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another-boids-in-rust/vendor/pxfm/src/bessel/k1ef.rs

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/*
* // Copyright (c) Radzivon Bartoshyk 7/2025. All rights reserved.
* //
* // Redistribution and use in source and binary forms, with or without modification,
* // are permitted provided that the following conditions are met:
* //
* // 1. Redistributions of source code must retain the above copyright notice, this
* // list of conditions and the following disclaimer.
* //
* // 2. Redistributions in binary form must reproduce the above copyright notice,
* // this list of conditions and the following disclaimer in the documentation
* // and/or other materials provided with the distribution.
* //
* // 3. Neither the name of the copyright holder nor the names of its
* // contributors may be used to endorse or promote products derived from
* // this software without specific prior written permission.
* //
* // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
* // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
* // DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
* // FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* // DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
* // SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
* // CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
* // OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
use crate::bessel::j0f::j1f_rsqrt;
use crate::common::f_fmla;
use crate::exponents::core_expf;
use crate::logs::fast_logf;
use crate::polyeval::{f_estrin_polyeval8, f_polyeval3, f_polyeval4};
/// Modified exponentially scaled Bessel of the second kind of order 1
///
/// Computes K1(x)exp(x)
///
/// Max ULP 0.5
pub fn f_k1ef(x: f32) -> f32 {
let ux = x.to_bits();
if ux >= 0xffu32 << 23 || ux == 0 {
// |x| == 0, |x| == inf, |x| == NaN, x < 0
if ux.wrapping_shl(1) == 0 {
return f32::INFINITY;
}
if x.is_infinite() {
return if x.is_sign_positive() { 0. } else { f32::NAN };
}
return x + f32::NAN; // x == NaN
}
let xb = x.to_bits();
if xb <= 0x3f800000u32 {
// x <= 1.0
if xb <= 0x34000000u32 {
// |x| <= f32::EPSILON
let dx = x as f64;
let leading_term = 1. / dx + 1.;
if xb <= 0x3109705fu32 {
// |x| <= 2e-9
// taylor series for tiny K1(x)exp(x) ~ 1/x + 1 + O(x)
return leading_term as f32;
}
// taylor series for small K1(x)exp(x) ~ 1/x+1+1/4 (1+2 EulerGamma-2 Log[2]+2 Log[x]) x + O(x^3)
const C: f64 = f64::from_bits(0xbffd8773039049e8); // 1 + 2 EulerGamma-2 Log[2]
let log_x = fast_logf(x);
let r = f_fmla(log_x, 2., C);
let w0 = f_fmla(dx * 0.25, r, leading_term);
return w0 as f32;
}
return k1ef_small(x);
}
k1ef_asympt(x)
}
/**
Computes
I1(x) = x/2 * (1 + 1 * (x/2)^2 + (x/2)^4 * P((x/2)^2))
Generated by Woflram Mathematica:
```text
<<FunctionApproximations`
ClearAll["Global`*"]
f[x_]:=(BesselI[1,x]*2/x-1-1/2(x/2)^2)/(x/2)^4
g[z_]:=f[2 Sqrt[z]]
{err, approx}=MiniMaxApproximation[g[z],{z,{0.000000001,1},3,2},WorkingPrecision->60]
poly=Numerator[approx][[1]];
coeffs=CoefficientList[poly,z];
TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50}, ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
poly=Denominator[approx][[1]];
coeffs=CoefficientList[poly,z];
TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50}, ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
```
**/
#[inline]
fn i1f_small(x: f32) -> f64 {
let dx = x as f64;
let x_over_two = dx * 0.5;
let x_over_two_sqr = x_over_two * x_over_two;
let x_over_two_p4 = x_over_two_sqr * x_over_two_sqr;
let p_num = f_polyeval4(
x_over_two_sqr,
f64::from_bits(0x3fb5555555555355),
f64::from_bits(0x3f6ebf07f0dbc49b),
f64::from_bits(0x3f1fdc02bf28a8d9),
f64::from_bits(0x3ebb5e7574c700a6),
);
let p_den = f_polyeval3(
x_over_two_sqr,
f64::from_bits(0x3ff0000000000000),
f64::from_bits(0xbfa39b64b6135b5a),
f64::from_bits(0x3f3fa729bbe951f9),
);
let p = p_num / p_den;
let p1 = f_fmla(0.5, x_over_two_sqr, 1.);
let p2 = f_fmla(x_over_two_p4, p, p1);
p2 * x_over_two
}
/**
Series for
f(x) := BesselK(1, x) - Log(x)*BesselI(1, x) - 1/x
Generated by Wolfram Mathematica:
```text
<<FunctionApproximations`
ClearAll["Global`*"]
f[x_]:=(BesselK[1, x]-Log[x]BesselI[1,x]-1/x)/x
g[z_]:=f[Sqrt[z]]
{err, approx}=MiniMaxApproximation[g[z],{z,{0.000000001,1},3,3},WorkingPrecision->60]
poly=Numerator[approx][[1]];
coeffs=CoefficientList[poly,z];
TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50}, ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
poly=Denominator[approx][[1]];
coeffs=CoefficientList[poly,z];
TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50}, ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
```
**/
#[inline]
fn k1ef_small(x: f32) -> f32 {
let dx = x as f64;
let rcp = 1. / dx;
let x2 = dx * dx;
let p_num = f_polyeval4(
x2,
f64::from_bits(0xbfd3b5b6028a83d6),
f64::from_bits(0xbfb3fde2c83f7cca),
f64::from_bits(0xbf662b2e5defbe8c),
f64::from_bits(0xbefa2a63cc5c4feb),
);
let p_den = f_polyeval4(
x2,
f64::from_bits(0x3ff0000000000000),
f64::from_bits(0xbf9833197207a7c6),
f64::from_bits(0x3f315663bc7330ef),
f64::from_bits(0xbeb9211958f6b8c3),
);
let p = p_num / p_den;
let v_exp = core_expf(x);
let lg = fast_logf(x);
let v_i = i1f_small(x);
let z = f_fmla(lg, v_i, rcp);
let z0 = f_fmla(p, dx, z);
(z0 * v_exp) as f32
}
/**
Generated by Wolfram Mathematica:
```text
<<FunctionApproximations`
ClearAll["Global`*"]
f[x_]:=Sqrt[x] Exp[x] BesselK[1,x]
g[z_]:=f[1/z]
{err, approx}=MiniMaxApproximation[g[z],{z,{0.000000001,1},7,7},WorkingPrecision->60]
poly=Numerator[approx][[1]];
coeffs=CoefficientList[poly,z];
TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50}, ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
poly=Denominator[approx][[1]];
coeffs=CoefficientList[poly,z];
TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50}, ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
```
**/
#[inline]
fn k1ef_asympt(x: f32) -> f32 {
let dx = x as f64;
let recip = 1. / dx;
let r_sqrt = j1f_rsqrt(dx);
let p_num = f_estrin_polyeval8(
recip,
f64::from_bits(0x3ff40d931ff6270d),
f64::from_bits(0x402d250670ed7a6c),
f64::from_bits(0x404e517b9b494d38),
f64::from_bits(0x405cb02b7433a838),
f64::from_bits(0x405a03e606a1b871),
f64::from_bits(0x4045c98d4308dbcd),
f64::from_bits(0x401d115c4ce0540c),
f64::from_bits(0x3fd4213e72b24b3a),
);
let p_den = f_estrin_polyeval8(
recip,
f64::from_bits(0x3ff0000000000000),
f64::from_bits(0x402681096aa3a87d),
f64::from_bits(0x404623ab8d72ceea),
f64::from_bits(0x40530af06ea802b2),
f64::from_bits(0x404d526906fb9cec),
f64::from_bits(0x403281caca389f1b),
f64::from_bits(0x3ffdb93996948bb4),
f64::from_bits(0x3f9a009da07eb989),
);
let v = p_num / p_den;
let pp = v * r_sqrt;
pp as f32
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn test_k1f() {
assert_eq!(f_k1ef(0.00000000005423), 18439980000.0);
assert_eq!(f_k1ef(0.0000000043123), 231894820.0);
assert_eq!(f_k1ef(0.3), 4.125158);
assert_eq!(f_k1ef(1.89), 1.0710458);
assert_eq!(f_k1ef(5.89), 0.5477655);
assert_eq!(f_k1ef(101.89), 0.12461915);
assert_eq!(f_k1ef(0.), f32::INFINITY);
assert_eq!(f_k1ef(-0.), f32::INFINITY);
assert!(f_k1ef(-0.5).is_nan());
assert!(f_k1ef(f32::NEG_INFINITY).is_nan());
assert_eq!(f_k1ef(f32::INFINITY), 0.);
}
}
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