pxfm/bessel/
jincpif.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::bessel::j0f::j1f_rsqrt;
30use crate::bessel::j1_coeffs::{J1_ZEROS, J1_ZEROS_VALUE};
31use crate::bessel::j1f::{j1f_asympt_alpha, j1f_asympt_beta};
32use crate::bessel::j1f_coeffs::J1F_COEFFS;
33use crate::bessel::trigo_bessel::sin_small;
34use crate::common::f_fmla;
35use crate::double_double::DoubleDouble;
36use crate::polyeval::{f_polyeval6, f_polyeval14};
37use crate::round::RoundFinite;
38
39/// Normalized jinc 2*J1(PI\*x)/(pi\*x)
40///
41/// ULP 0.5
42pub fn f_jincpif(x: f32) -> f32 {
43    let ux = x.to_bits().wrapping_shl(1);
44    if ux >= 0xffu32 << 24 || ux <= 0x6800_0000u32 {
45        // |x| <= f32::EPSILON, |x| == inf, |x| == NaN
46        if ux <= 0x6800_0000u32 {
47            // |x| == 0
48            return 1.;
49        }
50        if x.is_infinite() {
51            return 0.;
52        }
53        return x + f32::NAN; // x == NaN
54    }
55
56    let ax = x.to_bits() & 0x7fff_ffff;
57
58    if ax < 0x429533c2u32 {
59        // |x| < 74.60109
60        if ax <= 0x3e800000u32 {
61            // |x| < 0.25
62            return jincf_near_zero(f32::from_bits(ax));
63        }
64        let scaled_pix = f32::from_bits(ax) * std::f32::consts::PI; // just test boundaries
65        if scaled_pix < 74.60109 {
66            return jincpif_small_argument(f32::from_bits(ax));
67        }
68    }
69
70    jincpif_asympt(f32::from_bits(ax)) as f32
71}
72
73#[inline]
74fn jincf_near_zero(x: f32) -> f32 {
75    let dx = x as f64;
76    // Generated in Wolfram Mathematica:
77    // <<FunctionApproximations`
78    // ClearAll["Global`*"]
79    // f[x_]:=BesselJ[1,x*Pi]/(x*Pi)
80    // {err,approx}=MiniMaxApproximation[f[z],{z,{2^-23,0.3},6,0},WorkingPrecision->60]
81    // poly=Numerator[approx][[1]];
82    // coeffs=CoefficientList[poly,z];
83    // TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
84    let p_num = f_polyeval6(
85        dx,
86        f64::from_bits(0x3fe0000000000002),
87        f64::from_bits(0xbfd46cd1822a5aa0),
88        f64::from_bits(0xbfde583c923dc6f4),
89        f64::from_bits(0x3fd3834f47496519),
90        f64::from_bits(0x3fb8118468756e6f),
91        f64::from_bits(0xbfafaff09f13df88),
92    );
93    let p_den = f_polyeval6(
94        dx,
95        f64::from_bits(0x3ff0000000000000),
96        f64::from_bits(0xbfe46cd1822a4cb0),
97        f64::from_bits(0x3fd2447a026f477a),
98        f64::from_bits(0xbfc6bdf2192404e5),
99        f64::from_bits(0x3fa0cf182218e448),
100        f64::from_bits(0xbf939ab46c3f7a7d),
101    );
102    (p_num / p_den * 2.) as f32
103}
104
105/// This method on small range searches for nearest zero or extremum.
106/// Then picks stored series expansion at the point end evaluates the poly at the point.
107#[inline]
108fn jincpif_small_argument(ox: f32) -> f32 {
109    const PI: f64 = f64::from_bits(0x400921fb54442d18);
110    let x = ox as f64 * PI;
111    let x_abs = f64::from_bits(x.to_bits() & 0x7fff_ffff_ffff_ffff);
112
113    // let avg_step = 74.60109 / 47.0;
114    // let inv_step = 1.0 / avg_step;
115
116    const INV_STEP: f64 = 0.6300176043004198;
117
118    let inv_scale = x;
119
120    let fx = x_abs * INV_STEP;
121    const J1_ZEROS_COUNT: f64 = (J1_ZEROS.len() - 1) as f64;
122    let idx0 = unsafe { fx.min(J1_ZEROS_COUNT).to_int_unchecked::<usize>() };
123    let idx1 = unsafe { fx.ceil().min(J1_ZEROS_COUNT).to_int_unchecked::<usize>() };
124
125    let found_zero0 = DoubleDouble::from_bit_pair(J1_ZEROS[idx0]);
126    let found_zero1 = DoubleDouble::from_bit_pair(J1_ZEROS[idx1]);
127
128    let dist0 = (found_zero0.hi - x_abs).abs();
129    let dist1 = (found_zero1.hi - x_abs).abs();
130
131    let (found_zero, idx, dist) = if dist0 < dist1 {
132        (found_zero0, idx0, dist0)
133    } else {
134        (found_zero1, idx1, dist1)
135    };
136
137    if idx == 0 {
138        return jincf_near_zero(ox);
139    }
140
141    // We hit exact zero, value, better to return it directly
142    if dist == 0. {
143        return (f64::from_bits(J1_ZEROS_VALUE[idx]) / inv_scale * 2.) as f32;
144    }
145
146    let c = &J1F_COEFFS[idx - 1];
147
148    let r = (x_abs - found_zero.hi) - found_zero.lo;
149
150    let p = f_polyeval14(
151        r,
152        f64::from_bits(c[0]),
153        f64::from_bits(c[1]),
154        f64::from_bits(c[2]),
155        f64::from_bits(c[3]),
156        f64::from_bits(c[4]),
157        f64::from_bits(c[5]),
158        f64::from_bits(c[6]),
159        f64::from_bits(c[7]),
160        f64::from_bits(c[8]),
161        f64::from_bits(c[9]),
162        f64::from_bits(c[10]),
163        f64::from_bits(c[11]),
164        f64::from_bits(c[12]),
165        f64::from_bits(c[13]),
166    );
167
168    (p / inv_scale * 2.) as f32
169}
170
171/*
172   Evaluates:
173   J1 = sqrt(2/(PI*x)) * beta(x) * cos(x - 3*PI/4 - alpha(x))
174   discarding 1*PI/2 using identities gives:
175   J1 = sqrt(2/(PI*x)) * beta(x) * sin(x - PI/4 - alpha(x))
176
177   to avoid squashing small (-PI/4 - alpha(x)) into a large x actual expansion is:
178
179   J1 = sqrt(2/(PI*x)) * beta(x) * sin((x mod 2*PI) - PI/4 - alpha(x))
180*/
181#[inline]
182pub(crate) fn jincpif_asympt(x: f32) -> f64 {
183    const PI: f64 = f64::from_bits(0x400921fb54442d18);
184
185    let dox = x as f64;
186    let dx = dox * PI;
187
188    let inv_scale = dx;
189
190    let alpha = j1f_asympt_alpha(dx);
191    let beta = j1f_asympt_beta(dx);
192
193    // argument reduction assuming x here value is already multiple of PI.
194    // k = round((x*Pi) / (pi*2))
195    let kd = (dox * 0.5).round_finite();
196    //  y = (x * Pi) - k * 2
197    let angle = f_fmla(kd, -2., dox) * PI;
198
199    const SQRT_2_OVER_PI: f64 = f64::from_bits(0x3fe9884533d43651);
200    const MPI_OVER_4: f64 = f64::from_bits(0xbfe921fb54442d18);
201
202    let x0pi34 = MPI_OVER_4 - alpha;
203    let r0 = angle + x0pi34;
204
205    let m_sin = sin_small(r0);
206
207    let z0 = beta * m_sin;
208    let scale = SQRT_2_OVER_PI * j1f_rsqrt(dx);
209
210    let j1pix = scale * z0;
211    (j1pix / inv_scale) * 2.
212}
213
214#[cfg(test)]
215mod tests {
216    use super::*;
217
218    #[test]
219    fn test_jincpif() {
220        assert_eq!(f_jincpif(-102.59484), 0.00024380769);
221        assert_eq!(f_jincpif(102.59484), 0.00024380769);
222        assert_eq!(f_jincpif(100.08199), -0.00014386141);
223        assert_eq!(f_jincpif(0.27715185), 0.9081822);
224        assert_eq!(f_jincpif(0.007638072), 0.99992806);
225        assert_eq!(f_jincpif(-f32::EPSILON), 1.0);
226        assert_eq!(f_jincpif(f32::EPSILON), 1.0);
227        assert_eq!(
228            f_jincpif(0.000000000000000000000000000000000000008827127),
229            1.0
230        );
231        assert_eq!(f_jincpif(5.4), -0.010821743);
232        assert_eq!(
233            f_jincpif(77.743162408196766932633181568235159),
234            -0.00041799102
235        );
236        assert_eq!(
237            f_jincpif(-77.743162408196766932633181568235159),
238            -0.00041799102
239        );
240        assert_eq!(
241            f_jincpif(84.027189586293545175976760219782591),
242            -0.00023927793
243        );
244        assert_eq!(f_jincpif(f32::INFINITY), 0.);
245        assert_eq!(f_jincpif(f32::NEG_INFINITY), 0.);
246        assert!(f_jincpif(f32::NAN).is_nan());
247        assert_eq!(f_jincpif(-1.7014118e38), -0.0);
248    }
249}
pxfm/bessel/jincpif.rs

pxfm/bessel/
jincpif.rs
