1 /* From: @(#)k_tan.c 1.5 04/04/22 SMI */
2
3 /*
4 * ====================================================
5 * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
6 * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
7 *
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 */
13
14
15 /*
16 * ld80 version of k_tan.c. See ../src/k_tan.c for most comments.
17 */
18
19
20
21 /*
22 * Domain [-0.67434, 0.67434], range ~[-2.25e-22, 1.921e-22]
23 * |tan(x)/x - t(x)| < 2**-71.9
24 *
25 * See k_cosl.c for more details about the polynomial.
26 */
27 #if defined(__amd64__) || defined(__i386__)
28 /* Long double constants are slow on these arches, and broken on i386. */
29 static const volatile double
30 T3hi = 0.33333333333333331, /* 0x15555555555555.0p-54 */
31 T3lo = 1.8350121769317163e-17, /* 0x15280000000000.0p-108 */
32 T5hi = 0.13333333333333336, /* 0x11111111111112.0p-55 */
33 T5lo = 1.3051083651294260e-17, /* 0x1e180000000000.0p-109 */
34 T7hi = 0.053968253968250494, /* 0x1ba1ba1ba1b827.0p-57 */
35 T7lo = 3.1509625637859973e-18, /* 0x1d100000000000.0p-111 */
36 pio4_hi = 0.78539816339744828, /* 0x1921fb54442d18.0p-53 */
37 pio4_lo = 3.0628711372715500e-17, /* 0x11a80000000000.0p-107 */
38 pio4lo_hi = -1.2541394031670831e-20, /* -0x1d9cceba3f91f2.0p-119 */
39 pio4lo_lo = 6.1493048227390915e-37; /* 0x1a280000000000.0p-173 */
40 #define T3 ((long double)T3hi + (long double)T3lo)
41 #define T5 ((long double)T5hi + (long double)T5lo)
42 #define T7 ((long double)T7hi + (long double)T7lo)
43 #define pio4 ((long double)pio4_hi + (long double)pio4_lo)
44 #define pio4lo ((long double)pio4lo_hi + (long double)pio4lo_lo)
45 #else
46 static const long double
47 T3 = 0.333333333333333333180L, /* 0xaaaaaaaaaaaaaaa5.0p-65 */
48 T5 = 0.133333333333333372290L, /* 0x88888888888893c3.0p-66 */
49 T7 = 0.0539682539682504975744L, /* 0xdd0dd0dd0dc13ba2.0p-68 */
50 pio4 = 0.785398163397448309628L, /* 0xc90fdaa22168c235.0p-64 */
51 pio4lo = -1.25413940316708300586e-20L; /* -0xece675d1fc8f8cbb.0p-130 */
52 #endif
53
54 static const double
55 T9 = 0.021869488536312216, /* 0x1664f4882cc1c2.0p-58 */
56 T11 = 0.0088632355256619590, /* 0x1226e355c17612.0p-59 */
57 T13 = 0.0035921281113786528, /* 0x1d6d3d185d7ff8.0p-61 */
58 T15 = 0.0014558334756312418, /* 0x17da354aa3f96b.0p-62 */
59 T17 = 0.00059003538700862256, /* 0x13559358685b83.0p-63 */
60 T19 = 0.00023907843576635544, /* 0x1f56242026b5be.0p-65 */
61 T21 = 0.000097154625656538905, /* 0x1977efc26806f4.0p-66 */
62 T23 = 0.000038440165747303162, /* 0x14275a09b3ceac.0p-67 */
63 T25 = 0.000018082171885432524, /* 0x12f5e563e5487e.0p-68 */
64 T27 = 0.0000024196006108814377, /* 0x144c0d80cc6896.0p-71 */
65 T29 = 0.0000078293456938132840, /* 0x106b59141a6cb3.0p-69 */
66 T31 = -0.0000032609076735050182, /* -0x1b5abef3ba4b59.0p-71 */
67 T33 = 0.0000023261313142559411; /* 0x13835436c0c87f.0p-71 */
68
69 long double
__kernel_tanl(long double x,long double y,int iy)70 __kernel_tanl(long double x, long double y, int iy) {
71 long double z, r, v, w, s;
72 long double osign;
73 int i;
74
75 iy = (iy == 1 ? -1 : 1); /* XXX recover original interface */
76 osign = (x >= 0 ? 1.0l : -1.0l); /* XXX slow, probably wrong for -0 */
77 if (fabsl(x) >= 0.67434l) {
78 if (x < 0) {
79 x = -x;
80 y = -y;
81 }
82 z = pio4 - x;
83 w = pio4lo - y;
84 x = z + w;
85 y = 0.0l;
86 i = 1;
87 } else
88 i = 0;
89 z = x * x;
90 w = z * z;
91 r = T5 + w * ((long double)T9 + w * ((long double)T13 + w * ((long double)T17 + w * ((long double)T21 +
92 w * ((long double)T25 + w * ((long double)T29 + w * (long double)T33))))));
93 v = z * ((long double)T7 + w * ((long double)T11 + w * ((long double)T15 + w * ((long double)T19 + w * ((long double)T23 +
94 w * ((long double)T27 + w * (long double)T31))))));
95 s = z * x;
96 r = y + z * (s * (r + v) + y);
97 r += T3 * s;
98 w = x + r;
99 if (i == 1) {
100 v = (long double) iy;
101 return osign *
102 (v - 2.0L * (x - (w * w / (w + v) - r)));
103 }
104 if (iy == 1)
105 return w;
106 else {
107 /*
108 * if allow error up to 2 ulp, simply return
109 * -1.0 / (x+r) here
110 */
111 /* compute -1.0 / (x+r) accurately */
112 long double a, t;
113 z = w;
114 z = z + 0x1p32L - 0x1p32L;
115 v = r - (z - x); /* z+v = r+x */
116 t = a = -1.0L / w; /* a = -1.0/w */
117 t = t + 0x1p32L - 0x1p32L;
118 s = 1.0L + t * z;
119 return t + a * (s + t * v);
120 }
121 }
122
