libm/k_tan.c

   1 /*
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  *
   5  * Developed at SunPro, a Sun Microsystems, Inc. business.
   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 /* __kernel_tan( x, y, k )
  13  * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
  14  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  15  * Input y is the tail of x.
  16  * Input k indicates whether tan (if k=1) or
  17  * -1/tan (if k= -1) is returned.
  18  *
  19  * Algorithm
  20  *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
  21  *      2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
  22  *      3. tan(x) is approximated by a odd polynomial of degree 27 on
  23  *         [0,0.67434]
  24  *                               3             27
  25  *              tan(x) ~ x + T1*x + ... + T13*x
  26  *         where
  27  *
  28  *              |tan(x)         2     4            26   |     -59.2
  29  *              |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
  30  *              |  x                                    |
  31  *
  32  *         Note: tan(x+y) = tan(x) + tan'(x)*y
  33  *                        ~ tan(x) + (1+x*x)*y
  34  *         Therefore, for better accuracy in computing tan(x+y), let
  35  *                   3      2      2       2       2
  36  *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
  37  *         then
  38  *                                  3    2
  39  *              tan(x+y) = x + (T1*x + (x *(r+y)+y))
  40  *
  41  *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
  42  *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
  43  *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
  44  */
  45
  46 #include "math.h"
  47 #include "math_private.h"
  48
  49 static const double
  50 one   =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  51 pio4  =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
  52 pio4lo=  3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
  53 T[] =  {
  54   3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
  55   1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
  56   5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
  57   2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
  58   8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
  59   3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
  60   1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
  61   5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
  62   2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
  63   7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
  64   7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
  65  -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
  66   2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
  67 };
  68
  69 double __kernel_tan(double x, double y, int iy)
  70 {
  71         double z,r,v,w,s;
  72         int32_t ix,hx;
  73         GET_HIGH_WORD(hx,x);
  74         ix = hx&0x7fffffff;     /* high word of |x| */
  75         if(ix<0x3e300000)                       /* x < 2**-28 */
  76             {if((int)x==0) {                    /* generate inexact */
  77                 u_int32_t low;
  78                 GET_LOW_WORD(low,x);
  79                 if(((ix|low)|(iy+1))==0) return one/fabs(x);
  80                 else return (iy==1)? x: -one/x;
  81             }
  82             }
  83         if(ix>=0x3FE59428) {                    /* |x|>=0.6744 */
  84             if(hx<0) {x = -x; y = -y;}
  85             z = pio4-x;
  86             w = pio4lo-y;
  87             x = z+w; y = 0.0;
  88         }
  89         z       =  x*x;
  90         w       =  z*z;
  91     /* Break x^5*(T[1]+x^2*T[2]+...) into
  92      *    x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
  93      *    x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
  94      */
  95         r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
  96         v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
  97         s = z*x;
  98         r = y + z*(s*(r+v)+y);
  99         r += T[0]*s;
 100         w = x+r;
 101         if(ix>=0x3FE59428) {
 102             v = (double)iy;
 103             return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
 104         }
 105         if(iy==1) return w;
 106         else {          /* if allow error up to 2 ulp,
 107                            simply return -1.0/(x+r) here */
 108      /*  compute -1.0/(x+r) accurately */
 109             double a,t;
 110             z  = w;
 111             SET_LOW_WORD(z,0);
 112             v  = r-(z - x);     /* z+v = r+x */
 113             t = a  = -1.0/w;    /* a = -1.0/w */
 114             SET_LOW_WORD(t,0);
 115             s  = 1.0+t*z;
 116             return t+a*(s+t*v);
 117         }
 118 }