libm/math/jnf.rs
1/* origin: FreeBSD /usr/src/lib/msun/src/e_jnf.c */
2/*
3 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
4 */
5/*
6 * ====================================================
7 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
8 *
9 * Developed at SunPro, a Sun Microsystems, Inc. business.
10 * Permission to use, copy, modify, and distribute this
11 * software is freely granted, provided that this notice
12 * is preserved.
13 * ====================================================
14 */
15
16use super::{fabsf, j0f, j1f, logf, y0f, y1f};
17
18pub fn jnf(n: i32, mut x: f32) -> f32 {
19 let mut ix: u32;
20 let mut nm1: i32;
21 let mut sign: bool;
22 let mut i: i32;
23 let mut a: f32;
24 let mut b: f32;
25 let mut temp: f32;
26
27 ix = x.to_bits();
28 sign = (ix >> 31) != 0;
29 ix &= 0x7fffffff;
30 if ix > 0x7f800000 {
31 /* nan */
32 return x;
33 }
34
35 /* J(-n,x) = J(n,-x), use |n|-1 to avoid overflow in -n */
36 if n == 0 {
37 return j0f(x);
38 }
39 if n < 0 {
40 nm1 = -(n + 1);
41 x = -x;
42 sign = !sign;
43 } else {
44 nm1 = n - 1;
45 }
46 if nm1 == 0 {
47 return j1f(x);
48 }
49
50 sign &= (n & 1) != 0; /* even n: 0, odd n: signbit(x) */
51 x = fabsf(x);
52 if ix == 0 || ix == 0x7f800000 {
53 /* if x is 0 or inf */
54 b = 0.0;
55 } else if (nm1 as f32) < x {
56 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
57 a = j0f(x);
58 b = j1f(x);
59 i = 0;
60 while i < nm1 {
61 i += 1;
62 temp = b;
63 b = b * (2.0 * (i as f32) / x) - a;
64 a = temp;
65 }
66 } else {
67 if ix < 0x35800000 {
68 /* x < 2**-20 */
69 /* x is tiny, return the first Taylor expansion of J(n,x)
70 * J(n,x) = 1/n!*(x/2)^n - ...
71 */
72 if nm1 > 8 {
73 /* underflow */
74 nm1 = 8;
75 }
76 temp = 0.5 * x;
77 b = temp;
78 a = 1.0;
79 i = 2;
80 while i <= nm1 + 1 {
81 a *= i as f32; /* a = n! */
82 b *= temp; /* b = (x/2)^n */
83 i += 1;
84 }
85 b = b / a;
86 } else {
87 /* use backward recurrence */
88 /* x x^2 x^2
89 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
90 * 2n - 2(n+1) - 2(n+2)
91 *
92 * 1 1 1
93 * (for large x) = ---- ------ ------ .....
94 * 2n 2(n+1) 2(n+2)
95 * -- - ------ - ------ -
96 * x x x
97 *
98 * Let w = 2n/x and h=2/x, then the above quotient
99 * is equal to the continued fraction:
100 * 1
101 * = -----------------------
102 * 1
103 * w - -----------------
104 * 1
105 * w+h - ---------
106 * w+2h - ...
107 *
108 * To determine how many terms needed, let
109 * Q(0) = w, Q(1) = w(w+h) - 1,
110 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
111 * When Q(k) > 1e4 good for single
112 * When Q(k) > 1e9 good for double
113 * When Q(k) > 1e17 good for quadruple
114 */
115 /* determine k */
116 let mut t: f32;
117 let mut q0: f32;
118 let mut q1: f32;
119 let mut w: f32;
120 let h: f32;
121 let mut z: f32;
122 let mut tmp: f32;
123 let nf: f32;
124 let mut k: i32;
125
126 nf = (nm1 as f32) + 1.0;
127 w = 2.0 * (nf as f32) / x;
128 h = 2.0 / x;
129 z = w + h;
130 q0 = w;
131 q1 = w * z - 1.0;
132 k = 1;
133 while q1 < 1.0e4 {
134 k += 1;
135 z += h;
136 tmp = z * q1 - q0;
137 q0 = q1;
138 q1 = tmp;
139 }
140 t = 0.0;
141 i = k;
142 while i >= 0 {
143 t = 1.0 / (2.0 * ((i as f32) + nf) / x - t);
144 i -= 1;
145 }
146 a = t;
147 b = 1.0;
148 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
149 * Hence, if n*(log(2n/x)) > ...
150 * single 8.8722839355e+01
151 * double 7.09782712893383973096e+02
152 * long double 1.1356523406294143949491931077970765006170e+04
153 * then recurrent value may overflow and the result is
154 * likely underflow to zero
155 */
156 tmp = nf * logf(fabsf(w));
157 if tmp < 88.721679688 {
158 i = nm1;
159 while i > 0 {
160 temp = b;
161 b = 2.0 * (i as f32) * b / x - a;
162 a = temp;
163 i -= 1;
164 }
165 } else {
166 i = nm1;
167 while i > 0 {
168 temp = b;
169 b = 2.0 * (i as f32) * b / x - a;
170 a = temp;
171 /* scale b to avoid spurious overflow */
172 let x1p60 = f32::from_bits(0x5d800000); // 0x1p60 == 2^60
173 if b > x1p60 {
174 a /= b;
175 t /= b;
176 b = 1.0;
177 }
178 i -= 1;
179 }
180 }
181 z = j0f(x);
182 w = j1f(x);
183 if fabsf(z) >= fabsf(w) {
184 b = t * z / b;
185 } else {
186 b = t * w / a;
187 }
188 }
189 }
190
191 if sign {
192 -b
193 } else {
194 b
195 }
196}
197
198pub fn ynf(n: i32, x: f32) -> f32 {
199 let mut ix: u32;
200 let mut ib: u32;
201 let nm1: i32;
202 let mut sign: bool;
203 let mut i: i32;
204 let mut a: f32;
205 let mut b: f32;
206 let mut temp: f32;
207
208 ix = x.to_bits();
209 sign = (ix >> 31) != 0;
210 ix &= 0x7fffffff;
211 if ix > 0x7f800000 {
212 /* nan */
213 return x;
214 }
215 if sign && ix != 0 {
216 /* x < 0 */
217 return 0.0 / 0.0;
218 }
219 if ix == 0x7f800000 {
220 return 0.0;
221 }
222
223 if n == 0 {
224 return y0f(x);
225 }
226 if n < 0 {
227 nm1 = -(n + 1);
228 sign = (n & 1) != 0;
229 } else {
230 nm1 = n - 1;
231 sign = false;
232 }
233 if nm1 == 0 {
234 if sign {
235 return -y1f(x);
236 } else {
237 return y1f(x);
238 }
239 }
240
241 a = y0f(x);
242 b = y1f(x);
243 /* quit if b is -inf */
244 ib = b.to_bits();
245 i = 0;
246 while i < nm1 && ib != 0xff800000 {
247 i += 1;
248 temp = b;
249 b = (2.0 * (i as f32) / x) * b - a;
250 ib = b.to_bits();
251 a = temp;
252 }
253
254 if sign {
255 -b
256 } else {
257 b
258 }
259}