pxfm/bessel/
beta0.rs

1/*
2 * // Copyright (c) Radzivon Bartoshyk 8/2025. 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::double_double::DoubleDouble;
30use crate::dyadic_float::{DyadicFloat128, DyadicSign};
31use crate::polyeval::f_polyeval9;
32
33/**
34Beta series
35
36Generated by SageMath:
37```python
38#generate b series
39def binomial_like(n, m):
40    prod = QQ(1)
41    z = QQ(4)*(n**2)
42    for k in range(1,m + 1):
43        prod *= (z - (2*k - 1)**2)
44    return prod / (QQ(2)**(2*m) * (ZZ(m).factorial()))
45
46R = LaurentSeriesRing(RealField(300), 'x',default_prec=300)
47x = R.gen()
48
49def Pn_asymptotic(n, y, terms=10):
50    # now y = 1/x
51    return sum( (-1)**m * binomial_like(n, 2*m) / (QQ(2)**(2*m)) * y**(QQ(2)*m) for m in range(terms) )
52
53def Qn_asymptotic(n, y, terms=10):
54    return sum( (-1)**m * binomial_like(n, 2*m + 1) / (QQ(2)**(2*m + 1)) * y**(QQ(2)*m + 1) for m in range(terms) )
55
56P = Pn_asymptotic(0, x, 50)
57Q = Qn_asymptotic(0, x, 50)
58
59def sqrt_series(s):
60    val = S.valuation()
61    lc = S[val]  # Leading coefficient
62    b = lc.sqrt() * x**(val // 2)
63
64    for _ in range(5):
65        b = (b + S / b) / 2
66        b = b
67    return b
68
69S = (P**2 + Q**2).truncate(50)
70
71b_series = sqrt_series(S).truncate(30)
72#see the series
73print(b_series)
74```
75**/
76#[inline]
77pub(crate) fn bessel_0_asympt_beta(recip: DoubleDouble) -> DoubleDouble {
78    const C: [(u64, u64); 10] = [
79        (0x0000000000000000, 0x3ff0000000000000),
80        (0x0000000000000000, 0xbfb0000000000000),
81        (0x0000000000000000, 0x3fba800000000000),
82        (0x0000000000000000, 0xbfe15f0000000000),
83        (0x0000000000000000, 0x4017651180000000),
84        (0x0000000000000000, 0xc05ab8c13b800000),
85        (0x0000000000000000, 0x40a730492f262000),
86        (0x0000000000000000, 0xc0fc73a7acd696f0),
87        (0xbdf3a00000000000, 0x41577458dd9fce68),
88        (0xbe4ba6b000000000, 0xc1b903ab9b27e18f),
89    ];
90
91    // Doing (1/x)*(1/x) instead (1/(x*x)) to avoid spurious overflow/underflow
92    let x2 = DoubleDouble::quick_mult(recip, recip);
93
94    let mut p = DoubleDouble::mul_add(
95        x2,
96        DoubleDouble::from_bit_pair(C[9]),
97        DoubleDouble::from_bit_pair(C[8]),
98    );
99
100    p = DoubleDouble::mul_add_f64(x2, p, f64::from_bits(C[7].1));
101    p = DoubleDouble::mul_add_f64(x2, p, f64::from_bits(C[6].1));
102    p = DoubleDouble::mul_add_f64(x2, p, f64::from_bits(C[5].1));
103    p = DoubleDouble::mul_add_f64(x2, p, f64::from_bits(C[4].1));
104    p = DoubleDouble::mul_add_f64(x2, p, f64::from_bits(C[3].1));
105    p = DoubleDouble::mul_add_f64(x2, p, f64::from_bits(C[2].1));
106    p = DoubleDouble::mul_add_f64(x2, p, f64::from_bits(C[1].1));
107    p = DoubleDouble::mul_add_f64(x2, p, f64::from_bits(C[0].1));
108    p
109}
110
111/**
112Beta series
113
114Generated by SageMath:
115```python
116#generate b series
117def binomial_like(n, m):
118    prod = QQ(1)
119    z = QQ(4)*(n**2)
120    for k in range(1,m + 1):
121        prod *= (z - (2*k - 1)**2)
122    return prod / (QQ(2)**(2*m) * (ZZ(m).factorial()))
123
124R = LaurentSeriesRing(RealField(300), 'x',default_prec=300)
125x = R.gen()
126
127def Pn_asymptotic(n, y, terms=10):
128    # now y = 1/x
129    return sum( (-1)**m * binomial_like(n, 2*m) / (QQ(2)**(2*m)) * y**(QQ(2)*m) for m in range(terms) )
130
131def Qn_asymptotic(n, y, terms=10):
132    return sum( (-1)**m * binomial_like(n, 2*m + 1) / (QQ(2)**(2*m + 1)) * y**(QQ(2)*m + 1) for m in range(terms) )
133
134P = Pn_asymptotic(0, x, 50)
135Q = Qn_asymptotic(0, x, 50)
136
137def sqrt_series(s):
138    val = S.valuation()
139    lc = S[val]  # Leading coefficient
140    b = lc.sqrt() * x**(val // 2)
141
142    for _ in range(5):
143        b = (b + S / b) / 2
144        b = b
145    return b
146
147S = (P**2 + Q**2).truncate(50)
148
149b_series = sqrt_series(S).truncate(30)
150#see the series
151print(b_series)
152```
153**/
154#[inline]
155pub(crate) fn bessel_0_asympt_beta_fast(recip: DoubleDouble) -> DoubleDouble {
156    const C: [u64; 10] = [
157        0x3ff0000000000000,
158        0xbfb0000000000000,
159        0x3fba800000000000,
160        0xbfe15f0000000000,
161        0x4017651180000000,
162        0xc05ab8c13b800000,
163        0x40a730492f262000,
164        0xc0fc73a7acd696f0,
165        0x41577458dd9fce68,
166        0xc1b903ab9b27e18f,
167    ];
168
169    // Doing (1/x)*(1/x) instead (1/(x*x)) to avoid spurious overflow/underflow
170    let x2 = DoubleDouble::quick_mult(recip, recip);
171
172    let p = f_polyeval9(
173        x2.hi,
174        f64::from_bits(C[1]),
175        f64::from_bits(C[2]),
176        f64::from_bits(C[3]),
177        f64::from_bits(C[4]),
178        f64::from_bits(C[5]),
179        f64::from_bits(C[6]),
180        f64::from_bits(C[7]),
181        f64::from_bits(C[8]),
182        f64::from_bits(C[9]),
183    );
184
185    DoubleDouble::mul_f64_add_f64(x2, p, f64::from_bits(C[0]))
186}
187
188/// see [bessel_0_asympt_beta] for more info
189pub(crate) fn bessel_0_asympt_beta_hard(recip: DyadicFloat128) -> DyadicFloat128 {
190    static C: [DyadicFloat128; 12] = [
191        DyadicFloat128 {
192            sign: DyadicSign::Pos,
193            exponent: -127,
194            mantissa: 0x80000000_00000000_00000000_00000000_u128,
195        },
196        DyadicFloat128 {
197            sign: DyadicSign::Neg,
198            exponent: -131,
199            mantissa: 0x80000000_00000000_00000000_00000000_u128,
200        },
201        DyadicFloat128 {
202            sign: DyadicSign::Pos,
203            exponent: -131,
204            mantissa: 0xd4000000_00000000_00000000_00000000_u128,
205        },
206        DyadicFloat128 {
207            sign: DyadicSign::Neg,
208            exponent: -128,
209            mantissa: 0x8af80000_00000000_00000000_00000000_u128,
210        },
211        DyadicFloat128 {
212            sign: DyadicSign::Pos,
213            exponent: -125,
214            mantissa: 0xbb288c00_00000000_00000000_00000000_u128,
215        },
216        DyadicFloat128 {
217            sign: DyadicSign::Neg,
218            exponent: -121,
219            mantissa: 0xd5c609dc_00000000_00000000_00000000_u128,
220        },
221        DyadicFloat128 {
222            sign: DyadicSign::Pos,
223            exponent: -116,
224            mantissa: 0xb9824979_31000000_00000000_00000000_u128,
225        },
226        DyadicFloat128 {
227            sign: DyadicSign::Neg,
228            exponent: -111,
229            mantissa: 0xe39d3d66_b4b78000_00000000_00000000_u128,
230        },
231        DyadicFloat128 {
232            sign: DyadicSign::Pos,
233            exponent: -105,
234            mantissa: 0xbba2c6ec_fe733d8c_00000000_00000000_u128,
235        },
236        DyadicFloat128 {
237            sign: DyadicSign::Neg,
238            exponent: -99,
239            mantissa: 0xc81d5cd9_3f0c79ba_6b000000_00000000_u128,
240        },
241        DyadicFloat128 {
242            sign: DyadicSign::Pos,
243            exponent: -92,
244            mantissa: 0x86118ddf_c1ffc100_0ee1b000_00000000_u128,
245        },
246        DyadicFloat128 {
247            sign: DyadicSign::Neg,
248            exponent: -86,
249            mantissa: 0xdc7ccfa9_930b874d_52df3464_00000000_u128,
250        },
251    ];
252
253    let x2 = recip * recip;
254
255    let mut p = C[11];
256    for i in (0..11).rev() {
257        p = x2 * p + C[i];
258    }
259    p
260}
pxfm/bessel/beta0.rs

pxfm/bessel/
beta0.rs
