pxfm 0.1.29

Fast and accurate math
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240

/*
 * // 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.);
    }
}