newlib/libm/ld80/k_tanl.c

   1 /*
   2  * ====================================================
   3  * Copyright 2004 Sun Microsystems, Inc.  All Rights Reserved.
   4  * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
   5  *
   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  * ld80 version of k_tan.c.  See ../src/k_tan.c for most comments.
  14  */
  15
  16 #include "math.h"
  17 #include "../ld/math_private.h"
  18
  19 /*
  20  * Domain [-0.67434, 0.67434], range ~[-2.25e-22, 1.921e-22]
  21  * |tan(x)/x - t(x)| < 2**-71.9
  22  *
  23  * See k_cosl.c for more details about the polynomial.
  24  */
  25 #if defined(__amd64__) || defined(__i386__)
  26 /* Long double constants are slow on these arches, and broken on i386. */
  27 static const volatile double
  28 T3hi =  0.33333333333333331,            /*  0x15555555555555.0p-54 */
  29 T3lo =  1.8350121769317163e-17,         /*  0x15280000000000.0p-108 */
  30 T5hi =  0.13333333333333336,            /*  0x11111111111112.0p-55 */
  31 T5lo =  1.3051083651294260e-17,         /*  0x1e180000000000.0p-109 */
  32 T7hi =  0.053968253968250494,           /*  0x1ba1ba1ba1b827.0p-57 */
  33 T7lo =  3.1509625637859973e-18,         /*  0x1d100000000000.0p-111 */
  34 pio4_hi =  0.78539816339744828,         /*  0x1921fb54442d18.0p-53 */
  35 pio4_lo =  3.0628711372715500e-17,      /*  0x11a80000000000.0p-107 */
  36 pio4lo_hi = -1.2541394031670831e-20,    /* -0x1d9cceba3f91f2.0p-119 */
  37 pio4lo_lo =  6.1493048227390915e-37;    /*  0x1a280000000000.0p-173 */
  38 #define T3      ((long double)T3hi + T3lo)
  39 #define T5      ((long double)T5hi + T5lo)
  40 #define T7      ((long double)T7hi + T7lo)
  41 #define pio4    ((long double)pio4_hi + pio4_lo)
  42 #define pio4lo  ((long double)pio4lo_hi + pio4lo_lo)
  43 #else
  44 static const long double
  45 T3 =   0.333333333333333333180L,        /*  0xaaaaaaaaaaaaaaa5.0p-65 */
  46 T5 =   0.133333333333333372290L,        /*  0x88888888888893c3.0p-66 */
  47 T7 =   0.0539682539682504975744L,       /*  0xdd0dd0dd0dc13ba2.0p-68 */
  48 pio4 = 0.785398163397448309628L,        /*  0xc90fdaa22168c235.0p-64 */
  49 pio4lo = -1.25413940316708300586e-20L;  /* -0xece675d1fc8f8cbb.0p-130 */
  50 #endif
  51
  52 static const double
  53 T9  =  0.021869488536312216,            /*  0x1664f4882cc1c2.0p-58 */
  54 T11 =  0.0088632355256619590,           /*  0x1226e355c17612.0p-59 */
  55 T13 =  0.0035921281113786528,           /*  0x1d6d3d185d7ff8.0p-61 */
  56 T15 =  0.0014558334756312418,           /*  0x17da354aa3f96b.0p-62 */
  57 T17 =  0.00059003538700862256,          /*  0x13559358685b83.0p-63 */
  58 T19 =  0.00023907843576635544,          /*  0x1f56242026b5be.0p-65 */
  59 T21 =  0.000097154625656538905,         /*  0x1977efc26806f4.0p-66 */
  60 T23 =  0.000038440165747303162,         /*  0x14275a09b3ceac.0p-67 */
  61 T25 =  0.000018082171885432524,         /*  0x12f5e563e5487e.0p-68 */
  62 T27 =  0.0000024196006108814377,        /*  0x144c0d80cc6896.0p-71 */
  63 T29 =  0.0000078293456938132840,        /*  0x106b59141a6cb3.0p-69 */
  64 T31 = -0.0000032609076735050182,        /* -0x1b5abef3ba4b59.0p-71 */
  65 T33 =  0.0000023261313142559411;        /*  0x13835436c0c87f.0p-71 */
  66
  67 long double
  68 __kernel_tanl(long double x, long double y, int iy) {
  69         long double z, r, v, w, s;
  70         long double osign;
  71         int i;
  72
  73         iy = (iy == 1 ? -1 : 1);        /* XXX recover original interface */
  74         osign = (x >= 0 ? 1.0 : -1.0);  /* XXX slow, probably wrong for -0 */
  75         if (fabsl(x) >= 0.67434) {
  76                 if (x < 0) {
  77                         x = -x;
  78                         y = -y;
  79                 }
  80                 z = pio4 - x;
  81                 w = pio4lo - y;
  82                 x = z + w;
  83                 y = 0.0;
  84                 i = 1;
  85         } else
  86                 i = 0;
  87         z = x * x;
  88         w = z * z;
  89         r = T5 + w * (T9 + w * (T13 + w * (T17 + w * (T21 +
  90             w * (T25 + w * (T29 + w * T33))))));
  91         v = z * (T7 + w * (T11 + w * (T15 + w * (T19 + w * (T23 +
  92             w * (T27 + w * T31))))));
  93         s = z * x;
  94         r = y + z * (s * (r + v) + y);
  95         r += T3 * s;
  96         w = x + r;
  97         if (i == 1) {
  98                 v = (long double) iy;
  99                 return osign *
 100                         (v - 2.0 * (x - (w * w / (w + v) - r)));
 101         }
 102         if (iy == 1)
 103                 return w;
 104         else {
 105                 /*
 106                  * if allow error up to 2 ulp, simply return
 107                  * -1.0 / (x+r) here
 108                  */
 109                 /* compute -1.0 / (x+r) accurately */
 110                 long double a, t;
 111                 z = w;
 112                 z = z + 0x1p32 - 0x1p32;
 113                 v = r - (z - x);        /* z+v = r+x */
 114                 t = a = -1.0 / w;       /* a = -1.0/w */
 115                 t = t + 0x1p32 - 0x1p32;
 116                 s = 1.0 + t * z;
 117                 return t + a * (s + t * v);
 118         }
 119 }