libjava/classpath/native/fdlibm/k_tan.c

   1 #pragma ident "@(#)k_tan.c 1.5 04/04/22 SMI"
   2
   3 /*
   4  * ====================================================
   5  * Copyright 2004 Sun Microsystems, Inc.  All Rights Reserved.
   6  *
   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 /* INDENT OFF */
  14 /* __kernel_tan( x, y, k )
  15  * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
  16  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  17  * Input y is the tail of x.
  18  * Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned.
  19  *
  20  * Algorithm
  21  *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
  22  *      2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
  23  *      3. tan(x) is approximated by a odd polynomial of degree 27 on
  24  *         [0,0.67434]
  25  *                               3             27
  26  *              tan(x) ~ x + T1*x + ... + T13*x
  27  *         where
  28  *
  29  *              |tan(x)         2     4            26   |     -59.2
  30  *              |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
  31  *              |  x                                    |
  32  *
  33  *         Note: tan(x+y) = tan(x) + tan'(x)*y
  34  *                        ~ tan(x) + (1+x*x)*y
  35  *         Therefore, for better accuracy in computing tan(x+y), let
  36  *                   3      2      2       2       2
  37  *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
  38  *         then
  39  *                                  3    2
  40  *              tan(x+y) = x + (T1*x + (x *(r+y)+y))
  41  *
  42  *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
  43  *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
  44  *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
  45  */
  46
  47 #include "fdlibm.h"
  48
  49 #ifndef _DOUBLE_IS_32BITS
  50
  51 static const double xxx[] = {
  52                  3.33333333333334091986e-01,    /* 3FD55555, 55555563 */
  53                  1.33333333333201242699e-01,    /* 3FC11111, 1110FE7A */
  54                  5.39682539762260521377e-02,    /* 3FABA1BA, 1BB341FE */
  55                  2.18694882948595424599e-02,    /* 3F9664F4, 8406D637 */
  56                  8.86323982359930005737e-03,    /* 3F8226E3, E96E8493 */
  57                  3.59207910759131235356e-03,    /* 3F6D6D22, C9560328 */
  58                  1.45620945432529025516e-03,    /* 3F57DBC8, FEE08315 */
  59                  5.88041240820264096874e-04,    /* 3F4344D8, F2F26501 */
  60                  2.46463134818469906812e-04,    /* 3F3026F7, 1A8D1068 */
  61                  7.81794442939557092300e-05,    /* 3F147E88, A03792A6 */
  62                  7.14072491382608190305e-05,    /* 3F12B80F, 32F0A7E9 */
  63                 -1.85586374855275456654e-05,    /* BEF375CB, DB605373 */
  64                  2.59073051863633712884e-05,    /* 3EFB2A70, 74BF7AD4 */
  65 /* one */        1.00000000000000000000e+00,    /* 3FF00000, 00000000 */
  66 /* pio4 */       7.85398163397448278999e-01,    /* 3FE921FB, 54442D18 */
  67 /* pio4lo */     3.06161699786838301793e-17     /* 3C81A626, 33145C07 */
  68 };
  69 #define one     xxx[13]
  70 #define pio4    xxx[14]
  71 #define pio4lo  xxx[15]
  72 #define T       xxx
  73 /* INDENT ON */
  74
  75 double
  76 __kernel_tan(double x, double y, int iy) {
  77         double z, r, v, w, s;
  78         int32_t ix, hx;
  79
  80         GET_HIGH_WORD(hx,x); /* high word of x */
  81         ix = hx & 0x7fffffff;                   /* high word of |x| */
  82         if (ix < 0x3e300000) {                  /* x < 2**-28 */
  83                 if ((int) x == 0) {             /* generate inexact */
  84                         uint32_t low;
  85                         GET_LOW_WORD(low,x);
  86                         if (((ix | low) | (iy + 1)) == 0)
  87                                 return one / fabs(x);
  88                         else {
  89                                 if (iy == 1)
  90                                         return x;
  91                                 else {  /* compute -1 / (x+y) carefully */
  92                                         double a, t;
  93
  94                                         z = w = x + y;
  95                                         SET_LOW_WORD(z,0);
  96                                         v = y - (z - x);
  97                                         t = a = -one / w;
  98                                         SET_LOW_WORD(t,0);
  99                                         s = one + t * z;
 100                                         return t + a * (s + t * v);
 101                                 }
 102                         }
 103                 }
 104         }
 105         if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */
 106                 if (hx < 0) {
 107                         x = -x;
 108                         y = -y;
 109                 }
 110                 z = pio4 - x;
 111                 w = pio4lo - y;
 112                 x = z + w;
 113                 y = 0.0;
 114         }
 115         z = x * x;
 116         w = z * z;
 117         /*
 118          * Break x^5*(T[1]+x^2*T[2]+...) into
 119          * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
 120          * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
 121          */
 122         r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +
 123                 w * T[11]))));
 124         v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +
 125                 w * T[12])))));
 126         s = z * x;
 127         r = y + z * (s * (r + v) + y);
 128         r += T[0] * s;
 129         w = x + r;
 130         if (ix >= 0x3FE59428) {
 131                 v = (double) iy;
 132                 return (double) (1 - ((hx >> 30) & 2)) *
 133                         (v - 2.0 * (x - (w * w / (w + v) - r)));
 134         }
 135         if (iy == 1)
 136                 return w;
 137         else {
 138                 /*
 139                  * if allow error up to 2 ulp, simply return
 140                  * -1.0 / (x+r) here
 141                  */
 142                 /* compute -1.0 / (x+r) accurately */
 143                 double a, t;
 144                 z = w;
 145                 SET_LOW_WORD(z,0);
 146                 v = r - (z - x);        /* z+v = r+x */
 147                 t = a = -1.0 / w;       /* a = -1.0/w */
 148                 SET_LOW_WORD(t,0);
 149                 s = 1.0 + t * z;
 150                 return t + a * (s + t * v);
 151         }
 152 }
 153 #endif /* defined(_DOUBLE_IS_32BITS) */