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// origin: FreeBSD /usr/src/lib/msun/src/k_tan.c */
//
// ====================================================
// Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
//
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
// kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854
// Input x is assumed to be bounded by ~pi/4 in magnitude.
// Input y is the tail of x.
// Input odd indicates whether tan (if odd = 0) or -1/tan (if odd = 1) is returned.
//
// Algorithm
// 1. Since tan(-x) = -tan(x), we need only to consider positive x.
// 2. Callers must return tan(-0) = -0 without calling here since our
// odd polynomial is not evaluated in a way that preserves -0.
// Callers may do the optimization tan(x) ~ x for tiny x.
// 3. tan(x) is approximated by a odd polynomial of degree 27 on
// [0,0.67434]
// 3 27
// tan(x) ~ x + T1*x + ... + T13*x
// where
//
// |tan(x) 2 4 26 | -59.2
// |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
// | x |
//
// Note: tan(x+y) = tan(x) + tan'(x)*y
// ~ tan(x) + (1+x*x)*y
// Therefore, for better accuracy in computing tan(x+y), let
// 3 2 2 2 2
// r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
// then
// 3 2
// tan(x+y) = x + (T1*x + (x *(r+y)+y))
//
// 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
// tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
// = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
static T: [f64; 13] = [
3.33333333333334091986e-01, /* 3FD55555, 55555563 */
1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
-1.85586374855275456654e-05, /* BEF375CB, DB605373 */
2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
];
const PIO4: f64 = 7.85398163397448278999e-01; /* 3FE921FB, 54442D18 */
const PIO4_LO: f64 = 3.06161699786838301793e-17; /* 3C81A626, 33145C07 */
#[inline]
#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
pub(crate) fn k_tan(mut x: f64, mut y: f64, odd: i32) -> f64 {
let hx = (f64::to_bits(x) >> 32) as u32;
let big = (hx & 0x7fffffff) >= 0x3FE59428; /* |x| >= 0.6744 */
if big {
let sign = hx >> 31;
if sign != 0 {
x = -x;
y = -y;
}
x = (PIO4 - x) + (PIO4_LO - y);
y = 0.0;
}
let z = x * x;
let w = z * z;
/*
* Break x^5*(T[1]+x^2*T[2]+...) into
* x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
* x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
*/
let r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + w * T[11]))));
let v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + w * T[12])))));
let s = z * x;
let r = y + z * (s * (r + v) + y) + s * T[0];
let w = x + r;
if big {
let sign = hx >> 31;
let s = 1.0 - 2.0 * odd as f64;
let v = s - 2.0 * (x + (r - w * w / (w + s)));
return if sign != 0 { -v } else { v };
}
if odd == 0 {
return w;
}
/* -1.0/(x+r) has up to 2ulp error, so compute it accurately */
let w0 = zero_low_word(w);
let v = r - (w0 - x); /* w0+v = r+x */
let a = -1.0 / w;
let a0 = zero_low_word(a);
a0 + a * (1.0 + a0 * w0 + a0 * v)
}
#[inline]
fn zero_low_word(x: f64) -> f64 {
f64::from_bits(f64::to_bits(x) & 0xFFFF_FFFF_0000_0000)
}
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