libquadmath/math/tanq.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 /*
  13   Long double expansions are
  14   Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
  15   and are incorporated herein by permission of the author.  The author
  16   reserves the right to distribute this material elsewhere under different
  17   copying permissions.  These modifications are distributed here under
  18   the following terms:
  19
  20     This library is free software; you can redistribute it and/or
  21     modify it under the terms of the GNU Lesser General Public
  22     License as published by the Free Software Foundation; either
  23     version 2.1 of the License, or (at your option) any later version.
  24
  25     This library is distributed in the hope that it will be useful,
  26     but WITHOUT ANY WARRANTY; without even the implied warranty of
  27     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  28     Lesser General Public License for more details.
  29
  30     You should have received a copy of the GNU Lesser General Public
  31     License along with this library; if not, write to the Free Software
  32     Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307  USA */
  33
  34 /* __quadmath_kernel_tanq( x, y, k )
  35  * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
  36  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  37  * Input y is the tail of x.
  38  * Input k indicates whether tan (if k=1) or
  39  * -1/tan (if k= -1) is returned.
  40  *
  41  * Algorithm
  42  *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
  43  *      2. if x < 2^-57, return x with inexact if x!=0.
  44  *      3. tan(x) is approximated by a rational form x + x^3 / 3 + x^5 R(x^2)
  45  *          on [0,0.67433].
  46  *
  47  *         Note: tan(x+y) = tan(x) + tan'(x)*y
  48  *                        ~ tan(x) + (1+x*x)*y
  49  *         Therefore, for better accuracy in computing tan(x+y), let
  50  *              r = x^3 * R(x^2)
  51  *         then
  52  *              tan(x+y) = x + (x^3 / 3 + (x^2 *(r+y)+y))
  53  *
  54  *      4. For x in [0.67433,pi/4],  let y = pi/4 - x, then
  55  *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
  56  *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
  57  */
  58
  59 #include "quadmath-imp.h"
  60
  61
  62
  63 static const __float128
  64   one = 1.0Q,
  65   pio4hi = 7.8539816339744830961566084581987569936977E-1Q,
  66   pio4lo = 2.1679525325309452561992610065108379921906E-35Q,
  67
  68   /* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2)
  69      0 <= x <= 0.6743316650390625
  70      Peak relative error 8.0e-36  */
  71  TH =  3.333333333333333333333333333333333333333E-1Q,
  72  T0 = -1.813014711743583437742363284336855889393E7Q,
  73  T1 =  1.320767960008972224312740075083259247618E6Q,
  74  T2 = -2.626775478255838182468651821863299023956E4Q,
  75  T3 =  1.764573356488504935415411383687150199315E2Q,
  76  T4 = -3.333267763822178690794678978979803526092E-1Q,
  77
  78  U0 = -1.359761033807687578306772463253710042010E8Q,
  79  U1 =  6.494370630656893175666729313065113194784E7Q,
  80  U2 = -4.180787672237927475505536849168729386782E6Q,
  81  U3 =  8.031643765106170040139966622980914621521E4Q,
  82  U4 = -5.323131271912475695157127875560667378597E2Q;
  83   /* 1.000000000000000000000000000000000000000E0 */
  84
  85
  86 static __float128
  87 __quadmath_kernel_tanq (__float128 x, __float128 y, int iy)
  88 {
  89   __float128 z, r, v, w, s;
  90   int32_t ix, sign = 1;
  91   ieee854_float128 u, u1;
  92
  93   u.value = x;
  94   ix = u.words32.w0 & 0x7fffffff;
  95   if (ix < 0x3fc60000)          /* x < 2**-57 */
  96     {
  97       if ((int) x == 0)
  98         {                       /* generate inexact */
  99           if ((ix | u.words32.w1 | u.words32.w2 | u.words32.w3
 100                | (iy + 1)) == 0)
 101             return one / fabsq (x);
 102           else
 103             return (iy == 1) ? x : -one / x;
 104         }
 105     }
 106   if (ix >= 0x3ffe5942) /* |x| >= 0.6743316650390625 */
 107     {
 108       if ((u.words32.w0 & 0x80000000) != 0)
 109         {
 110           x = -x;
 111           y = -y;
 112           sign = -1;
 113         }
 114       else
 115         sign = 1;
 116       z = pio4hi - x;
 117       w = pio4lo - y;
 118       x = z + w;
 119       y = 0.0;
 120     }
 121   z = x * x;
 122   r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
 123   v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
 124   r = r / v;
 125
 126   s = z * x;
 127   r = y + z * (s * r + y);
 128   r += TH * s;
 129   w = x + r;
 130   if (ix >= 0x3ffe5942)
 131     {
 132       v = (__float128) iy;
 133       w = (v - 2.0Q * (x - (w * w / (w + v) - r)));
 134       if (sign < 0)
 135         w = -w;
 136       return w;
 137     }
 138   if (iy == 1)
 139     return w;
 140   else
 141     {                           /* if allow error up to 2 ulp,
 142                                    simply return -1.0/(x+r) here */
 143       /*  compute -1.0/(x+r) accurately */
 144       u1.value = w;
 145       u1.words32.w2 = 0;
 146       u1.words32.w3 = 0;
 147       v = r - (u1.value - x);           /* u1+v = r+x */
 148       z = -1.0 / w;
 149       u.value = z;
 150       u.words32.w2 = 0;
 151       u.words32.w3 = 0;
 152       s = 1.0 + u.value * u1.value;
 153       return u.value + z * (s + u.value * v);
 154     }
 155 }
 156
 157
 158
 159 \f
 160
 161
 162
 163 /* tanq.c -- __float128 version of s_tan.c.
 164  * Conversion to IEEE quad long double by Jakub Jelinek, jj@ultra.linux.cz.
 165  */
 166
 167 /* @(#)s_tan.c 5.1 93/09/24 */
 168 /*
 169  * ====================================================
 170  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 171  *
 172  * Developed at SunPro, a Sun Microsystems, Inc. business.
 173  * Permission to use, copy, modify, and distribute this
 174  * software is freely granted, provided that this notice
 175  * is preserved.
 176  * ====================================================
 177  */
 178
 179 /* tanl(x)
 180  * Return tangent function of x.
 181  *
 182  * kernel function:
 183  *      __quadmath_kernel_tanq  ... tangent function on [-pi/4,pi/4]
 184  *      __quadmath_rem_pio2q    ... argument reduction routine
 185  *
 186  * Method.
 187  *      Let S,C and T denote the sin, cos and tan respectively on
 188  *      [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
 189  *      in [-pi/4 , +pi/4], and let n = k mod 4.
 190  *      We have
 191  *
 192  *          n        sin(x)      cos(x)        tan(x)
 193  *     ----------------------------------------------------------
 194  *          0          S           C             T
 195  *          1          C          -S            -1/T
 196  *          2         -S          -C             T
 197  *          3         -C           S            -1/T
 198  *     ----------------------------------------------------------
 199  *
 200  * Special cases:
 201  *      Let trig be any of sin, cos, or tan.
 202  *      trig(+-INF)  is NaN, with signals;
 203  *      trig(NaN)    is that NaN;
 204  *
 205  * Accuracy:
 206  *      TRIG(x) returns trig(x) nearly rounded
 207  */
 208
 209
 210 __float128
 211 tanq (__float128 x)
 212 {
 213         __float128 y[2],z=0.0Q;
 214         int64_t n, ix;
 215
 216     /* High word of x. */
 217         GET_FLT128_MSW64(ix,x);
 218
 219     /* |x| ~< pi/4 */
 220         ix &= 0x7fffffffffffffffLL;
 221         if(ix <= 0x3ffe921fb54442d1LL) return __quadmath_kernel_tanq(x,z,1);
 222
 223     /* tanl(Inf or NaN) is NaN */
 224         else if (ix>=0x7fff000000000000LL) {
 225             if (ix == 0x7fff000000000000LL) {
 226                 GET_FLT128_LSW64(n,x);
 227             }
 228             return x-x;         /* NaN */
 229         }
 230
 231     /* argument reduction needed */
 232         else {
 233             n = __quadmath_rem_pio2q(x,y);
 234                                         /*   1 -- n even, -1 -- n odd */
 235             return __quadmath_kernel_tanq(y[0],y[1],1-((n&1)<<1));
 236         }
 237 }