1 | /* origin: FreeBSD /usr/src/lib/msun/src/e_sqrt.c */ |
2 | /* |
3 | * ==================================================== |
4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
5 | * |
6 | * Developed at SunSoft, a Sun Microsystems, Inc. business. |
7 | * Permission to use, copy, modify, and distribute this |
8 | * software is freely granted, provided that this notice |
9 | * is preserved. |
10 | * ==================================================== |
11 | */ |
12 | /* sqrt(x) |
13 | * Return correctly rounded sqrt. |
14 | * ------------------------------------------ |
15 | * | Use the hardware sqrt if you have one | |
16 | * ------------------------------------------ |
17 | * Method: |
18 | * Bit by bit method using integer arithmetic. (Slow, but portable) |
19 | * 1. Normalization |
20 | * Scale x to y in [1,4) with even powers of 2: |
21 | * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then |
22 | * sqrt(x) = 2^k * sqrt(y) |
23 | * 2. Bit by bit computation |
24 | * Let q = sqrt(y) truncated to i bit after binary point (q = 1), |
25 | * i 0 |
26 | * i+1 2 |
27 | * s = 2*q , and y = 2 * ( y - q ). (1) |
28 | * i i i i |
29 | * |
30 | * To compute q from q , one checks whether |
31 | * i+1 i |
32 | * |
33 | * -(i+1) 2 |
34 | * (q + 2 ) <= y. (2) |
35 | * i |
36 | * -(i+1) |
37 | * If (2) is false, then q = q ; otherwise q = q + 2 . |
38 | * i+1 i i+1 i |
39 | * |
40 | * With some algebraic manipulation, it is not difficult to see |
41 | * that (2) is equivalent to |
42 | * -(i+1) |
43 | * s + 2 <= y (3) |
44 | * i i |
45 | * |
46 | * The advantage of (3) is that s and y can be computed by |
47 | * i i |
48 | * the following recurrence formula: |
49 | * if (3) is false |
50 | * |
51 | * s = s , y = y ; (4) |
52 | * i+1 i i+1 i |
53 | * |
54 | * otherwise, |
55 | * -i -(i+1) |
56 | * s = s + 2 , y = y - s - 2 (5) |
57 | * i+1 i i+1 i i |
58 | * |
59 | * One may easily use induction to prove (4) and (5). |
60 | * Note. Since the left hand side of (3) contain only i+2 bits, |
61 | * it does not necessary to do a full (53-bit) comparison |
62 | * in (3). |
63 | * 3. Final rounding |
64 | * After generating the 53 bits result, we compute one more bit. |
65 | * Together with the remainder, we can decide whether the |
66 | * result is exact, bigger than 1/2ulp, or less than 1/2ulp |
67 | * (it will never equal to 1/2ulp). |
68 | * The rounding mode can be detected by checking whether |
69 | * huge + tiny is equal to huge, and whether huge - tiny is |
70 | * equal to huge for some floating point number "huge" and "tiny". |
71 | * |
72 | * Special cases: |
73 | * sqrt(+-0) = +-0 ... exact |
74 | * sqrt(inf) = inf |
75 | * sqrt(-ve) = NaN ... with invalid signal |
76 | * sqrt(NaN) = NaN ... with invalid signal for signaling NaN |
77 | */ |
78 | |
79 | use core::f64; |
80 | |
81 | #[cfg_attr (all(test, assert_no_panic), no_panic::no_panic)] |
82 | pub fn sqrt(x: f64) -> f64 { |
83 | // On wasm32 we know that LLVM's intrinsic will compile to an optimized |
84 | // `f64.sqrt` native instruction, so we can leverage this for both code size |
85 | // and speed. |
86 | llvm_intrinsically_optimized! { |
87 | #[cfg(target_arch = "wasm32" )] { |
88 | return if x < 0.0 { |
89 | f64::NAN |
90 | } else { |
91 | unsafe { ::core::intrinsics::sqrtf64(x) } |
92 | } |
93 | } |
94 | } |
95 | #[cfg (target_feature = "sse2" )] |
96 | { |
97 | // Note: This path is unlikely since LLVM will usually have already |
98 | // optimized sqrt calls into hardware instructions if sse2 is available, |
99 | // but if someone does end up here they'll apprected the speed increase. |
100 | #[cfg (target_arch = "x86" )] |
101 | use core::arch::x86::*; |
102 | #[cfg (target_arch = "x86_64" )] |
103 | use core::arch::x86_64::*; |
104 | unsafe { |
105 | let m = _mm_set_sd(x); |
106 | let m_sqrt = _mm_sqrt_pd(m); |
107 | _mm_cvtsd_f64(m_sqrt) |
108 | } |
109 | } |
110 | #[cfg (not(target_feature = "sse2" ))] |
111 | { |
112 | use core::num::Wrapping; |
113 | |
114 | const TINY: f64 = 1.0e-300; |
115 | |
116 | let mut z: f64; |
117 | let sign: Wrapping<u32> = Wrapping(0x80000000); |
118 | let mut ix0: i32; |
119 | let mut s0: i32; |
120 | let mut q: i32; |
121 | let mut m: i32; |
122 | let mut t: i32; |
123 | let mut i: i32; |
124 | let mut r: Wrapping<u32>; |
125 | let mut t1: Wrapping<u32>; |
126 | let mut s1: Wrapping<u32>; |
127 | let mut ix1: Wrapping<u32>; |
128 | let mut q1: Wrapping<u32>; |
129 | |
130 | ix0 = (x.to_bits() >> 32) as i32; |
131 | ix1 = Wrapping(x.to_bits() as u32); |
132 | |
133 | /* take care of Inf and NaN */ |
134 | if (ix0 & 0x7ff00000) == 0x7ff00000 { |
135 | return x * x + x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf, sqrt(-inf)=sNaN */ |
136 | } |
137 | /* take care of zero */ |
138 | if ix0 <= 0 { |
139 | if ((ix0 & !(sign.0 as i32)) | ix1.0 as i32) == 0 { |
140 | return x; /* sqrt(+-0) = +-0 */ |
141 | } |
142 | if ix0 < 0 { |
143 | return (x - x) / (x - x); /* sqrt(-ve) = sNaN */ |
144 | } |
145 | } |
146 | /* normalize x */ |
147 | m = ix0 >> 20; |
148 | if m == 0 { |
149 | /* subnormal x */ |
150 | while ix0 == 0 { |
151 | m -= 21; |
152 | ix0 |= (ix1 >> 11).0 as i32; |
153 | ix1 <<= 21; |
154 | } |
155 | i = 0; |
156 | while (ix0 & 0x00100000) == 0 { |
157 | i += 1; |
158 | ix0 <<= 1; |
159 | } |
160 | m -= i - 1; |
161 | ix0 |= (ix1 >> (32 - i) as usize).0 as i32; |
162 | ix1 = ix1 << i as usize; |
163 | } |
164 | m -= 1023; /* unbias exponent */ |
165 | ix0 = (ix0 & 0x000fffff) | 0x00100000; |
166 | if (m & 1) == 1 { |
167 | /* odd m, double x to make it even */ |
168 | ix0 += ix0 + ((ix1 & sign) >> 31).0 as i32; |
169 | ix1 += ix1; |
170 | } |
171 | m >>= 1; /* m = [m/2] */ |
172 | |
173 | /* generate sqrt(x) bit by bit */ |
174 | ix0 += ix0 + ((ix1 & sign) >> 31).0 as i32; |
175 | ix1 += ix1; |
176 | q = 0; /* [q,q1] = sqrt(x) */ |
177 | q1 = Wrapping(0); |
178 | s0 = 0; |
179 | s1 = Wrapping(0); |
180 | r = Wrapping(0x00200000); /* r = moving bit from right to left */ |
181 | |
182 | while r != Wrapping(0) { |
183 | t = s0 + r.0 as i32; |
184 | if t <= ix0 { |
185 | s0 = t + r.0 as i32; |
186 | ix0 -= t; |
187 | q += r.0 as i32; |
188 | } |
189 | ix0 += ix0 + ((ix1 & sign) >> 31).0 as i32; |
190 | ix1 += ix1; |
191 | r >>= 1; |
192 | } |
193 | |
194 | r = sign; |
195 | while r != Wrapping(0) { |
196 | t1 = s1 + r; |
197 | t = s0; |
198 | if t < ix0 || (t == ix0 && t1 <= ix1) { |
199 | s1 = t1 + r; |
200 | if (t1 & sign) == sign && (s1 & sign) == Wrapping(0) { |
201 | s0 += 1; |
202 | } |
203 | ix0 -= t; |
204 | if ix1 < t1 { |
205 | ix0 -= 1; |
206 | } |
207 | ix1 -= t1; |
208 | q1 += r; |
209 | } |
210 | ix0 += ix0 + ((ix1 & sign) >> 31).0 as i32; |
211 | ix1 += ix1; |
212 | r >>= 1; |
213 | } |
214 | |
215 | /* use floating add to find out rounding direction */ |
216 | if (ix0 as u32 | ix1.0) != 0 { |
217 | z = 1.0 - TINY; /* raise inexact flag */ |
218 | if z >= 1.0 { |
219 | z = 1.0 + TINY; |
220 | if q1.0 == 0xffffffff { |
221 | q1 = Wrapping(0); |
222 | q += 1; |
223 | } else if z > 1.0 { |
224 | if q1.0 == 0xfffffffe { |
225 | q += 1; |
226 | } |
227 | q1 += Wrapping(2); |
228 | } else { |
229 | q1 += q1 & Wrapping(1); |
230 | } |
231 | } |
232 | } |
233 | ix0 = (q >> 1) + 0x3fe00000; |
234 | ix1 = q1 >> 1; |
235 | if (q & 1) == 1 { |
236 | ix1 |= sign; |
237 | } |
238 | ix0 += m << 20; |
239 | f64::from_bits((ix0 as u64) << 32 | ix1.0 as u64) |
240 | } |
241 | } |
242 | |
243 | #[cfg (test)] |
244 | mod tests { |
245 | use super::*; |
246 | use core::f64::*; |
247 | |
248 | #[test ] |
249 | fn sanity_check() { |
250 | assert_eq!(sqrt(100.0), 10.0); |
251 | assert_eq!(sqrt(4.0), 2.0); |
252 | } |
253 | |
254 | /// The spec: https://en.cppreference.com/w/cpp/numeric/math/sqrt |
255 | #[test ] |
256 | fn spec_tests() { |
257 | // Not Asserted: FE_INVALID exception is raised if argument is negative. |
258 | assert!(sqrt(-1.0).is_nan()); |
259 | assert!(sqrt(NAN).is_nan()); |
260 | for f in [0.0, -0.0, INFINITY].iter().copied() { |
261 | assert_eq!(sqrt(f), f); |
262 | } |
263 | } |
264 | |
265 | #[test ] |
266 | fn conformance_tests() { |
267 | let values = [3.14159265359, 10000.0, f64::from_bits(0x0000000f), INFINITY]; |
268 | let results = [ |
269 | 4610661241675116657u64, |
270 | 4636737291354636288u64, |
271 | 2197470602079456986u64, |
272 | 9218868437227405312u64, |
273 | ]; |
274 | |
275 | for i in 0..values.len() { |
276 | let bits = f64::to_bits(sqrt(values[i])); |
277 | assert_eq!(results[i], bits); |
278 | } |
279 | } |
280 | } |
281 | |