gonzui

    1: /* @(#)k_tan.c 5.1 93/09/24 */
    2: /*
    3:  * ====================================================
    4:  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
    5:  *
    6:  * Developed at SunPro, 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: 
   13: /* __kernel_tan( x, y, k )
   14:  * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
   15:  * Input x is assumed to be bounded by ~pi/4 in magnitude.
   16:  * Input y is the tail of x.
   17:  * Input k indicates whether tan (if k=1) or
   18:  * -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 "math.h"
   48: #include "math_private.h"
   49: 
   50: static const double xxx[] = {
   51:                  3.33333333333334091986e-01,  /* 3FD55555, 55555563 */
   52:                  1.33333333333201242699e-01,  /* 3FC11111, 1110FE7A */
   53:                  5.39682539762260521377e-02,  /* 3FABA1BA, 1BB341FE */
   54:                  2.18694882948595424599e-02,  /* 3F9664F4, 8406D637 */
   55:                  8.86323982359930005737e-03,  /* 3F8226E3, E96E8493 */
   56:                  3.59207910759131235356e-03,  /* 3F6D6D22, C9560328 */
   57:                  1.45620945432529025516e-03,  /* 3F57DBC8, FEE08315 */
   58:                  5.88041240820264096874e-04,  /* 3F4344D8, F2F26501 */
   59:                  2.46463134818469906812e-04,  /* 3F3026F7, 1A8D1068 */
   60:                  7.81794442939557092300e-05,  /* 3F147E88, A03792A6 */
   61:                  7.14072491382608190305e-05,  /* 3F12B80F, 32F0A7E9 */
   62:                 -1.85586374855275456654e-05,  /* BEF375CB, DB605373 */
   63:                  2.59073051863633712884e-05,  /* 3EFB2A70, 74BF7AD4 */
   64: /* one */        1.00000000000000000000e+00,  /* 3FF00000, 00000000 */
   65: /* pio4 */       7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
   66: /* pio4lo */     3.06161699786838301793e-17        /* 3C81A626, 33145C07 */
   67: };
   68: #define one     xxx[13]
   69: #define pio4    xxx[14]
   70: #define pio4lo  xxx[15]
   71: #define T       xxx
   72: 
   73: double
   74: __kernel_tan(double x, double y, int iy)
   75: {
   76:         double z, r, v, w, s;
   77:         int32_t ix, hx;
   78: 
   79:         GET_HIGH_WORD(hx, x);  /* high word of x */
   80:         ix = hx & 0x7fffffff;                  /* high word of |x| */
   81:         if (ix < 0x3e300000) {                 /* x < 2**-28 */
   82:                 if ((int) x == 0) {           /* generate inexact */
   83:                         u_int32_t low;
   84:                         GET_LOW_WORD(low, x);
   85:                         if(((ix | low) | (iy + 1)) == 0)
   86:                                 return one / fabs(x);
   87:                         else {
   88:                                 if (iy == 1)
   89:                                         return x;
   90:                                 else {      /* compute -1 / (x+y) carefully */
   91:                                         double a, t;
   92: 
   93:                                         z = w = x + y;
   94:                                         SET_LOW_WORD(z, 0);
   95:                                         v = y - (z - x);
   96:                                         t = a = -one / w;
   97:                                         SET_LOW_WORD(t, 0);
   98:                                         s = one + t * z;
   99:                                         return t + a * (s + t * v);
  100:                                 }
  101:                         }
  102:                 }
  103:         }
  104:         if (ix >= 0x3FE59428) {        /* |x| >= 0.6744 */
  105:                 if (hx < 0) {
  106:                         x = -x;
  107:                         y = -y;
  108:                 }
  109:                 z = pio4 - x;
  110:                 w = pio4lo - y;
  111:                 x = z + w;
  112:                 y = 0.0;
  113:         }
  114:         z = x * x;
  115:         w = z * z;
  116:         /*
  117:          * Break x^5*(T[1]+x^2*T[2]+...) into
  118:          * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
  119:          * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
  120:          */
  121:         r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +
  122:                 w * T[11]))));
  123:         v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +
  124:                 w * T[12])))));
  125:         s = z * x;
  126:         r = y + z * (s * (r + v) + y);
  127:         r += T[0] * s;
  128:         w = x + r;
  129:         if (ix >= 0x3FE59428) {
  130:                 v = (double) iy;
  131:                 return (double) (1 - ((hx >> 30) & 2)) *
  132:                         (v - 2.0 * (x - (w * w / (w + v) - r)));
  133:         }
  134:         if (iy == 1)
  135:                 return w;
  136:         else {
  137:                 /*
  138:                  * if allow error up to 2 ulp, simply return
  139:                  * -1.0 / (x+r) here
  140:                  */
  141:                 /* compute -1.0 / (x+r) accurately */
  142:                 double a, t;
  143:                 z = w;
  144:                 SET_LOW_WORD(z, 0);
  145:                 v = r - (z - x);      /* z+v = r+x */
  146:                 t = a = -1.0 / w;     /* a = -1.0/w */
  147:                 SET_LOW_WORD(t, 0);
  148:                 s = 1.0 + t * z;
  149:                 return t + a * (s + t * v);
  150:         }
  151: }