lib/msun/src/k_tan.c

   1 /* @(#)k_tan.c 5.1 93/09/24 */
   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
  12 #ifndef lint
  13 static char rcsid[] = "$FreeBSD$";
  14 #endif
  15
  16 /* __kernel_tan( x, y, k )
  17  * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
  18  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  19  * Input y is the tail of x.
  20  * Input k indicates whether tan (if k=1) or
  21  * -1/tan (if k= -1) is returned.
  22  *
  23  * Algorithm
  24  *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
  25  *      2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
  26  *      3. tan(x) is approximated by an odd polynomial of degree 27 on
  27  *         [0,0.67434]
  28  *                               3             27
  29  *              tan(x) ~ x + T1*x + ... + T13*x
  30  *         where
  31  *
  32  *              |tan(x)         2     4            26   |     -59.2
  33  *              |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
  34  *              |  x                                    |
  35  *
  36  *         Note: tan(x+y) = tan(x) + tan'(x)*y
  37  *                        ~ tan(x) + (1+x*x)*y
  38  *         Therefore, for better accuracy in computing tan(x+y), let
  39  *                   3      2      2       2       2
  40  *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
  41  *         then
  42  *                                  3    2
  43  *              tan(x+y) = x + (T1*x + (x *(r+y)+y))
  44  *
  45  *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
  46  *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
  47  *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
  48  */
  49
  50 #include "math.h"
  51 #include "math_private.h"
  52 static const double
  53 one   =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  54 pio4  =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
  55 pio4lo=  3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
  56 T[] =  {
  57   3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
  58   1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
  59   5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
  60   2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
  61   8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
  62   3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
  63   1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
  64   5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
  65   2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
  66   7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
  67   7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
  68  -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
  69   2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
  70 };
  71
  72 double
  73 __kernel_tan(double x, double y, int iy)
  74 {
  75         double z,r,v,w,s;
  76         int32_t ix,hx;
  77         GET_HIGH_WORD(hx,x);
  78         ix = hx&0x7fffffff;     /* high word of |x| */
  79         if(ix<0x3e300000) {                     /* x < 2**-28 */
  80                 if ((int) x == 0) {             /* generate inexact */
  81                         u_int32_t low;
  82                         GET_LOW_WORD(low,x);
  83                         if (((ix | low) | (iy + 1)) == 0)
  84                                 return one / fabs(x);
  85                         else {
  86                                 if (iy == 1)
  87                                         return x;
  88                                 else {  /* compute -1 / (x+y) carefully */
  89                                         double a, t;
  90
  91                                         z = w = x + y;
  92                                         SET_LOW_WORD(z, 0);
  93                                         v = y - (z - x);
  94                                         t = a = -one / w;
  95                                         SET_LOW_WORD(t, 0);
  96                                         s = one + t * z;
  97                                         return t + a * (s + t * v);
  98                                 }
  99                         }
 100                 }
 101         }
 102         if(ix>=0x3FE59428) {                    /* |x|>=0.6744 */
 103             if(hx<0) {x = -x; y = -y;}
 104             z = pio4-x;
 105             w = pio4lo-y;
 106             x = z+w; y = 0.0;
 107         }
 108         z       =  x*x;
 109         w       =  z*z;
 110     /* Break x^5*(T[1]+x^2*T[2]+...) into
 111      *    x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
 112      *    x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
 113      */
 114         r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
 115         v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
 116         s = z*x;
 117         r = y + z*(s*(r+v)+y);
 118         r += T[0]*s;
 119         w = x+r;
 120         if(ix>=0x3FE59428) {
 121             v = (double)iy;
 122             return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
 123         }
 124         if(iy==1) return w;
 125         else {          /* if allow error up to 2 ulp,
 126                            simply return -1.0/(x+r) here */
 127      /*  compute -1.0/(x+r) accurately */
 128             double a,t;
 129             z  = w;
 130             SET_LOW_WORD(z,0);
 131             v  = r-(z - x);     /* z+v = r+x */
 132             t = a  = -1.0/w;    /* a = -1.0/w */
 133             SET_LOW_WORD(t,0);
 134             s  = 1.0+t*z;
 135             return t+a*(s+t*v);
 136         }
 137 }