libm/math/sqrt.rs
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
79use core::f64;
80
81#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
82pub 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)]
244mod 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}