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  1. /* Copyright (C) 1994 DJ Delorie, see COPYING.DJ for details */
  2. /* @(#)k_tan.c 5.1 93/09/24 */
  3. /*
  4.  * ====================================================
  5.  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  6.  *
  7.  * Developed at SunPro, a Sun Microsystems, Inc. business.
  8.  * Permission to use, copy, modify, and distribute this
  9.  * software is freely granted, provided that this notice
  10.  * is preserved.
  11.  * ====================================================
  12.  */
  13.  
  14. #if defined(LIBM_SCCS) && !defined(lint)
  15. static char rcsid[] = "$Id: k_tan.c,v 1.6 1994/08/18 23:06:16 jtc 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. #ifdef __STDC__
  55. static const double
  56. #else
  57. static double
  58. #endif
  59. one   =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  60. pio4  =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
  61. pio4lo=  3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
  62. T[] =  {
  63.   3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
  64.   1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
  65.   5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
  66.   2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
  67.   8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
  68.   3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
  69.   1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
  70.   5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
  71.   2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
  72.   7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
  73.   7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
  74.  -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
  75.   2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
  76. };
  77.  
  78. #ifdef __STDC__
  79.         double __kernel_tan(double x, double y, int iy)
  80. #else
  81.         double __kernel_tan(x, y, iy)
  82.         double x,y; int iy;
  83. #endif
  84. {
  85.         double z,r,v,w,s;
  86.         int32_t ix,hx;
  87.         GET_HIGH_WORD(hx,x);
  88.         ix = hx&0x7fffffff;     /* high word of |x| */
  89.         if(ix<0x3e300000)                       /* x < 2**-28 */
  90.             {if((int)x==0) {                    /* generate inexact */
  91.                 u_int32_t low;
  92.                 GET_LOW_WORD(low,x);
  93.                 if(((ix|low)|(iy+1))==0) return one/fabs(x);
  94.                 else return (iy==1)? x: -one/x;
  95.             }
  96.             }
  97.         if(ix>=0x3FE59428) {                    /* |x|>=0.6744 */
  98.             if(hx<0) {x = -x; y = -y;}
  99.             z = pio4-x;
  100.             w = pio4lo-y;
  101.             x = z+w; y = 0.0;
  102.         }
  103.         z       =  x*x;
  104.         w       =  z*z;
  105.     /* Break x^5*(T[1]+x^2*T[2]+...) into
  106.      *    x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
  107.      *    x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
  108.      */
  109.         r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
  110.         v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
  111.         s = z*x;
  112.         r = y + z*(s*(r+v)+y);
  113.         r += T[0]*s;
  114.         w = x+r;
  115.         if(ix>=0x3FE59428) {
  116.             v = (double)iy;
  117.             return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
  118.         }
  119.         if(iy==1) return w;
  120.         else {          /* if allow error up to 2 ulp,
  121.                            simply return -1.0/(x+r) here */
  122.      /*  compute -1.0/(x+r) accurately */
  123.             double a,t;
  124.             z  = w;
  125.             SET_LOW_WORD(z,0);
  126.             v  = r-(z - x);     /* z+v = r+x */
  127.             t = a  = -1.0/w;    /* a = -1.0/w */
  128.             SET_LOW_WORD(t,0);
  129.             s  = 1.0+t*z;
  130.             return t+a*(s+t*v);
  131.         }
  132. }
  133.