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