sysdeps/ieee754/ldbl-128/k_tanl.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     <https://www.gnu.org/licenses/>.  */
  33
  34 /* __kernel_tanl( 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 <float.h>
  60 #include <math.h>
  61 #include <math_private.h>
  62 #include <math-underflow.h>
  63 #include <libc-diag.h>
  64
  65 static const _Float128
  66   one = 1,
  67   pio4hi = L(7.8539816339744830961566084581987569936977E-1),
  68   pio4lo = L(2.1679525325309452561992610065108379921906E-35),
  69
  70   /* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2)
  71      0 <= x <= 0.6743316650390625
  72      Peak relative error 8.0e-36  */
  73  TH =  L(3.333333333333333333333333333333333333333E-1),
  74  T0 = L(-1.813014711743583437742363284336855889393E7),
  75  T1 =  L(1.320767960008972224312740075083259247618E6),
  76  T2 = L(-2.626775478255838182468651821863299023956E4),
  77  T3 =  L(1.764573356488504935415411383687150199315E2),
  78  T4 = L(-3.333267763822178690794678978979803526092E-1),
  79
  80  U0 = L(-1.359761033807687578306772463253710042010E8),
  81  U1 =  L(6.494370630656893175666729313065113194784E7),
  82  U2 = L(-4.180787672237927475505536849168729386782E6),
  83  U3 =  L(8.031643765106170040139966622980914621521E4),
  84  U4 = L(-5.323131271912475695157127875560667378597E2);
  85   /* 1.000000000000000000000000000000000000000E0 */
  86
  87
  88 _Float128
  89 __kernel_tanl (_Float128 x, _Float128 y, int iy)
  90 {
  91   _Float128 z, r, v, w, s;
  92   int32_t ix, sign;
  93   ieee854_long_double_shape_type u, u1;
  94
  95   u.value = x;
  96   ix = u.parts32.w0 & 0x7fffffff;
  97   if (ix < 0x3fc60000)          /* x < 2**-57 */
  98     {
  99       if ((int) x == 0)
 100         {                       /* generate inexact */
 101           if ((ix | u.parts32.w1 | u.parts32.w2 | u.parts32.w3
 102                | (iy + 1)) == 0)
 103             return one / fabsl (x);
 104           else if (iy == 1)
 105             {
 106               math_check_force_underflow (x);
 107               return x;
 108             }
 109           else
 110             return -one / x;
 111         }
 112     }
 113   if (ix >= 0x3ffe5942) /* |x| >= 0.6743316650390625 */
 114     {
 115       if ((u.parts32.w0 & 0x80000000) != 0)
 116         {
 117           x = -x;
 118           y = -y;
 119           sign = -1;
 120         }
 121       else
 122         sign = 1;
 123       z = pio4hi - x;
 124       w = pio4lo - y;
 125       x = z + w;
 126       y = 0.0;
 127     }
 128   z = x * x;
 129   r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
 130   v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
 131   r = r / v;
 132
 133   s = z * x;
 134   r = y + z * (s * r + y);
 135   r += TH * s;
 136   w = x + r;
 137   if (ix >= 0x3ffe5942)
 138     {
 139       v = (_Float128) iy;
 140       w = (v - 2.0 * (x - (w * w / (w + v) - r)));
 141       /* SIGN is set for arguments that reach this code, but not
 142          otherwise, resulting in warnings that it may be used
 143          uninitialized although in the cases where it is used it has
 144          always been set.  */
 145       DIAG_PUSH_NEEDS_COMMENT;
 146       DIAG_IGNORE_NEEDS_COMMENT (5, "-Wmaybe-uninitialized");
 147       if (sign < 0)
 148         w = -w;
 149       DIAG_POP_NEEDS_COMMENT;
 150       return w;
 151     }
 152   if (iy == 1)
 153     return w;
 154   else
 155     {                           /* if allow error up to 2 ulp,
 156                                    simply return -1.0/(x+r) here */
 157       /*  compute -1.0/(x+r) accurately */
 158       u1.value = w;
 159       u1.parts32.w2 = 0;
 160       u1.parts32.w3 = 0;
 161       v = r - (u1.value - x);           /* u1+v = r+x */
 162       z = -1.0 / w;
 163       u.value = z;
 164       u.parts32.w2 = 0;
 165       u.parts32.w3 = 0;
 166       s = 1.0 + u.value * u1.value;
 167       return u.value + z * (s + u.value * v);
 168     }
 169 }