compiler/mlib/k_tan.c

   1 /* @(#)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 #ifndef lint
  15 static char rcsid[] = "$FreeBSD: src/lib/msun/src/k_tan.c,v 1.12 2005/11/02 14:01:45 bde Exp $";
  16 #endif
  17
  18 /* __kernel_tan( x, y, k )
  19  * kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854
  20  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  21  * Input y is the tail of x.
  22  * Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned.
  23  *
  24  * Algorithm
  25  *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
  26  *      2. Callers must return tan(-0) = -0 without calling here since our
  27  *         odd polynomial is not evaluated in a way that preserves -0.
  28  *         Callers may do the optimization tan(x) ~ x for tiny x.
  29  *      3. tan(x) is approximated by a odd polynomial of degree 27 on
  30  *         [0,0.67434]
  31  *                               3             27
  32  *              tan(x) ~ x + T1*x + ... + T13*x
  33  *         where
  34  *
  35  *              |tan(x)         2     4            26   |     -59.2
  36  *              |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
  37  *              |  x                                    |
  38  *
  39  *         Note: tan(x+y) = tan(x) + tan'(x)*y
  40  *                        ~ tan(x) + (1+x*x)*y
  41  *         Therefore, for better accuracy in computing tan(x+y), let
  42  *                   3      2      2       2       2
  43  *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
  44  *         then
  45  *                                  3    2
  46  *              tan(x+y) = x + (T1*x + (x *(r+y)+y))
  47  *
  48  *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
  49  *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
  50  *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
  51  */
  52
  53 #include "math.h"
  54 #include "math_private.h"
  55 static const double xxx[] = {
  56                  3.33333333333334091986e-01,    /* 3FD55555, 55555563 */
  57                  1.33333333333201242699e-01,    /* 3FC11111, 1110FE7A */
  58                  5.39682539762260521377e-02,    /* 3FABA1BA, 1BB341FE */
  59                  2.18694882948595424599e-02,    /* 3F9664F4, 8406D637 */
  60                  8.86323982359930005737e-03,    /* 3F8226E3, E96E8493 */
  61                  3.59207910759131235356e-03,    /* 3F6D6D22, C9560328 */
  62                  1.45620945432529025516e-03,    /* 3F57DBC8, FEE08315 */
  63                  5.88041240820264096874e-04,    /* 3F4344D8, F2F26501 */
  64                  2.46463134818469906812e-04,    /* 3F3026F7, 1A8D1068 */
  65                  7.81794442939557092300e-05,    /* 3F147E88, A03792A6 */
  66                  7.14072491382608190305e-05,    /* 3F12B80F, 32F0A7E9 */
  67                 -1.85586374855275456654e-05,    /* BEF375CB, DB605373 */
  68                  2.59073051863633712884e-05,    /* 3EFB2A70, 74BF7AD4 */
  69 /* one */        1.00000000000000000000e+00,    /* 3FF00000, 00000000 */
  70 /* pio4 */       7.85398163397448278999e-01,    /* 3FE921FB, 54442D18 */
  71 /* pio4lo */     3.06161699786838301793e-17     /* 3C81A626, 33145C07 */
  72 };
  73 #define one     xxx[13]
  74 #define pio4    xxx[14]
  75 #define pio4lo  xxx[15]
  76 #define T       xxx
  77 /* INDENT ON */
  78
  79 double
  80 __kernel_tan(double x, double y, int iy) {
  81         double z, r, v, w, s;
  82         int32_t ix, hx;
  83
  84         GET_HIGH_WORD(hx,x);
  85         ix = hx & 0x7fffffff;                   /* high word of |x| */
  86         if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */
  87                 if (hx < 0) {
  88                         x = -x;
  89                         y = -y;
  90             }
  91                 z = pio4 - x;
  92                 w = pio4lo - y;
  93                 x = z + w;
  94                 y = 0.0;
  95             }
  96         z = x * x;
  97         w = z * z;
  98         /*
  99          * Break x^5*(T[1]+x^2*T[2]+...) into
 100      *    x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
 101      *    x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
 102      */
 103         r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +
 104                 w * T[11]))));
 105         v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +
 106                 w * T[12])))));
 107         s = z * x;
 108         r = y + z * (s * (r + v) + y);
 109         r += T[0] * s;
 110         w = x + r;
 111         if (ix >= 0x3FE59428) {
 112                 v = (double) iy;
 113                 return (double) (1 - ((hx >> 30) & 2)) *
 114                         (v - 2.0 * (x - (w * w / (w + v) - r)));
 115         }
 116         if (iy == 1)
 117                 return w;
 118         else {
 119                 /*
 120                  * if allow error up to 2 ulp, simply return
 121                  * -1.0 / (x+r) here
 122                  */
 123                 /* compute -1.0 / (x+r) accurately */
 124                 double a, t;
 125             z  = w;
 126             SET_LOW_WORD(z,0);
 127                 v = r - (z - x);        /* z+v = r+x */
 128                 t = a = -1.0 / w;       /* a = -1.0/w */
 129             SET_LOW_WORD(t,0);
 130                 s = 1.0 + t * z;
 131                 return t + a * (s + t * v);
 132         }
 133 }