src/math/__rem_pio2.c

   1 /* origin: FreeBSD /usr/src/lib/msun/src/e_rem_pio2.c */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunSoft, 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  * Optimized by Bruce D. Evans.
  13  */
  14 /* __rem_pio2(x,y)
  15  *
  16  * return the remainder of x rem pi/2 in y[0]+y[1]
  17  * use __rem_pio2_large() for large x
  18  */
  19
  20 #include "libm.h"
  21
  22 #if FLT_EVAL_METHOD==0 || FLT_EVAL_METHOD==1
  23 #define EPS DBL_EPSILON
  24 #elif FLT_EVAL_METHOD==2
  25 #define EPS LDBL_EPSILON
  26 #endif
  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 static const double
  38 toint   = 1.5/EPS,
  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 /* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */
  48 int __rem_pio2(double x, double *y)
  49 {
  50         union {double f; uint64_t i;} u = {x};
  51         double_t z,w,t,r,fn;
  52         double tx[3],ty[2];
  53         uint32_t ix;
  54         int sign, n, ex, ey, i;
  55
  56         sign = u.i>>63;
  57         ix = u.i>>32 & 0x7fffffff;
  58         if (ix <= 0x400f6a7a) {  /* |x| ~<= 5pi/4 */
  59                 if ((ix & 0xfffff) == 0x921fb)  /* |x| ~= pi/2 or 2pi/2 */
  60                         goto medium;  /* cancellation -- use medium case */
  61                 if (ix <= 0x4002d97c) {  /* |x| ~<= 3pi/4 */
  62                         if (!sign) {
  63                                 z = x - pio2_1;  /* one round good to 85 bits */
  64                                 y[0] = z - pio2_1t;
  65                                 y[1] = (z-y[0]) - pio2_1t;
  66                                 return 1;
  67                         } else {
  68                                 z = x + pio2_1;
  69                                 y[0] = z + pio2_1t;
  70                                 y[1] = (z-y[0]) + pio2_1t;
  71                                 return -1;
  72                         }
  73                 } else {
  74                         if (!sign) {
  75                                 z = x - 2*pio2_1;
  76                                 y[0] = z - 2*pio2_1t;
  77                                 y[1] = (z-y[0]) - 2*pio2_1t;
  78                                 return 2;
  79                         } else {
  80                                 z = x + 2*pio2_1;
  81                                 y[0] = z + 2*pio2_1t;
  82                                 y[1] = (z-y[0]) + 2*pio2_1t;
  83                                 return -2;
  84                         }
  85                 }
  86         }
  87         if (ix <= 0x401c463b) {  /* |x| ~<= 9pi/4 */
  88                 if (ix <= 0x4015fdbc) {  /* |x| ~<= 7pi/4 */
  89                         if (ix == 0x4012d97c)  /* |x| ~= 3pi/2 */
  90                                 goto medium;
  91                         if (!sign) {
  92                                 z = x - 3*pio2_1;
  93                                 y[0] = z - 3*pio2_1t;
  94                                 y[1] = (z-y[0]) - 3*pio2_1t;
  95                                 return 3;
  96                         } else {
  97                                 z = x + 3*pio2_1;
  98                                 y[0] = z + 3*pio2_1t;
  99                                 y[1] = (z-y[0]) + 3*pio2_1t;
 100                                 return -3;
 101                         }
 102                 } else {
 103                         if (ix == 0x401921fb)  /* |x| ~= 4pi/2 */
 104                                 goto medium;
 105                         if (!sign) {
 106                                 z = x - 4*pio2_1;
 107                                 y[0] = z - 4*pio2_1t;
 108                                 y[1] = (z-y[0]) - 4*pio2_1t;
 109                                 return 4;
 110                         } else {
 111                                 z = x + 4*pio2_1;
 112                                 y[0] = z + 4*pio2_1t;
 113                                 y[1] = (z-y[0]) + 4*pio2_1t;
 114                                 return -4;
 115                         }
 116                 }
 117         }
 118         if (ix < 0x413921fb) {  /* |x| ~< 2^20*(pi/2), medium size */
 119 medium:
 120                 /* rint(x/(pi/2)), Assume round-to-nearest. */
 121                 fn = (double_t)x*invpio2 + toint - toint;
 122                 n = (int32_t)fn;
 123                 r = x - fn*pio2_1;
 124                 w = fn*pio2_1t;  /* 1st round, good to 85 bits */
 125                 y[0] = r - w;
 126                 u.f = y[0];
 127                 ey = u.i>>52 & 0x7ff;
 128                 ex = ix>>20;
 129                 if (ex - ey > 16) { /* 2nd round, good to 118 bits */
 130                         t = r;
 131                         w = fn*pio2_2;
 132                         r = t - w;
 133                         w = fn*pio2_2t - ((t-r)-w);
 134                         y[0] = r - w;
 135                         u.f = y[0];
 136                         ey = u.i>>52 & 0x7ff;
 137                         if (ex - ey > 49) {  /* 3rd round, good to 151 bits, covers all cases */
 138                                 t = r;
 139                                 w = fn*pio2_3;
 140                                 r = t - w;
 141                                 w = fn*pio2_3t - ((t-r)-w);
 142                                 y[0] = r - w;
 143                         }
 144                 }
 145                 y[1] = (r - y[0]) - w;
 146                 return n;
 147         }
 148         /*
 149          * all other (large) arguments
 150          */
 151         if (ix >= 0x7ff00000) {  /* x is inf or NaN */
 152                 y[0] = y[1] = x - x;
 153                 return 0;
 154         }
 155         /* set z = scalbn(|x|,-ilogb(x)+23) */
 156         u.f = x;
 157         u.i &= (uint64_t)-1>>12;
 158         u.i |= (uint64_t)(0x3ff + 23)<<52;
 159         z = u.f;
 160         for (i=0; i < 2; i++) {
 161                 tx[i] = (double)(int32_t)z;
 162                 z     = (z-tx[i])*0x1p24;
 163         }
 164         tx[i] = z;
 165         /* skip zero terms, first term is non-zero */
 166         while (tx[i] == 0.0)
 167                 i--;
 168         n = __rem_pio2_large(tx,ty,(int)(ix>>20)-(0x3ff+23),i+1,1);
 169         if (sign) {
 170                 y[0] = -ty[0];
 171                 y[1] = -ty[1];
 172                 return -n;
 173         }
 174         y[0] = ty[0];
 175         y[1] = ty[1];
 176         return n;
 177 }