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  1.  
  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. /* __kernel_tan( x, y, k )
  15.  * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
  16.  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  17.  * Input y is the tail of x.
  18.  * Input k indicates whether tan (if k=1) or
  19.  * -1/tan (if k= -1) is returned.
  20.  *
  21.  * Algorithm
  22.  *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
  23.  *      2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
  24.  *      3. tan(x) is approximated by a odd polynomial of degree 27 on
  25.  *         [0,0.67434]
  26.  *                               3             27
  27.  *              tan(x) ~ x + T1*x + ... + T13*x
  28.  *         where
  29.  *     
  30.  *              |tan(x)         2     4            26   |     -59.2
  31.  *              |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
  32.  *              |  x                                    |
  33.  *
  34.  *         Note: tan(x+y) = tan(x) + tan'(x)*y
  35.  *                        ~ tan(x) + (1+x*x)*y
  36.  *         Therefore, for better accuracy in computing tan(x+y), let
  37.  *                   3      2      2       2       2
  38.  *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
  39.  *         then
  40.  *                                  3    2
  41.  *              tan(x+y) = x + (T1*x + (x *(r+y)+y))
  42.  *
  43.  *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
  44.  *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
  45.  *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
  46.  */
  47.  
  48. #include "fdlibm.h"
  49.  
  50. #ifndef _DOUBLE_IS_32BITS
  51.  
  52. #ifdef __STDC__
  53. static const double
  54. #else
  55. static double
  56. #endif
  57. one   =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  58. pio4  =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
  59. pio4lo=  3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
  60. T[] =  {
  61.   3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
  62.   1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
  63.   5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
  64.   2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
  65.   8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
  66.   3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
  67.   1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
  68.   5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
  69.   2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
  70.   7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
  71.   7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
  72.  -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
  73.   2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
  74. };
  75.  
  76. #ifdef __STDC__
  77.         double __kernel_tan(double x, double y, int iy)
  78. #else
  79.         double __kernel_tan(x, y, iy)
  80.         double x,y; int iy;
  81. #endif
  82. {
  83.         double z,r,v,w,s;
  84.         __int32_t ix,hx;
  85.         GET_HIGH_WORD(hx,x);
  86.         ix = hx&0x7fffffff;     /* high word of |x| */
  87.         if(ix<0x3e300000)                       /* x < 2**-28 */
  88.             {if((int)x==0) {                    /* generate inexact */
  89.                 __uint32_t low;
  90.                 GET_LOW_WORD(low,x);
  91.                 if(((ix|low)|(iy+1))==0) return one/fabs(x);
  92.                 else return (iy==1)? x: -one/x;
  93.             }
  94.             }
  95.         if(ix>=0x3FE59428) {                    /* |x|>=0.6744 */
  96.             if(hx<0) {x = -x; y = -y;}
  97.             z = pio4-x;
  98.             w = pio4lo-y;
  99.             x = z+w; y = 0.0;
  100.         }
  101.         z       =  x*x;
  102.         w       =  z*z;
  103.     /* Break x^5*(T[1]+x^2*T[2]+...) into
  104.      *    x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
  105.      *    x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
  106.      */
  107.         r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
  108.         v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
  109.         s = z*x;
  110.         r = y + z*(s*(r+v)+y);
  111.         r += T[0]*s;
  112.         w = x+r;
  113.         if(ix>=0x3FE59428) {
  114.             v = (double)iy;
  115.             return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
  116.         }
  117.         if(iy==1) return w;
  118.         else {          /* if allow error up to 2 ulp,
  119.                            simply return -1.0/(x+r) here */
  120.      /*  compute -1.0/(x+r) accurately */
  121.             double a,t;
  122.             z  = w;
  123.             SET_LOW_WORD(z,0);
  124.             v  = r-(z - x);     /* z+v = r+x */
  125.             t = a  = -1.0/w;    /* a = -1.0/w */
  126.             SET_LOW_WORD(t,0);
  127.             s  = 1.0+t*z;
  128.             return t+a*(s+t*v);
  129.         }
  130. }
  131.  
  132. #endif /* defined(_DOUBLE_IS_32BITS) */
  133.