libquadmath/math/tanq_kernel.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, see
  32     <http://www.gnu.org/licenses/>.  */
  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 static const __float128
  62   one = 1,
  63   pio4hi = 7.8539816339744830961566084581987569936977E-1Q,
  64   pio4lo = 2.1679525325309452561992610065108379921906E-35Q,
  65
  66   /* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2)
  67      0 <= x <= 0.6743316650390625
  68      Peak relative error 8.0e-36  */
  69  TH =  3.333333333333333333333333333333333333333E-1Q,
  70  T0 = -1.813014711743583437742363284336855889393E7Q,
  71  T1 =  1.320767960008972224312740075083259247618E6Q,
  72  T2 = -2.626775478255838182468651821863299023956E4Q,
  73  T3 =  1.764573356488504935415411383687150199315E2Q,
  74  T4 = -3.333267763822178690794678978979803526092E-1Q,
  75
  76  U0 = -1.359761033807687578306772463253710042010E8Q,
  77  U1 =  6.494370630656893175666729313065113194784E7Q,
  78  U2 = -4.180787672237927475505536849168729386782E6Q,
  79  U3 =  8.031643765106170040139966622980914621521E4Q,
  80  U4 = -5.323131271912475695157127875560667378597E2Q;
  81   /* 1.000000000000000000000000000000000000000E0 */
  82
  83
  84 __float128
  85 __quadmath_kernel_tanq (__float128 x, __float128 y, int iy)
  86 {
  87   __float128 z, r, v, w, s;
  88   int32_t ix, sign;
  89   ieee854_float128 u, u1;
  90
  91   u.value = x;
  92   ix = u.words32.w0 & 0x7fffffff;
  93   if (ix < 0x3fc60000)          /* x < 2**-57 */
  94     {
  95       if ((int) x == 0)
  96         {                       /* generate inexact */
  97           if ((ix | u.words32.w1 | u.words32.w2 | u.words32.w3
  98                | (iy + 1)) == 0)
  99             return one / fabsq (x);
 100           else if (iy == 1)
 101             {
 102               math_check_force_underflow (x);
 103               return x;
 104             }
 105           else
 106             return -one / x;
 107         }
 108     }
 109   if (ix >= 0x3ffe5942) /* |x| >= 0.6743316650390625 */
 110     {
 111       if ((u.words32.w0 & 0x80000000) != 0)
 112         {
 113           x = -x;
 114           y = -y;
 115           sign = -1;
 116         }
 117       else
 118         sign = 1;
 119       z = pio4hi - x;
 120       w = pio4lo - y;
 121       x = z + w;
 122       y = 0.0;
 123     }
 124   z = x * x;
 125   r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
 126   v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
 127   r = r / v;
 128
 129   s = z * x;
 130   r = y + z * (s * r + y);
 131   r += TH * s;
 132   w = x + r;
 133   if (ix >= 0x3ffe5942)
 134     {
 135       v = (__float128) iy;
 136       w = (v - 2.0 * (x - (w * w / (w + v) - r)));
 137       /* SIGN is set for arguments that reach this code, but not
 138          otherwise, resulting in warnings that it may be used
 139          uninitialized although in the cases where it is used it has
 140          always been set.  */
 141
 142
 143       if (sign < 0)
 144         w = -w;
 145
 146       return w;
 147     }
 148   if (iy == 1)
 149     return w;
 150   else
 151     {                           /* if allow error up to 2 ulp,
 152                                    simply return -1.0/(x+r) here */
 153       /*  compute -1.0/(x+r) accurately */
 154       u1.value = w;
 155       u1.words32.w2 = 0;
 156       u1.words32.w3 = 0;
 157       v = r - (u1.value - x);           /* u1+v = r+x */
 158       z = -1.0 / w;
 159       u.value = z;
 160       u.words32.w2 = 0;
 161       u.words32.w3 = 0;
 162       s = 1.0 + u.value * u1.value;
 163       return u.value + z * (s + u.value * v);
 164     }
 165 }