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 pio4    = 0x1.921fb54442d18p-1,
  40 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
  41 pio2_1  = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
  42 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
  43 pio2_2  = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
  44 pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
  45 pio2_3  = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
  46 pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
  47
  48 /* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */
  49 int __rem_pio2(double x, double *y)
  50 {
  51         union {double f; uint64_t i;} u = {x};
  52         double_t z,w,t,r,fn;
  53         double tx[3],ty[2];
  54         uint32_t ix;
  55         int sign, n, ex, ey, i;
  56
  57         sign = u.i>>63;
  58         ix = u.i>>32 & 0x7fffffff;
  59         if (ix <= 0x400f6a7a) {  /* |x| ~<= 5pi/4 */
  60                 if ((ix & 0xfffff) == 0x921fb)  /* |x| ~= pi/2 or 2pi/2 */
  61                         goto medium;  /* cancellation -- use medium case */
  62                 if (ix <= 0x4002d97c) {  /* |x| ~<= 3pi/4 */
  63                         if (!sign) {
  64                                 z = x - pio2_1;  /* one round good to 85 bits */
  65                                 y[0] = z - pio2_1t;
  66                                 y[1] = (z-y[0]) - pio2_1t;
  67                                 return 1;
  68                         } else {
  69                                 z = x + pio2_1;
  70                                 y[0] = z + pio2_1t;
  71                                 y[1] = (z-y[0]) + pio2_1t;
  72                                 return -1;
  73                         }
  74                 } else {
  75                         if (!sign) {
  76                                 z = x - 2*pio2_1;
  77                                 y[0] = z - 2*pio2_1t;
  78                                 y[1] = (z-y[0]) - 2*pio2_1t;
  79                                 return 2;
  80                         } else {
  81                                 z = x + 2*pio2_1;
  82                                 y[0] = z + 2*pio2_1t;
  83                                 y[1] = (z-y[0]) + 2*pio2_1t;
  84                                 return -2;
  85                         }
  86                 }
  87         }
  88         if (ix <= 0x401c463b) {  /* |x| ~<= 9pi/4 */
  89                 if (ix <= 0x4015fdbc) {  /* |x| ~<= 7pi/4 */
  90                         if (ix == 0x4012d97c)  /* |x| ~= 3pi/2 */
  91                                 goto medium;
  92                         if (!sign) {
  93                                 z = x - 3*pio2_1;
  94                                 y[0] = z - 3*pio2_1t;
  95                                 y[1] = (z-y[0]) - 3*pio2_1t;
  96                                 return 3;
  97                         } else {
  98                                 z = x + 3*pio2_1;
  99                                 y[0] = z + 3*pio2_1t;
 100                                 y[1] = (z-y[0]) + 3*pio2_1t;
 101                                 return -3;
 102                         }
 103                 } else {
 104                         if (ix == 0x401921fb)  /* |x| ~= 4pi/2 */
 105                                 goto medium;
 106                         if (!sign) {
 107                                 z = x - 4*pio2_1;
 108                                 y[0] = z - 4*pio2_1t;
 109                                 y[1] = (z-y[0]) - 4*pio2_1t;
 110                                 return 4;
 111                         } else {
 112                                 z = x + 4*pio2_1;
 113                                 y[0] = z + 4*pio2_1t;
 114                                 y[1] = (z-y[0]) + 4*pio2_1t;
 115                                 return -4;
 116                         }
 117                 }
 118         }
 119         if (ix < 0x413921fb) {  /* |x| ~< 2^20*(pi/2), medium size */
 120 medium:
 121                 /* rint(x/(pi/2)) */
 122                 fn = (double_t)x*invpio2 + toint - toint;
 123                 n = (int32_t)fn;
 124                 r = x - fn*pio2_1;
 125                 w = fn*pio2_1t;  /* 1st round, good to 85 bits */
 126                 /* Matters with directed rounding. */
 127                 if (predict_false(r - w < -pio4)) {
 128                         n--;
 129                         fn--;
 130                         r = x - fn*pio2_1;
 131                         w = fn*pio2_1t;
 132                 } else if (predict_false(r - w > pio4)) {
 133                         n++;
 134                         fn++;
 135                         r = x - fn*pio2_1;
 136                         w = fn*pio2_1t;
 137                 }
 138                 y[0] = r - w;
 139                 u.f = y[0];
 140                 ey = u.i>>52 & 0x7ff;
 141                 ex = ix>>20;
 142                 if (ex - ey > 16) { /* 2nd round, good to 118 bits */
 143                         t = r;
 144                         w = fn*pio2_2;
 145                         r = t - w;
 146                         w = fn*pio2_2t - ((t-r)-w);
 147                         y[0] = r - w;
 148                         u.f = y[0];
 149                         ey = u.i>>52 & 0x7ff;
 150                         if (ex - ey > 49) {  /* 3rd round, good to 151 bits, covers all cases */
 151                                 t = r;
 152                                 w = fn*pio2_3;
 153                                 r = t - w;
 154                                 w = fn*pio2_3t - ((t-r)-w);
 155                                 y[0] = r - w;
 156                         }
 157                 }
 158                 y[1] = (r - y[0]) - w;
 159                 return n;
 160         }
 161         /*
 162          * all other (large) arguments
 163          */
 164         if (ix >= 0x7ff00000) {  /* x is inf or NaN */
 165                 y[0] = y[1] = x - x;
 166                 return 0;
 167         }
 168         /* set z = scalbn(|x|,-ilogb(x)+23) */
 169         u.f = x;
 170         u.i &= (uint64_t)-1>>12;
 171         u.i |= (uint64_t)(0x3ff + 23)<<52;
 172         z = u.f;
 173         for (i=0; i < 2; i++) {
 174                 tx[i] = (double)(int32_t)z;
 175                 z     = (z-tx[i])*0x1p24;
 176         }
 177         tx[i] = z;
 178         /* skip zero terms, first term is non-zero */
 179         while (tx[i] == 0.0)
 180                 i--;
 181         n = __rem_pio2_large(tx,ty,(int)(ix>>20)-(0x3ff+23),i+1,1);
 182         if (sign) {
 183                 y[0] = -ty[0];
 184                 y[1] = -ty[1];
 185                 return -n;
 186         }
 187         y[0] = ty[0];
 188         y[1] = ty[1];
 189         return n;
 190 }