js/src/fdlibm/k_tan.c

   1 /* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*-
   2  *
   3  * ***** BEGIN LICENSE BLOCK *****
   4  * Version: MPL 1.1/GPL 2.0/LGPL 2.1
   5  *
   6  * The contents of this file are subject to the Mozilla Public License Version
   7  * 1.1 (the "License"); you may not use this file except in compliance with
   8  * the License. You may obtain a copy of the License at
   9  * http://www.mozilla.org/MPL/
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  12  * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
  13  * for the specific language governing rights and limitations under the
  14  * License.
  15  *
  16  * The Original Code is Mozilla Communicator client code, released
  17  * March 31, 1998.
  18  *
  19  * The Initial Developer of the Original Code is
  20  * Sun Microsystems, Inc.
  21  * Portions created by the Initial Developer are Copyright (C) 1998
  22  * the Initial Developer. All Rights Reserved.
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  26  * Alternatively, the contents of this file may be used under the terms of
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  38  * ***** END LICENSE BLOCK ***** */
  39
  40 /* @(#)k_tan.c 1.3 95/01/18 */
  41 /*
  42  * ====================================================
  43  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  44  *
  45  * Developed at SunSoft, a Sun Microsystems, Inc. business.
  46  * Permission to use, copy, modify, and distribute this
  47  * software is freely granted, provided that this notice
  48  * is preserved.
  49  * ====================================================
  50  */
  51
  52 /* __kernel_tan( x, y, k )
  53  * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
  54  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  55  * Input y is the tail of x.
  56  * Input k indicates whether tan (if k=1) or
  57  * -1/tan (if k= -1) is returned.
  58  *
  59  * Algorithm
  60  *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
  61  *      2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
  62  *      3. tan(x) is approximated by a odd polynomial of degree 27 on
  63  *         [0,0.67434]
  64  *                               3             27
  65  *              tan(x) ~ x + T1*x + ... + T13*x
  66  *         where
  67  *
  68  *              |tan(x)         2     4            26   |     -59.2
  69  *              |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
  70  *              |  x                                    |
  71  *
  72  *         Note: tan(x+y) = tan(x) + tan'(x)*y
  73  *                        ~ tan(x) + (1+x*x)*y
  74  *         Therefore, for better accuracy in computing tan(x+y), let
  75  *                   3      2      2       2       2
  76  *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
  77  *         then
  78  *                                  3    2
  79  *              tan(x+y) = x + (T1*x + (x *(r+y)+y))
  80  *
  81  *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
  82  *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
  83  *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
  84  */
  85
  86 #include "fdlibm.h"
  87 #ifdef __STDC__
  88 static const double
  89 #else
  90 static double
  91 #endif
  92 one   =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  93 pio4  =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
  94 pio4lo=  3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
  95 T[] =  {
  96   3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
  97   1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
  98   5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
  99   2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
 100   8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
 101   3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
 102   1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
 103   5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
 104   2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
 105   7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
 106   7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
 107  -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
 108   2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
 109 };
 110
 111 #ifdef __STDC__
 112         double __kernel_tan(double x, double y, int iy)
 113 #else
 114         double __kernel_tan(x, y, iy)
 115         double x,y; int iy;
 116 #endif
 117 {
 118         fd_twoints u;
 119         double z,r,v,w,s;
 120         int ix,hx;
 121         u.d = x;
 122         hx = __HI(u);   /* high word of x */
 123         ix = hx&0x7fffffff;     /* high word of |x| */
 124         if(ix<0x3e300000)                       /* x < 2**-28 */
 125             {if((int)x==0) {                    /* generate inexact */
 126                 u.d =x;
 127                 if(((ix|__LO(u))|(iy+1))==0) return one/fd_fabs(x);
 128                 else return (iy==1)? x: -one/x;
 129             }
 130             }
 131         if(ix>=0x3FE59428) {                    /* |x|>=0.6744 */
 132             if(hx<0) {x = -x; y = -y;}
 133             z = pio4-x;
 134             w = pio4lo-y;
 135             x = z+w; y = 0.0;
 136         }
 137         z       =  x*x;
 138         w       =  z*z;
 139     /* Break x^5*(T[1]+x^2*T[2]+...) into
 140      *    x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
 141      *    x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
 142      */
 143         r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
 144         v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
 145         s = z*x;
 146         r = y + z*(s*(r+v)+y);
 147         r += T[0]*s;
 148         w = x+r;
 149         if(ix>=0x3FE59428) {
 150             v = (double)iy;
 151             return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
 152         }
 153         if(iy==1) return w;
 154         else {          /* if allow error up to 2 ulp,
 155                            simply return -1.0/(x+r) here */
 156      /*  compute -1.0/(x+r) accurately */
 157             double a,t;
 158             z  = w;
 159             u.d = z;
 160             __LO(u) = 0;
 161             z = u.d;
 162             v  = r-(z - x);     /* z+v = r+x */
 163             t = a  = -1.0/w;    /* a = -1.0/w */
 164             u.d = t;
 165             __LO(u) = 0;
 166             t = u.d;
 167             s  = 1.0+t*z;
 168             return t+a*(s+t*v);
 169         }
 170 }