github.com/afumu/libc@v0.0.6/musl/src/math/__tan.c (about)

     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  }