[388] | 1 | /* origin: FreeBSD /usr/src/lib/msun/src/k_tan.c */
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| 2 | /*
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| 3 | * ====================================================
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| 4 | * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
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| 5 | *
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| 6 | * Permission to use, copy, modify, and distribute this
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| 7 | * software is freely granted, provided that this notice
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| 8 | * is preserved.
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| 9 | * ====================================================
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| 10 | */
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| 11 | /* __tan( x, y, k )
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| 12 | * kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854
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| 13 | * Input x is assumed to be bounded by ~pi/4 in magnitude.
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| 14 | * Input y is the tail of x.
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| 15 | * Input odd indicates whether tan (if odd = 0) or -1/tan (if odd = 1) is returned.
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| 16 | *
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| 17 | * Algorithm
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| 18 | * 1. Since tan(-x) = -tan(x), we need only to consider positive x.
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| 19 | * 2. Callers must return tan(-0) = -0 without calling here since our
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| 20 | * odd polynomial is not evaluated in a way that preserves -0.
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| 21 | * Callers may do the optimization tan(x) ~ x for tiny x.
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| 22 | * 3. tan(x) is approximated by a odd polynomial of degree 27 on
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| 23 | * [0,0.67434]
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| 24 | * 3 27
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| 25 | * tan(x) ~ x + T1*x + ... + T13*x
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| 26 | * where
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| 27 | *
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| 28 | * |tan(x) 2 4 26 | -59.2
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| 29 | * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
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| 30 | * | x |
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| 31 | *
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| 32 | * Note: tan(x+y) = tan(x) + tan'(x)*y
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| 33 | * ~ tan(x) + (1+x*x)*y
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| 34 | * Therefore, for better accuracy in computing tan(x+y), let
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| 35 | * 3 2 2 2 2
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| 36 | * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
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| 37 | * then
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| 38 | * 3 2
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| 39 | * tan(x+y) = x + (T1*x + (x *(r+y)+y))
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| 40 | *
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| 41 | * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
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| 42 | * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
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| 43 | * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
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| 44 | */
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| 45 |
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| 46 | #include "libm.h"
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| 47 |
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| 48 | static const double T[] = {
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| 49 | 3.33333333333334091986e-01, /* 3FD55555, 55555563 */
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| 50 | 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
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| 51 | 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
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| 52 | 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
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| 53 | 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
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| 54 | 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
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| 55 | 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
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| 56 | 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
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| 57 | 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
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| 58 | 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
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| 59 | 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
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| 60 | -1.85586374855275456654e-05, /* BEF375CB, DB605373 */
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| 61 | 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
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| 62 | },
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| 63 | pio4 = 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
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| 64 | pio4lo = 3.06161699786838301793e-17; /* 3C81A626, 33145C07 */
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| 65 |
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| 66 | double __tan(double x, double y, int odd)
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| 67 | {
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| 68 | double_t z, r, v, w, s, a;
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| 69 | double w0, a0;
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| 70 | uint32_t hx;
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| 71 | int big, sign;
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| 72 |
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| 73 | GET_HIGH_WORD(hx,x);
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| 74 | big = (hx&0x7fffffff) >= 0x3FE59428; /* |x| >= 0.6744 */
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| 75 | if (big) {
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| 76 | sign = hx>>31;
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| 77 | if (sign) {
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| 78 | x = -x;
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| 79 | y = -y;
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| 80 | }
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| 81 | x = (pio4 - x) + (pio4lo - y);
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| 82 | y = 0.0;
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| 83 | }
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| 84 | z = x * x;
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| 85 | w = z * z;
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| 86 | /*
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| 87 | * Break x^5*(T[1]+x^2*T[2]+...) into
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| 88 | * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
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| 89 | * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
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| 90 | */
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| 91 | r = T[1] + w*(T[3] + w*(T[5] + w*(T[7] + w*(T[9] + w*T[11]))));
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| 92 | v = z*(T[2] + w*(T[4] + w*(T[6] + w*(T[8] + w*(T[10] + w*T[12])))));
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| 93 | s = z * x;
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| 94 | r = y + z*(s*(r + v) + y) + s*T[0];
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| 95 | w = x + r;
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| 96 | if (big) {
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| 97 | s = 1 - 2*odd;
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| 98 | v = s - 2.0 * (x + (r - w*w/(w + s)));
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| 99 | return sign ? -v : v;
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| 100 | }
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| 101 | if (!odd)
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| 102 | return w;
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| 103 | /* -1.0/(x+r) has up to 2ulp error, so compute it accurately */
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| 104 | w0 = w;
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| 105 | SET_LOW_WORD(w0, 0);
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| 106 | v = r - (w0 - x); /* w0+v = r+x */
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| 107 | a0 = a = -1.0 / w;
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| 108 | SET_LOW_WORD(a0, 0);
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| 109 | return a0 + a*(1.0 + a0*w0 + a0*v);
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| 110 | }
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