z-y[0])+4*pio2_1t;
 122                     return -4;
 123                 }
 124             }
 125         }
 126         if(ix<0x413921fb) {     /* |x| ~< 2^20*(pi/2), medium size */
 127 medium:
 128             /* Use a specialized rint() to get fn.  Assume round-to-nearest. */
 129             STRICT_ASSIGN(double,fn,x*invpio2+0x1.8p52);
 130             fn = fn-0x1.8p52;
 131 #ifdef HAVE_EFFICIENT_IRINT
 132             n  = irint(fn);
 133 #else
 134             n  = (int32_t)fn;
 135 #endif
 136             r  = x-fn*pio2_1;
 137             w  = fn*pio2_1t;    /* 1st round good to 85 bit */
 138             {
 139                 u_int32_t high;
 140                 j  = ix>>20;
 141                 y[0] = r-w;
 142                 GET_HIGH_WORD(high,y[0]);
 143                 i = j-((high>>20)&0x7ff);
 144                 if(i>16) {  /* 2nd iteration needed, good to 118 */
 145                     t  = r;
 146                     w  = fn*pio2_2;
 147                     r  = t-w;
 148                     w  = fn*pio2_2t-((t-r)-w);
 149                     y[0] = r-w;
 150                     GET_HIGH_WORD(high,y[0]);
 151                     i = j-((high>>20)&0x7ff);
 152                     if(i>49)  { /* 3rd iteration need, 151 bits acc */
 153                         t  = r; /* will cover all possible cases */
 154                         w  = fn*pio2_3;
 155                         r  = t-w;
 156                         w  = fn*pio2_3t-((t-r)-w);
 157                         y[0] = r-w;
 158                     }
 159                 }
 160             }
 161             y[1] = (r-y[0])-w;
 162             return n;
 163         }
 164     /*
 165      * all other (large) arguments
 166      */
 167         if(ix>=0x7ff00000) {            /* x is inf or NaN */
 168             y[0]=y[1]=x-x; return 0;
 169         }
 170     /* set z = scalbn(|x|,ilogb(x)-23) */
 171         GET_LOW_WORD(low,x);
 172         e0      = (ix>>20)-1046;        /* e0 = ilogb(z)-23; */
 173         INSERT_WORDS(z, ix - ((int32_t)(e0<<20)), low);
 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,ty,e0,nx,1);
 182         if(hx<0) {y[0] = -ty[0]; y[1] = -ty[1]; return -n;}
 183         y[0] = ty[0]; y[1] = ty[1]; return n;
 184 }