pxfm/tangent/
tanpi.rs

1/*
2 * // Copyright (c) Radzivon Bartoshyk 7/2025. All rights reserved.
3 * //
4 * // Redistribution and use in source and binary forms, with or without modification,
5 * // are permitted provided that the following conditions are met:
6 * //
7 * // 1.  Redistributions of source code must retain the above copyright notice, this
8 * // list of conditions and the following disclaimer.
9 * //
10 * // 2.  Redistributions in binary form must reproduce the above copyright notice,
11 * // this list of conditions and the following disclaimer in the documentation
12 * // and/or other materials provided with the distribution.
13 * //
14 * // 3.  Neither the name of the copyright holder nor the names of its
15 * // contributors may be used to endorse or promote products derived from
16 * // this software without specific prior written permission.
17 * //
18 * // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
19 * // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
20 * // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
21 * // DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
22 * // FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
23 * // DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
24 * // SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
25 * // CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
26 * // OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
27 * // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
28 */
29use crate::common::f_fmla;
30use crate::double_double::DoubleDouble;
31use crate::sincospi::reduce_pi_64;
32use crate::tangent::tanpi_table::TANPI_K_PI_OVER_64;
33
34#[inline]
35pub(crate) fn tanpi_eval(x: f64) -> DoubleDouble {
36    let x2 = DoubleDouble::from_exact_mult(x, x);
37    // tan(pi*x) generated by Sollya:
38    // d = [0, 0.0078128];
39    // f_tan = tan(y*pi)/y;
40    // Q = fpminimax(f_tan, [|0, 2, 4, 6, 8|], [|107, D...|], d, relative, floating);
41    const C: [u64; 4] = [
42        0x4024abbce625be51,
43        0x404466bc677698e5,
44        0x40645fff70379ae3,
45        0x4084626b091b7fd0,
46    ];
47    const C0: DoubleDouble = DoubleDouble::from_bit_pair((0x3ca1a6444aa5b996, 0x400921fb54442d18));
48
49    // polyeval 4, estrin scheme
50    let u0 = f_fmla(x2.hi, f64::from_bits(C[1]), f64::from_bits(C[0]));
51    let u1 = f_fmla(x2.hi, f64::from_bits(C[3]), f64::from_bits(C[2]));
52    let tan_poly_lo = f_fmla(x2.hi * x2.hi, u1, u0);
53
54    // We're splitting polynomial in two parts, since first term dominates
55    // we compute: (a0_lo + a0_hi) * x + x * (a1 * x^2 + a2 + x^4) ...
56    let r_lo = DoubleDouble::quick_mult_f64(x2, tan_poly_lo);
57    let tan_lo = f_fmla(r_lo.lo, x, r_lo.hi * x);
58    let tan_hi = DoubleDouble::quick_mult_f64(C0, x);
59    DoubleDouble::full_add_f64(tan_hi, tan_lo)
60}
61
62#[cold]
63fn tanpi_hard(x: f64, tan_k: DoubleDouble) -> DoubleDouble {
64    const C: [(u64, u64); 6] = [
65        (0x3ca1a62632712fc8, 0x400921fb54442d18),
66        (0xbcc052338fbb4528, 0x4024abbce625be53),
67        (0x3ced42454c5f85b3, 0x404466bc6775aad9),
68        (0xbd00c7d6a971a560, 0x40645fff9b4b244d),
69        (0x3d205970eff53274, 0x40845f46e96c3a0b),
70        (0xbd3589489ad24fc4, 0x40a4630551cd123d),
71    ];
72    let x2 = DoubleDouble::from_exact_mult(x, x);
73    let mut tan_y = DoubleDouble::quick_mul_add(
74        x2,
75        DoubleDouble::from_bit_pair(C[5]),
76        DoubleDouble::from_bit_pair(C[4]),
77    );
78    tan_y = DoubleDouble::quick_mul_add(x2, tan_y, DoubleDouble::from_bit_pair(C[3]));
79    tan_y = DoubleDouble::quick_mul_add(x2, tan_y, DoubleDouble::from_bit_pair(C[2]));
80    tan_y = DoubleDouble::quick_mul_add(x2, tan_y, DoubleDouble::from_bit_pair(C[1]));
81    tan_y = DoubleDouble::quick_mul_add(x2, tan_y, DoubleDouble::from_bit_pair(C[0]));
82    tan_y = DoubleDouble::quick_mult_f64(tan_y, x);
83
84    // num = tan(y*pi/64) + tan(k*pi/64)
85    let num = DoubleDouble::full_dd_add(tan_y, tan_k);
86    // den = 1 - tan(y*pi/64)*tan(k*pi/64)
87    let den = DoubleDouble::mul_add_f64(tan_y, -tan_k, 1.);
88    // tan = num / den
89    DoubleDouble::div(num, den)
90}
91
92/// Computes tan(PI*x)
93///
94/// Max found ULP 0.5
95pub fn f_tanpi(x: f64) -> f64 {
96    if x == 0. {
97        return x;
98    }
99    let ax = x.to_bits() & 0x7fff_ffff_ffff_ffff;
100    if ax >= (0x7ffu64 << 52) {
101        // NaN, Inf
102        if ax > (0x7ffu64 << 52) {
103            return x + x;
104        } // NaN
105        return f64::NAN; // x=Inf
106    }
107    let e: i32 = (ax >> 52) as i32 - 1023;
108    if e > 0 {
109        if e >= 52 {
110            // when |x| > 2^53 it's always an integer
111            return f64::copysign(0., x);
112        }
113        // |x| > 1 and |x| < 2^53
114        let m = (ax & ((1u64 << 52) - 1)) | (1u64 << 52); // mantissa with hidden 1
115        let shift = 52 - e;
116
117        let frac = m & ((1u64 << shift) - 1);
118        if frac == (1u64 << (shift - 1)) {
119            // |x| is always integer.5 means it's inf
120            return f64::INFINITY;
121        }
122    }
123
124    if ax <= 0x3cb0000000000000 {
125        // for tiny x ( |x| < f64::EPSILON ) just small taylor expansion
126        // tan(PI*x) ~ PI*x + PI^3*x^3/3 + O(x^5)
127        const PI: DoubleDouble =
128            DoubleDouble::from_bit_pair((0x3ca1a62633145c07, 0x400921fb54442d18));
129        if ax <= 0x3ca0000000000000 {
130            // |x| <= 2^-53, renormalize value
131            let e: i32 = (ax >> 52) as i32;
132            let sc = f64::from_bits((2045i64 - e as i64).wrapping_shl(52) as u64);
133            let isc = f64::from_bits(1i64.wrapping_add(e as i64).wrapping_shl(52) as u64);
134            let dx = x * sc;
135            let q0 = DoubleDouble::quick_mult_f64(PI, dx);
136            let r = q0.to_f64() * isc;
137            return r;
138        }
139        let q0 = DoubleDouble::quick_mult_f64(PI, x);
140        let r = q0.to_f64();
141        return r;
142    }
143
144    // argument reduction
145    let (y, k) = reduce_pi_64(x);
146
147    if y == 0.0 {
148        let km = (k.abs() & 63) as i32; // k mod 64
149
150        match km {
151            0 => return f64::copysign(0f64, x),           // tanpi(n) = 0
152            32 => return f64::copysign(f64::INFINITY, x), // tanpi(n+0.5) = ±∞
153            16 => return f64::copysign(1.0, x),           // tanpi(n+0.25) = ±1
154            48 => return -f64::copysign(1.0, x),          // tanpi(n+0.75) = ∓1
155            _ => {}
156        }
157    }
158
159    let tan_k = DoubleDouble::from_bit_pair(TANPI_K_PI_OVER_64[((k as u64) & 127) as usize]);
160
161    // Computes tan(pi*x) through identities.
162    // tan(a+b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b)) = (tan(y*pi/64) + tan(k*pi/64)) / (1 - tan(y*pi/64)*tan(k*pi/64))
163    let tan_y = tanpi_eval(y);
164    // num = tan(y*pi/64) + tan(k*pi/64)
165    let num = DoubleDouble::add(tan_k, tan_y);
166    // den = 1 - tan(y*pi/64)*tan(k*pi/64)
167    let den = DoubleDouble::mul_add_f64(tan_y, -tan_k, 1.);
168    // tan = num / den
169    let tan_value = DoubleDouble::div(num, den);
170    let err = f_fmla(
171        tan_value.hi,
172        f64::from_bits(0x3bf0000000000000), // 2^-64
173        f64::from_bits(0x3b60000000000000), // 2^-73
174    );
175    let ub = tan_value.hi + (tan_value.lo + err);
176    let lb = tan_value.hi + (tan_value.lo - err);
177    if ub == lb {
178        return tan_value.to_f64();
179    }
180    tanpi_hard(y, tan_k).to_f64()
181}
182
183#[cfg(test)]
184mod tests {
185    use super::*;
186
187    #[test]
188    fn test_tanpi() {
189        assert_eq!(f_tanpi(0.4999999999119535), 3615246871.564404);
190        assert_eq!(f_tanpi(7119681148991743.0), 0.);
191        assert_eq!(f_tanpi(63.5), f64::INFINITY);
192        assert_eq!(f_tanpi(63.99935913085936), -0.0020133525045719896);
193        assert_eq!(f_tanpi(3.3821122649309461E-306), 1.0625219045122997E-305);
194        assert_eq!(f_tanpi(1.8010707049867402E-255), 5.6582304953821333E-255);
195        assert_eq!(f_tanpi(1.001000000061801), 0.0031416031832113213);
196        assert_eq!(f_tanpi(-0.5000000000000226), 14054316517702.594);
197        assert_eq!(f_tanpi(0.5000000000000001), -2867080569611329.5);
198        assert_eq!(f_tanpi(0.02131), 0.06704753721009375);
199        assert!(f_tanpi(f64::INFINITY).is_nan());
200        assert!(f_tanpi(f64::NAN).is_nan());
201        assert!(f_tanpi(f64::NEG_INFINITY).is_nan());
202    }
203}
pxfm/tangent/tanpi.rs

pxfm/tangent/
tanpi.rs
