lib/libm/src/k_tan.c

   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 #include <sys/cdefs.h>
  14 #if defined(LIBM_SCCS) && !defined(lint)
  15 __RCSID("$NetBSD: k_tan.c,v 1.12 2004/07/22 18:24:09 drochner Exp $");
  16 #endif
  17
  18 /* __kernel_tan( x, y, k )
  19  * kernel tan function on [-pi/4, pi/4], 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
  23  * -1/tan (if k= -1) is returned.
  24  *
  25  * Algorithm
  26  *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
  27  *      2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
  28  *      3. tan(x) is approximated by a odd polynomial of degree 27 on
  29  *         [0,0.67434]
  30  *                               3             27
  31  *              tan(x) ~ x + T1*x + ... + T13*x
  32  *         where
  33  *
  34  *              |tan(x)         2     4            26   |     -59.2
  35  *              |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
  36  *              |  x                                    |
  37  *
  38  *         Note: tan(x+y) = tan(x) + tan'(x)*y
  39  *                        ~ tan(x) + (1+x*x)*y
  40  *         Therefore, for better accuracy in computing tan(x+y), let
  41  *                   3      2      2       2       2
  42  *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
  43  *         then
  44  *                                  3    2
  45  *              tan(x+y) = x + (T1*x + (x *(r+y)+y))
  46  *
  47  *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
  48  *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
  49  *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
  50  */
  51
  52 #include "math.h"
  53 #include "math_private.h"
  54
  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
  78 double
  79 __kernel_tan(double x, double y, int iy)
  80 {
  81         double z, r, v, w, s;
  82         int32_t ix, hx;
  83
  84         GET_HIGH_WORD(hx, x);   /* high word of x */
  85         ix = hx & 0x7fffffff;                   /* high word of |x| */
  86         if (ix < 0x3e300000) {                  /* x < 2**-28 */
  87                 if ((int) x == 0) {             /* generate inexact */
  88                         u_int32_t low;
  89                         GET_LOW_WORD(low, x);
  90                         if(((ix | low) | (iy + 1)) == 0)
  91                                 return one / fabs(x);
  92                         else {
  93                                 if (iy == 1)
  94                                         return x;
  95                                 else {  /* compute -1 / (x+y) carefully */
  96                                         double a, t;
  97
  98                                         z = w = x + y;
  99                                         SET_LOW_WORD(z, 0);
 100                                         v = y - (z - x);
 101                                         t = a = -one / w;
 102                                         SET_LOW_WORD(t, 0);
 103                                         s = one + t * z;
 104                                         return t + a * (s + t * v);
 105                                 }
 106                         }
 107                 }
 108         }
 109         if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */
 110                 if (hx < 0) {
 111                         x = -x;
 112                         y = -y;
 113                 }
 114                 z = pio4 - x;
 115                 w = pio4lo - y;
 116                 x = z + w;
 117                 y = 0.0;
 118         }
 119         z = x * x;
 120         w = z * z;
 121         /*
 122          * Break x^5*(T[1]+x^2*T[2]+...) into
 123          * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
 124          * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
 125          */
 126         r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +
 127                 w * T[11]))));
 128         v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +
 129                 w * T[12])))));
 130         s = z * x;
 131         r = y + z * (s * (r + v) + y);
 132         r += T[0] * s;
 133         w = x + r;
 134         if (ix >= 0x3FE59428) {
 135                 v = (double) iy;
 136                 return (double) (1 - ((hx >> 30) & 2)) *
 137                         (v - 2.0 * (x - (w * w / (w + v) - r)));
 138         }
 139         if (iy == 1)
 140                 return w;
 141         else {
 142                 /*
 143                  * if allow error up to 2 ulp, simply return
 144                  * -1.0 / (x+r) here
 145                  */
 146                 /* compute -1.0 / (x+r) accurately */
 147                 double a, t;
 148                 z = w;
 149                 SET_LOW_WORD(z, 0);
 150                 v = r - (z - x);        /* z+v = r+x */
 151                 t = a = -1.0 / w;       /* a = -1.0/w */
 152                 SET_LOW_WORD(t, 0);
 153                 s = 1.0 + t * z;
 154                 return t + a * (s + t * v);
 155         }
 156 }