sysdeps/libm-ieee754/k_tan.c

   1 /* @(#)k_tan.c 5.1 93/09/24 */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunPro, a Sun Microsystems, Inc. business.
   7  * Permission to use, copy, modify, and distribute this
   8  * software is freely granted, provided that this notice
   9  * is preserved.
  10  * ====================================================
  11  */
  12 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
  13    for performance improvement on pipelined processors.
  14 */
  15
  16 #if defined(LIBM_SCCS) && !defined(lint)
  17 static char rcsid[] = "$NetBSD: k_tan.c,v 1.8 1995/05/10 20:46:37 jtc Exp $";
  18 #endif
  19
  20 /* __kernel_tan( x, y, k )
  21  * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
  22  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  23  * Input y is the tail of x.
  24  * Input k indicates whether tan (if k=1) or
  25  * -1/tan (if k= -1) is returned.
  26  *
  27  * Algorithm
  28  *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
  29  *      2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
  30  *      3. tan(x) is approximated by a odd polynomial of degree 27 on
  31  *         [0,0.67434]
  32  *                               3             27
  33  *              tan(x) ~ x + T1*x + ... + T13*x
  34  *         where
  35  *
  36  *              |tan(x)         2     4            26   |     -59.2
  37  *              |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
  38  *              |  x                                    |
  39  *
  40  *         Note: tan(x+y) = tan(x) + tan'(x)*y
  41  *                        ~ tan(x) + (1+x*x)*y
  42  *         Therefore, for better accuracy in computing tan(x+y), let
  43  *                   3      2      2       2       2
  44  *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
  45  *         then
  46  *                                  3    2
  47  *              tan(x+y) = x + (T1*x + (x *(r+y)+y))
  48  *
  49  *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
  50  *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
  51  *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
  52  */
  53
  54 #include "math.h"
  55 #include "math_private.h"
  56 #ifdef __STDC__
  57 static const double
  58 #else
  59 static double
  60 #endif
  61 one   =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  62 pio4  =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
  63 pio4lo=  3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
  64 T[] =  {
  65   3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
  66   1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
  67   5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
  68   2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
  69   8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
  70   3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
  71   1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
  72   5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
  73   2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
  74   7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
  75   7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
  76  -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
  77   2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
  78 };
  79
  80 #ifdef __STDC__
  81         double __kernel_tan(double x, double y, int iy)
  82 #else
  83         double __kernel_tan(x, y, iy)
  84         double x,y; int iy;
  85 #endif
  86 {
  87         double z,r,v,w,s,r1,r2,r3,v1,v2,v3,w2,w4;
  88         int32_t ix,hx;
  89         GET_HIGH_WORD(hx,x);
  90         ix = hx&0x7fffffff;     /* high word of |x| */
  91         if(ix<0x3e300000)                       /* x < 2**-28 */
  92             {if((int)x==0) {                    /* generate inexact */
  93                 u_int32_t low;
  94                 GET_LOW_WORD(low,x);
  95                 if(((ix|low)|(iy+1))==0) return one/fabs(x);
  96                 else return (iy==1)? x: -one/x;
  97             }
  98             }
  99         if(ix>=0x3FE59428) {                    /* |x|>=0.6744 */
 100             if(hx<0) {x = -x; y = -y;}
 101             z = pio4-x;
 102             w = pio4lo-y;
 103             x = z+w; y = 0.0;
 104         }
 105         z       =  x*x;
 106         w       =  z*z;
 107     /* Break x^5*(T[1]+x^2*T[2]+...) into
 108      *    x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
 109      *    x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
 110      */
 111 #ifdef DO_NOT_USE_THIS
 112         r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
 113         v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
 114 #else
 115         v1 = T[10]+w*T[12]; w2=w*w;
 116         v2 = T[6]+w*T[8]; w4=w2*w2;
 117         v3 = T[2]+w*T[4]; v1=z*v1;
 118         r1 = T[9]+w*T[11]; v2=z*v2;
 119         r2 = T[5]+w*T[7]; v3=z*v3;
 120         r3 = T[1]+w*T[3];
 121         v  = v3 + w2*v2 + w4*v1;
 122         r = r3 + w2*r2 + w4*r1;
 123 #endif
 124         s = z*x;
 125         r = y + z*(s*(r+v)+y);
 126         r += T[0]*s;
 127         w = x+r;
 128         if(ix>=0x3FE59428) {
 129             v = (double)iy;
 130             return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
 131         }
 132         if(iy==1) return w;
 133         else {          /* if allow error up to 2 ulp,
 134                            simply return -1.0/(x+r) here */
 135      /*  compute -1.0/(x+r) accurately */
 136             double a,t;
 137             z  = w;
 138             SET_LOW_WORD(z,0);
 139             v  = r-(z - x);     /* z+v = r+x */
 140             t = a  = -1.0/w;    /* a = -1.0/w */
 141             SET_LOW_WORD(t,0);
 142             s  = 1.0+t*z;
 143             return t+a*(s+t*v);
 144         }
 145 }