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 #include <sys/cdefs.h> 15 __FBSDID("$FreeBSD$"); 16 17 /* 18 * ld80 version of k_tan.c. See ../src/k_tan.c for most comments. 19 */ 20 21 #include "math.h" 22 #include "math_private.h" 23 24 /* 25 * Domain [-0.67434, 0.67434], range ~[-2.25e-22, 1.921e-22] 26 * |tan(x)/x - t(x)| < 2**-71.9 27 * 28 * See k_cosl.c for more details about the polynomial. 29 */ 30 #if defined(__amd64__) || defined(__i386__) 31 /* Long double constants are slow on these arches, and broken on i386. */ 32 static const volatile double 33 T3hi = 0.33333333333333331, /* 0x15555555555555.0p-54 */ 34 T3lo = 1.8350121769317163e-17, /* 0x15280000000000.0p-108 */ 35 T5hi = 0.13333333333333336, /* 0x11111111111112.0p-55 */ 36 T5lo = 1.3051083651294260e-17, /* 0x1e180000000000.0p-109 */ 37 T7hi = 0.053968253968250494, /* 0x1ba1ba1ba1b827.0p-57 */ 38 T7lo = 3.1509625637859973e-18, /* 0x1d100000000000.0p-111 */ 39 pio4_hi = 0.78539816339744828, /* 0x1921fb54442d18.0p-53 */ 40 pio4_lo = 3.0628711372715500e-17, /* 0x11a80000000000.0p-107 */ 41 pio4lo_hi = -1.2541394031670831e-20, /* -0x1d9cceba3f91f2.0p-119 */ 42 pio4lo_lo = 6.1493048227390915e-37; /* 0x1a280000000000.0p-173 */ 43 #define T3 ((long double)T3hi + T3lo) 44 #define T5 ((long double)T5hi + T5lo) 45 #define T7 ((long double)T7hi + T7lo) 46 #define pio4 ((long double)pio4_hi + pio4_lo) 47 #define pio4lo ((long double)pio4lo_hi + pio4lo_lo) 48 #else 49 static const long double 50 T3 = 0.333333333333333333180L, /* 0xaaaaaaaaaaaaaaa5.0p-65 */ 51 T5 = 0.133333333333333372290L, /* 0x88888888888893c3.0p-66 */ 52 T7 = 0.0539682539682504975744L, /* 0xdd0dd0dd0dc13ba2.0p-68 */ 53 pio4 = 0.785398163397448309628L, /* 0xc90fdaa22168c235.0p-64 */ 54 pio4lo = -1.25413940316708300586e-20L; /* -0xece675d1fc8f8cbb.0p-130 */ 55 #endif 56 57 static const double 58 T9 = 0.021869488536312216, /* 0x1664f4882cc1c2.0p-58 */ 59 T11 = 0.0088632355256619590, /* 0x1226e355c17612.0p-59 */ 60 T13 = 0.0035921281113786528, /* 0x1d6d3d185d7ff8.0p-61 */ 61 T15 = 0.0014558334756312418, /* 0x17da354aa3f96b.0p-62 */ 62 T17 = 0.00059003538700862256, /* 0x13559358685b83.0p-63 */ 63 T19 = 0.00023907843576635544, /* 0x1f56242026b5be.0p-65 */ 64 T21 = 0.000097154625656538905, /* 0x1977efc26806f4.0p-66 */ 65 T23 = 0.000038440165747303162, /* 0x14275a09b3ceac.0p-67 */ 66 T25 = 0.000018082171885432524, /* 0x12f5e563e5487e.0p-68 */ 67 T27 = 0.0000024196006108814377, /* 0x144c0d80cc6896.0p-71 */ 68 T29 = 0.0000078293456938132840, /* 0x106b59141a6cb3.0p-69 */ 69 T31 = -0.0000032609076735050182, /* -0x1b5abef3ba4b59.0p-71 */ 70 T33 = 0.0000023261313142559411; /* 0x13835436c0c87f.0p-71 */ 71 72 long double 73 __kernel_tanl(long double x, long double y, int iy) { 74 long double z, r, v, w, s; 75 long double osign; 76 int i; 77 78 iy = (iy == 1 ? -1 : 1); /* XXX recover original interface */ 79 osign = (x >= 0 ? 1.0 : -1.0); /* XXX slow, probably wrong for -0 */ 80 if (fabsl(x) >= 0.67434) { 81 if (x < 0) { 82 x = -x; 83 y = -y; 84 } 85 z = pio4 - x; 86 w = pio4lo - y; 87 x = z + w; 88 y = 0.0; 89 i = 1; 90 } else 91 i = 0; 92 z = x * x; 93 w = z * z; 94 r = T5 + w * (T9 + w * (T13 + w * (T17 + w * (T21 + 95 w * (T25 + w * (T29 + w * T33)))))); 96 v = z * (T7 + w * (T11 + w * (T15 + w * (T19 + w * (T23 + 97 w * (T27 + w * T31)))))); 98 s = z * x; 99 r = y + z * (s * (r + v) + y); 100 r += T3 * s; 101 w = x + r; 102 if (i == 1) { 103 v = (long double) iy; 104 return osign * 105 (v - 2.0 * (x - (w * w / (w + v) - r))); 106 } 107 if (iy == 1) 108 return w; 109 else { 110 /* 111 * if allow error up to 2 ulp, simply return 112 * -1.0 / (x+r) here 113 */ 114 /* compute -1.0 / (x+r) accurately */ 115 long double a, t; 116 z = w; 117 z = z + 0x1p32 - 0x1p32; 118 v = r - (z - x); /* z+v = r+x */ 119 t = a = -1.0 / w; /* a = -1.0/w */ 120 t = t + 0x1p32 - 0x1p32; 121 s = 1.0 + t * z; 122 return t + a * (s + t * v); 123 } 124 } 125