| // 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) |
| } |
