newlib/libm/math/k_tan.c

   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 {
  93                                 if(iy==1)
  94                                         return x;
  95                                 else {
  96                                         double a, t;
  97                                         z = w = x + y;
  98                                         SET_LOW_WORD(z,0);
  99                                         v = y - (z - x);
 100                                         t = a = -one / w;
 101                                         SET_LOW_WORD(t,0);
 102                                         s = one + t * z;
 103                                         return t + a * (s + t * v);
 104                                 }
 105                         }
 106                 }
 107         }
 108         if(ix>=0x3FE59428) {                    /* |x|>=0.6744 */
 109             if(hx<0) {x = -x; y = -y;}
 110             z = pio4-x;
 111             w = pio4lo-y;
 112             x = z+w; y = 0.0;
 113         }
 114         z       =  x*x;
 115         w       =  z*z;
 116     /* Break x^5*(T[1]+x^2*T[2]+...) into
 117      *    x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
 118      *    x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
 119      */
 120         r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
 121         v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
 122         s = z*x;
 123         r = y + z*(s*(r+v)+y);
 124         r += T[0]*s;
 125         w = x+r;
 126         if(ix>=0x3FE59428) {
 127             v = (double)iy;
 128             return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
 129         }
 130         if(iy==1) return w;
 131         else {          /* if allow error up to 2 ulp,
 132                            simply return -1.0/(x+r) here */
 133      /*  compute -1.0/(x+r) accurately */
 134             double a,t;
 135             z  = w;
 136             SET_LOW_WORD(z,0);
 137             v  = r-(z - x);     /* z+v = r+x */
 138             t = a  = -1.0/w;    /* a = -1.0/w */
 139             SET_LOW_WORD(t,0);
 140             s  = 1.0+t*z;
 141             return t+a*(s+t*v);
 142         }
 143 }
 144
 145 #endif /* defined(_DOUBLE_IS_32BITS) */