src/math/__tan.c

   1 /* origin: FreeBSD /usr/src/lib/msun/src/k_tan.c */
   2 /*
   3  * ====================================================
   4  * Copyright 2004 Sun Microsystems, Inc.  All Rights Reserved.
   5  *
   6  * Permission to use, copy, modify, and distribute this
   7  * software is freely granted, provided that this notice
   8  * is preserved.
   9  * ====================================================
  10  */
  11 /* __tan( x, y, k )
  12  * kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854
  13  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  14  * Input y is the tail of x.
  15  * Input odd indicates whether tan (if odd = 0) or -1/tan (if odd = 1) is returned.
  16  *
  17  * Algorithm
  18  *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
  19  *      2. Callers must return tan(-0) = -0 without calling here since our
  20  *         odd polynomial is not evaluated in a way that preserves -0.
  21  *         Callers may do the optimization tan(x) ~ x for tiny x.
  22  *      3. tan(x) is approximated by a odd polynomial of degree 27 on
  23  *         [0,0.67434]
  24  *                               3             27
  25  *              tan(x) ~ x + T1*x + ... + T13*x
  26  *         where
  27  *
  28  *              |tan(x)         2     4            26   |     -59.2
  29  *              |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
  30  *              |  x                                    |
  31  *
  32  *         Note: tan(x+y) = tan(x) + tan'(x)*y
  33  *                        ~ tan(x) + (1+x*x)*y
  34  *         Therefore, for better accuracy in computing tan(x+y), let
  35  *                   3      2      2       2       2
  36  *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
  37  *         then
  38  *                                  3    2
  39  *              tan(x+y) = x + (T1*x + (x *(r+y)+y))
  40  *
  41  *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
  42  *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
  43  *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
  44  */
  45
  46 #include "libm.h"
  47
  48 static const double T[] = {
  49              3.33333333333334091986e-01, /* 3FD55555, 55555563 */
  50              1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
  51              5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
  52              2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
  53              8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
  54              3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
  55              1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
  56              5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
  57              2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
  58              7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
  59              7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
  60             -1.85586374855275456654e-05, /* BEF375CB, DB605373 */
  61              2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
  62 },
  63 pio4 =       7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
  64 pio4lo =     3.06161699786838301793e-17; /* 3C81A626, 33145C07 */
  65
  66 double __tan(double x, double y, int odd)
  67 {
  68         double_t z, r, v, w, s, a;
  69         double w0, a0;
  70         uint32_t hx;
  71         int big, sign;
  72
  73         GET_HIGH_WORD(hx,x);
  74         big = (hx&0x7fffffff) >= 0x3FE59428; /* |x| >= 0.6744 */
  75         if (big) {
  76                 sign = hx>>31;
  77                 if (sign) {
  78                         x = -x;
  79                         y = -y;
  80                 }
  81                 x = (pio4 - x) + (pio4lo - y);
  82                 y = 0.0;
  83         }
  84         z = x * x;
  85         w = z * z;
  86         /*
  87          * Break x^5*(T[1]+x^2*T[2]+...) into
  88          * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
  89          * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
  90          */
  91         r = T[1] + w*(T[3] + w*(T[5] + w*(T[7] + w*(T[9] + w*T[11]))));
  92         v = z*(T[2] + w*(T[4] + w*(T[6] + w*(T[8] + w*(T[10] + w*T[12])))));
  93         s = z * x;
  94         r = y + z*(s*(r + v) + y) + s*T[0];
  95         w = x + r;
  96         if (big) {
  97                 s = 1 - 2*odd;
  98                 v = s - 2.0 * (x + (r - w*w/(w + s)));
  99                 return sign ? -v : v;
 100         }
 101         if (!odd)
 102                 return w;
 103         /* -1.0/(x+r) has up to 2ulp error, so compute it accurately */
 104         w0 = w;
 105         SET_LOW_WORD(w0, 0);
 106         v = r - (w0 - x);       /* w0+v = r+x */
 107         a0 = a = -1.0 / w;
 108         SET_LOW_WORD(a0, 0);
 109         return a0 + a*(1.0 + a0*w0 + a0*v);
 110 }