1239 lines
38 KiB
Rust
1239 lines
38 KiB
Rust
//! A small number of math routines for floats and doubles.
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//!
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//! These are adapted from libm, a port of musl libc's libm to Rust.
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//! libm can be found online [here](https://github.com/rust-lang/libm),
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//! and is similarly licensed under an Apache2.0/MIT license
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#![cfg(all(not(feature = "std"), feature = "compact"))]
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#![doc(hidden)]
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/* origin: FreeBSD /usr/src/lib/msun/src/e_powf.c */
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/*
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* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
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*/
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/// # Safety
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///
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/// Safe if `index < array.len()`.
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macro_rules! i {
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($array:ident, $index:expr) => {
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// SAFETY: safe if `index < array.len()`.
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unsafe { *$array.get_unchecked($index) }
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};
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}
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pub fn powf(x: f32, y: f32) -> f32 {
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const BP: [f32; 2] = [1.0, 1.5];
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const DP_H: [f32; 2] = [0.0, 5.84960938e-01]; /* 0x3f15c000 */
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const DP_L: [f32; 2] = [0.0, 1.56322085e-06]; /* 0x35d1cfdc */
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const TWO24: f32 = 16777216.0; /* 0x4b800000 */
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const HUGE: f32 = 1.0e30;
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const TINY: f32 = 1.0e-30;
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const L1: f32 = 6.0000002384e-01; /* 0x3f19999a */
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const L2: f32 = 4.2857143283e-01; /* 0x3edb6db7 */
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const L3: f32 = 3.3333334327e-01; /* 0x3eaaaaab */
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const L4: f32 = 2.7272811532e-01; /* 0x3e8ba305 */
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const L5: f32 = 2.3066075146e-01; /* 0x3e6c3255 */
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const L6: f32 = 2.0697501302e-01; /* 0x3e53f142 */
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const P1: f32 = 1.6666667163e-01; /* 0x3e2aaaab */
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const P2: f32 = -2.7777778450e-03; /* 0xbb360b61 */
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const P3: f32 = 6.6137559770e-05; /* 0x388ab355 */
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const P4: f32 = -1.6533901999e-06; /* 0xb5ddea0e */
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const P5: f32 = 4.1381369442e-08; /* 0x3331bb4c */
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const LG2: f32 = 6.9314718246e-01; /* 0x3f317218 */
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const LG2_H: f32 = 6.93145752e-01; /* 0x3f317200 */
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const LG2_L: f32 = 1.42860654e-06; /* 0x35bfbe8c */
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const OVT: f32 = 4.2995665694e-08; /* -(128-log2(ovfl+.5ulp)) */
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const CP: f32 = 9.6179670095e-01; /* 0x3f76384f =2/(3ln2) */
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const CP_H: f32 = 9.6191406250e-01; /* 0x3f764000 =12b cp */
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const CP_L: f32 = -1.1736857402e-04; /* 0xb8f623c6 =tail of cp_h */
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const IVLN2: f32 = 1.4426950216e+00;
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const IVLN2_H: f32 = 1.4426879883e+00;
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const IVLN2_L: f32 = 7.0526075433e-06;
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let mut z: f32;
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let mut ax: f32;
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let z_h: f32;
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let z_l: f32;
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let mut p_h: f32;
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let mut p_l: f32;
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let y1: f32;
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let mut t1: f32;
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let t2: f32;
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let mut r: f32;
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let s: f32;
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let mut sn: f32;
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let mut t: f32;
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let mut u: f32;
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let mut v: f32;
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let mut w: f32;
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let i: i32;
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let mut j: i32;
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let mut k: i32;
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let mut yisint: i32;
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let mut n: i32;
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let hx: i32;
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let hy: i32;
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let mut ix: i32;
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let iy: i32;
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let mut is: i32;
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hx = x.to_bits() as i32;
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hy = y.to_bits() as i32;
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ix = hx & 0x7fffffff;
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iy = hy & 0x7fffffff;
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/* x**0 = 1, even if x is NaN */
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if iy == 0 {
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return 1.0;
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}
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/* 1**y = 1, even if y is NaN */
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if hx == 0x3f800000 {
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return 1.0;
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}
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/* NaN if either arg is NaN */
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if ix > 0x7f800000 || iy > 0x7f800000 {
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return x + y;
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}
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/* determine if y is an odd int when x < 0
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* yisint = 0 ... y is not an integer
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* yisint = 1 ... y is an odd int
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* yisint = 2 ... y is an even int
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*/
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yisint = 0;
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if hx < 0 {
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if iy >= 0x4b800000 {
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yisint = 2; /* even integer y */
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} else if iy >= 0x3f800000 {
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k = (iy >> 23) - 0x7f; /* exponent */
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j = iy >> (23 - k);
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if (j << (23 - k)) == iy {
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yisint = 2 - (j & 1);
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}
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}
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}
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/* special value of y */
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if iy == 0x7f800000 {
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/* y is +-inf */
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if ix == 0x3f800000 {
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/* (-1)**+-inf is 1 */
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return 1.0;
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} else if ix > 0x3f800000 {
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/* (|x|>1)**+-inf = inf,0 */
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return if hy >= 0 {
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y
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} else {
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0.0
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};
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} else {
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/* (|x|<1)**+-inf = 0,inf */
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return if hy >= 0 {
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0.0
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} else {
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-y
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};
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}
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}
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if iy == 0x3f800000 {
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/* y is +-1 */
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return if hy >= 0 {
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x
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} else {
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1.0 / x
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};
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}
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if hy == 0x40000000 {
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/* y is 2 */
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return x * x;
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}
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if hy == 0x3f000000
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/* y is 0.5 */
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&& hx >= 0
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{
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/* x >= +0 */
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return sqrtf(x);
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}
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ax = fabsf(x);
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/* special value of x */
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if ix == 0x7f800000 || ix == 0 || ix == 0x3f800000 {
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/* x is +-0,+-inf,+-1 */
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z = ax;
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if hy < 0 {
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/* z = (1/|x|) */
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z = 1.0 / z;
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}
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if hx < 0 {
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if ((ix - 0x3f800000) | yisint) == 0 {
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z = (z - z) / (z - z); /* (-1)**non-int is NaN */
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} else if yisint == 1 {
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z = -z; /* (x<0)**odd = -(|x|**odd) */
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}
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}
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return z;
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}
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sn = 1.0; /* sign of result */
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if hx < 0 {
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if yisint == 0 {
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/* (x<0)**(non-int) is NaN */
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return (x - x) / (x - x);
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}
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if yisint == 1 {
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/* (x<0)**(odd int) */
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sn = -1.0;
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}
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}
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/* |y| is HUGE */
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if iy > 0x4d000000 {
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/* if |y| > 2**27 */
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/* over/underflow if x is not close to one */
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if ix < 0x3f7ffff8 {
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return if hy < 0 {
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sn * HUGE * HUGE
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} else {
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sn * TINY * TINY
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};
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}
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if ix > 0x3f800007 {
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return if hy > 0 {
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sn * HUGE * HUGE
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} else {
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sn * TINY * TINY
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};
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}
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/* now |1-x| is TINY <= 2**-20, suffice to compute
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log(x) by x-x^2/2+x^3/3-x^4/4 */
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t = ax - 1.; /* t has 20 trailing zeros */
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w = (t * t) * (0.5 - t * (0.333333333333 - t * 0.25));
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u = IVLN2_H * t; /* IVLN2_H has 16 sig. bits */
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v = t * IVLN2_L - w * IVLN2;
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t1 = u + v;
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is = t1.to_bits() as i32;
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t1 = f32::from_bits(is as u32 & 0xfffff000);
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t2 = v - (t1 - u);
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} else {
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let mut s2: f32;
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let mut s_h: f32;
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let s_l: f32;
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let mut t_h: f32;
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let mut t_l: f32;
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n = 0;
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/* take care subnormal number */
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if ix < 0x00800000 {
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ax *= TWO24;
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n -= 24;
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ix = ax.to_bits() as i32;
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}
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n += ((ix) >> 23) - 0x7f;
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j = ix & 0x007fffff;
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/* determine interval */
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ix = j | 0x3f800000; /* normalize ix */
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if j <= 0x1cc471 {
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/* |x|<sqrt(3/2) */
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k = 0;
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} else if j < 0x5db3d7 {
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/* |x|<sqrt(3) */
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k = 1;
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} else {
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k = 0;
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n += 1;
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ix -= 0x00800000;
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}
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ax = f32::from_bits(ix as u32);
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/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
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u = ax - i!(BP, k as usize); /* bp[0]=1.0, bp[1]=1.5 */
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v = 1.0 / (ax + i!(BP, k as usize));
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s = u * v;
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s_h = s;
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is = s_h.to_bits() as i32;
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s_h = f32::from_bits(is as u32 & 0xfffff000);
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/* t_h=ax+bp[k] High */
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is = (((ix as u32 >> 1) & 0xfffff000) | 0x20000000) as i32;
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t_h = f32::from_bits(is as u32 + 0x00400000 + ((k as u32) << 21));
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t_l = ax - (t_h - i!(BP, k as usize));
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s_l = v * ((u - s_h * t_h) - s_h * t_l);
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/* compute log(ax) */
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s2 = s * s;
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r = s2 * s2 * (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
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r += s_l * (s_h + s);
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s2 = s_h * s_h;
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t_h = 3.0 + s2 + r;
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is = t_h.to_bits() as i32;
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t_h = f32::from_bits(is as u32 & 0xfffff000);
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t_l = r - ((t_h - 3.0) - s2);
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/* u+v = s*(1+...) */
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u = s_h * t_h;
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v = s_l * t_h + t_l * s;
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/* 2/(3log2)*(s+...) */
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p_h = u + v;
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is = p_h.to_bits() as i32;
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p_h = f32::from_bits(is as u32 & 0xfffff000);
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p_l = v - (p_h - u);
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z_h = CP_H * p_h; /* cp_h+cp_l = 2/(3*log2) */
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z_l = CP_L * p_h + p_l * CP + i!(DP_L, k as usize);
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/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
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t = n as f32;
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t1 = ((z_h + z_l) + i!(DP_H, k as usize)) + t;
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is = t1.to_bits() as i32;
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t1 = f32::from_bits(is as u32 & 0xfffff000);
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t2 = z_l - (((t1 - t) - i!(DP_H, k as usize)) - z_h);
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};
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/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
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is = y.to_bits() as i32;
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y1 = f32::from_bits(is as u32 & 0xfffff000);
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p_l = (y - y1) * t1 + y * t2;
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p_h = y1 * t1;
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z = p_l + p_h;
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j = z.to_bits() as i32;
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if j > 0x43000000 {
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/* if z > 128 */
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return sn * HUGE * HUGE; /* overflow */
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} else if j == 0x43000000 {
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/* if z == 128 */
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if p_l + OVT > z - p_h {
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return sn * HUGE * HUGE; /* overflow */
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}
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} else if (j & 0x7fffffff) > 0x43160000 {
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/* z < -150 */
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// FIXME: check should be (uint32_t)j > 0xc3160000
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return sn * TINY * TINY; /* underflow */
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} else if j as u32 == 0xc3160000
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/* z == -150 */
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&& p_l <= z - p_h
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{
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return sn * TINY * TINY; /* underflow */
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}
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/*
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* compute 2**(p_h+p_l)
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*/
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i = j & 0x7fffffff;
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k = (i >> 23) - 0x7f;
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n = 0;
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if i > 0x3f000000 {
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/* if |z| > 0.5, set n = [z+0.5] */
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n = j + (0x00800000 >> (k + 1));
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k = ((n & 0x7fffffff) >> 23) - 0x7f; /* new k for n */
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t = f32::from_bits(n as u32 & !(0x007fffff >> k));
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n = ((n & 0x007fffff) | 0x00800000) >> (23 - k);
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if j < 0 {
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n = -n;
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}
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p_h -= t;
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}
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t = p_l + p_h;
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is = t.to_bits() as i32;
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t = f32::from_bits(is as u32 & 0xffff8000);
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u = t * LG2_H;
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v = (p_l - (t - p_h)) * LG2 + t * LG2_L;
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z = u + v;
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w = v - (z - u);
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t = z * z;
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t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
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r = (z * t1) / (t1 - 2.0) - (w + z * w);
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z = 1.0 - (r - z);
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j = z.to_bits() as i32;
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j += n << 23;
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if (j >> 23) <= 0 {
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/* subnormal output */
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z = scalbnf(z, n);
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} else {
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z = f32::from_bits(j as u32);
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}
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sn * z
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}
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|
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/* origin: FreeBSD /usr/src/lib/msun/src/e_sqrtf.c */
|
|
/*
|
|
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
|
*/
|
|
/*
|
|
* ====================================================
|
|
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
|
*
|
|
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
|
* Permission to use, copy, modify, and distribute this
|
|
* software is freely granted, provided that this notice
|
|
* is preserved.
|
|
* ====================================================
|
|
*/
|
|
|
|
pub fn sqrtf(x: f32) -> f32 {
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#[cfg(target_feature = "sse")]
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{
|
|
// Note: This path is unlikely since LLVM will usually have already
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// optimized sqrt calls into hardware instructions if sse is available,
|
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// but if someone does end up here they'll apprected the speed increase.
|
|
#[cfg(target_arch = "x86")]
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use core::arch::x86::*;
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#[cfg(target_arch = "x86_64")]
|
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use core::arch::x86_64::*;
|
|
// SAFETY: safe, since `_mm_set_ss` takes a 32-bit float, and returns
|
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// a 128-bit type with the lowest 32-bits as `x`, `_mm_sqrt_ss` calculates
|
|
// the sqrt of this 128-bit vector, and `_mm_cvtss_f32` extracts the lower
|
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// 32-bits as a 32-bit float.
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unsafe {
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let m = _mm_set_ss(x);
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let m_sqrt = _mm_sqrt_ss(m);
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_mm_cvtss_f32(m_sqrt)
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}
|
|
}
|
|
#[cfg(not(target_feature = "sse"))]
|
|
{
|
|
const TINY: f32 = 1.0e-30;
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|
|
|
let mut z: f32;
|
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let sign: i32 = 0x80000000u32 as i32;
|
|
let mut ix: i32;
|
|
let mut s: i32;
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|
let mut q: i32;
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|
let mut m: i32;
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|
let mut t: i32;
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|
let mut i: i32;
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|
let mut r: u32;
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|
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ix = x.to_bits() as i32;
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|
|
/* take care of Inf and NaN */
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if (ix as u32 & 0x7f800000) == 0x7f800000 {
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return x * x + x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf, sqrt(-inf)=sNaN */
|
|
}
|
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|
|
/* take care of zero */
|
|
if ix <= 0 {
|
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if (ix & !sign) == 0 {
|
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return x; /* sqrt(+-0) = +-0 */
|
|
}
|
|
if ix < 0 {
|
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return (x - x) / (x - x); /* sqrt(-ve) = sNaN */
|
|
}
|
|
}
|
|
|
|
/* normalize x */
|
|
m = ix >> 23;
|
|
if m == 0 {
|
|
/* subnormal x */
|
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i = 0;
|
|
while ix & 0x00800000 == 0 {
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|
ix <<= 1;
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i = i + 1;
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}
|
|
m -= i - 1;
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}
|
|
m -= 127; /* unbias exponent */
|
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ix = (ix & 0x007fffff) | 0x00800000;
|
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if m & 1 == 1 {
|
|
/* odd m, double x to make it even */
|
|
ix += ix;
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}
|
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m >>= 1; /* m = [m/2] */
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|
|
|
/* generate sqrt(x) bit by bit */
|
|
ix += ix;
|
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q = 0;
|
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s = 0;
|
|
r = 0x01000000; /* r = moving bit from right to left */
|
|
|
|
while r != 0 {
|
|
t = s + r as i32;
|
|
if t <= ix {
|
|
s = t + r as i32;
|
|
ix -= t;
|
|
q += r as i32;
|
|
}
|
|
ix += ix;
|
|
r >>= 1;
|
|
}
|
|
|
|
/* use floating add to find out rounding direction */
|
|
if ix != 0 {
|
|
z = 1.0 - TINY; /* raise inexact flag */
|
|
if z >= 1.0 {
|
|
z = 1.0 + TINY;
|
|
if z > 1.0 {
|
|
q += 2;
|
|
} else {
|
|
q += q & 1;
|
|
}
|
|
}
|
|
}
|
|
|
|
ix = (q >> 1) + 0x3f000000;
|
|
ix += m << 23;
|
|
f32::from_bits(ix as u32)
|
|
}
|
|
}
|
|
|
|
/// Absolute value (magnitude) (f32)
|
|
/// Calculates the absolute value (magnitude) of the argument `x`,
|
|
/// by direct manipulation of the bit representation of `x`.
|
|
pub fn fabsf(x: f32) -> f32 {
|
|
f32::from_bits(x.to_bits() & 0x7fffffff)
|
|
}
|
|
|
|
pub fn scalbnf(mut x: f32, mut n: i32) -> f32 {
|
|
let x1p127 = f32::from_bits(0x7f000000); // 0x1p127f === 2 ^ 127
|
|
let x1p_126 = f32::from_bits(0x800000); // 0x1p-126f === 2 ^ -126
|
|
let x1p24 = f32::from_bits(0x4b800000); // 0x1p24f === 2 ^ 24
|
|
|
|
if n > 127 {
|
|
x *= x1p127;
|
|
n -= 127;
|
|
if n > 127 {
|
|
x *= x1p127;
|
|
n -= 127;
|
|
if n > 127 {
|
|
n = 127;
|
|
}
|
|
}
|
|
} else if n < -126 {
|
|
x *= x1p_126 * x1p24;
|
|
n += 126 - 24;
|
|
if n < -126 {
|
|
x *= x1p_126 * x1p24;
|
|
n += 126 - 24;
|
|
if n < -126 {
|
|
n = -126;
|
|
}
|
|
}
|
|
}
|
|
x * f32::from_bits(((0x7f + n) as u32) << 23)
|
|
}
|
|
|
|
/* origin: FreeBSD /usr/src/lib/msun/src/e_pow.c */
|
|
/*
|
|
* ====================================================
|
|
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
|
|
*
|
|
* Permission to use, copy, modify, and distribute this
|
|
* software is freely granted, provided that this notice
|
|
* is preserved.
|
|
* ====================================================
|
|
*/
|
|
|
|
// pow(x,y) return x**y
|
|
//
|
|
// n
|
|
// Method: Let x = 2 * (1+f)
|
|
// 1. Compute and return log2(x) in two pieces:
|
|
// log2(x) = w1 + w2,
|
|
// where w1 has 53-24 = 29 bit trailing zeros.
|
|
// 2. Perform y*log2(x) = n+y' by simulating muti-precision
|
|
// arithmetic, where |y'|<=0.5.
|
|
// 3. Return x**y = 2**n*exp(y'*log2)
|
|
//
|
|
// Special cases:
|
|
// 1. (anything) ** 0 is 1
|
|
// 2. 1 ** (anything) is 1
|
|
// 3. (anything except 1) ** NAN is NAN
|
|
// 4. NAN ** (anything except 0) is NAN
|
|
// 5. +-(|x| > 1) ** +INF is +INF
|
|
// 6. +-(|x| > 1) ** -INF is +0
|
|
// 7. +-(|x| < 1) ** +INF is +0
|
|
// 8. +-(|x| < 1) ** -INF is +INF
|
|
// 9. -1 ** +-INF is 1
|
|
// 10. +0 ** (+anything except 0, NAN) is +0
|
|
// 11. -0 ** (+anything except 0, NAN, odd integer) is +0
|
|
// 12. +0 ** (-anything except 0, NAN) is +INF, raise divbyzero
|
|
// 13. -0 ** (-anything except 0, NAN, odd integer) is +INF, raise divbyzero
|
|
// 14. -0 ** (+odd integer) is -0
|
|
// 15. -0 ** (-odd integer) is -INF, raise divbyzero
|
|
// 16. +INF ** (+anything except 0,NAN) is +INF
|
|
// 17. +INF ** (-anything except 0,NAN) is +0
|
|
// 18. -INF ** (+odd integer) is -INF
|
|
// 19. -INF ** (anything) = -0 ** (-anything), (anything except odd integer)
|
|
// 20. (anything) ** 1 is (anything)
|
|
// 21. (anything) ** -1 is 1/(anything)
|
|
// 22. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
|
|
// 23. (-anything except 0 and inf) ** (non-integer) is NAN
|
|
//
|
|
// Accuracy:
|
|
// pow(x,y) returns x**y nearly rounded. In particular
|
|
// pow(integer,integer)
|
|
// always returns the correct integer provided it is
|
|
// representable.
|
|
//
|
|
// Constants :
|
|
// The hexadecimal values are the intended ones for the following
|
|
// constants. The decimal values may be used, provided that the
|
|
// compiler will convert from decimal to binary accurately enough
|
|
// to produce the hexadecimal values shown.
|
|
|
|
pub fn powd(x: f64, y: f64) -> f64 {
|
|
const BP: [f64; 2] = [1.0, 1.5];
|
|
const DP_H: [f64; 2] = [0.0, 5.84962487220764160156e-01]; /* 0x3fe2b803_40000000 */
|
|
const DP_L: [f64; 2] = [0.0, 1.35003920212974897128e-08]; /* 0x3E4CFDEB, 0x43CFD006 */
|
|
const TWO53: f64 = 9007199254740992.0; /* 0x43400000_00000000 */
|
|
const HUGE: f64 = 1.0e300;
|
|
const TINY: f64 = 1.0e-300;
|
|
|
|
// poly coefs for (3/2)*(log(x)-2s-2/3*s**3:
|
|
const L1: f64 = 5.99999999999994648725e-01; /* 0x3fe33333_33333303 */
|
|
const L2: f64 = 4.28571428578550184252e-01; /* 0x3fdb6db6_db6fabff */
|
|
const L3: f64 = 3.33333329818377432918e-01; /* 0x3fd55555_518f264d */
|
|
const L4: f64 = 2.72728123808534006489e-01; /* 0x3fd17460_a91d4101 */
|
|
const L5: f64 = 2.30660745775561754067e-01; /* 0x3fcd864a_93c9db65 */
|
|
const L6: f64 = 2.06975017800338417784e-01; /* 0x3fca7e28_4a454eef */
|
|
const P1: f64 = 1.66666666666666019037e-01; /* 0x3fc55555_5555553e */
|
|
const P2: f64 = -2.77777777770155933842e-03; /* 0xbf66c16c_16bebd93 */
|
|
const P3: f64 = 6.61375632143793436117e-05; /* 0x3f11566a_af25de2c */
|
|
const P4: f64 = -1.65339022054652515390e-06; /* 0xbebbbd41_c5d26bf1 */
|
|
const P5: f64 = 4.13813679705723846039e-08; /* 0x3e663769_72bea4d0 */
|
|
const LG2: f64 = 6.93147180559945286227e-01; /* 0x3fe62e42_fefa39ef */
|
|
const LG2_H: f64 = 6.93147182464599609375e-01; /* 0x3fe62e43_00000000 */
|
|
const LG2_L: f64 = -1.90465429995776804525e-09; /* 0xbe205c61_0ca86c39 */
|
|
const OVT: f64 = 8.0085662595372944372e-017; /* -(1024-log2(ovfl+.5ulp)) */
|
|
const CP: f64 = 9.61796693925975554329e-01; /* 0x3feec709_dc3a03fd =2/(3ln2) */
|
|
const CP_H: f64 = 9.61796700954437255859e-01; /* 0x3feec709_e0000000 =(float)cp */
|
|
const CP_L: f64 = -7.02846165095275826516e-09; /* 0xbe3e2fe0_145b01f5 =tail of cp_h*/
|
|
const IVLN2: f64 = 1.44269504088896338700e+00; /* 0x3ff71547_652b82fe =1/ln2 */
|
|
const IVLN2_H: f64 = 1.44269502162933349609e+00; /* 0x3ff71547_60000000 =24b 1/ln2*/
|
|
const IVLN2_L: f64 = 1.92596299112661746887e-08; /* 0x3e54ae0b_f85ddf44 =1/ln2 tail*/
|
|
|
|
let t1: f64;
|
|
let t2: f64;
|
|
|
|
let (hx, lx): (i32, u32) = ((x.to_bits() >> 32) as i32, x.to_bits() as u32);
|
|
let (hy, ly): (i32, u32) = ((y.to_bits() >> 32) as i32, y.to_bits() as u32);
|
|
|
|
let mut ix: i32 = (hx & 0x7fffffff) as i32;
|
|
let iy: i32 = (hy & 0x7fffffff) as i32;
|
|
|
|
/* x**0 = 1, even if x is NaN */
|
|
if ((iy as u32) | ly) == 0 {
|
|
return 1.0;
|
|
}
|
|
|
|
/* 1**y = 1, even if y is NaN */
|
|
if hx == 0x3ff00000 && lx == 0 {
|
|
return 1.0;
|
|
}
|
|
|
|
/* NaN if either arg is NaN */
|
|
if ix > 0x7ff00000
|
|
|| (ix == 0x7ff00000 && lx != 0)
|
|
|| iy > 0x7ff00000
|
|
|| (iy == 0x7ff00000 && ly != 0)
|
|
{
|
|
return x + y;
|
|
}
|
|
|
|
/* determine if y is an odd int when x < 0
|
|
* yisint = 0 ... y is not an integer
|
|
* yisint = 1 ... y is an odd int
|
|
* yisint = 2 ... y is an even int
|
|
*/
|
|
let mut yisint: i32 = 0;
|
|
let mut k: i32;
|
|
let mut j: i32;
|
|
if hx < 0 {
|
|
if iy >= 0x43400000 {
|
|
yisint = 2; /* even integer y */
|
|
} else if iy >= 0x3ff00000 {
|
|
k = (iy >> 20) - 0x3ff; /* exponent */
|
|
|
|
if k > 20 {
|
|
j = (ly >> (52 - k)) as i32;
|
|
|
|
if (j << (52 - k)) == (ly as i32) {
|
|
yisint = 2 - (j & 1);
|
|
}
|
|
} else if ly == 0 {
|
|
j = iy >> (20 - k);
|
|
|
|
if (j << (20 - k)) == iy {
|
|
yisint = 2 - (j & 1);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
if ly == 0 {
|
|
/* special value of y */
|
|
if iy == 0x7ff00000 {
|
|
/* y is +-inf */
|
|
|
|
return if ((ix - 0x3ff00000) | (lx as i32)) == 0 {
|
|
/* (-1)**+-inf is 1 */
|
|
1.0
|
|
} else if ix >= 0x3ff00000 {
|
|
/* (|x|>1)**+-inf = inf,0 */
|
|
if hy >= 0 {
|
|
y
|
|
} else {
|
|
0.0
|
|
}
|
|
} else {
|
|
/* (|x|<1)**+-inf = 0,inf */
|
|
if hy >= 0 {
|
|
0.0
|
|
} else {
|
|
-y
|
|
}
|
|
};
|
|
}
|
|
|
|
if iy == 0x3ff00000 {
|
|
/* y is +-1 */
|
|
return if hy >= 0 {
|
|
x
|
|
} else {
|
|
1.0 / x
|
|
};
|
|
}
|
|
|
|
if hy == 0x40000000 {
|
|
/* y is 2 */
|
|
return x * x;
|
|
}
|
|
|
|
if hy == 0x3fe00000 {
|
|
/* y is 0.5 */
|
|
if hx >= 0 {
|
|
/* x >= +0 */
|
|
return sqrtd(x);
|
|
}
|
|
}
|
|
}
|
|
|
|
let mut ax: f64 = fabsd(x);
|
|
if lx == 0 {
|
|
/* special value of x */
|
|
if ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000 {
|
|
/* x is +-0,+-inf,+-1 */
|
|
let mut z: f64 = ax;
|
|
|
|
if hy < 0 {
|
|
/* z = (1/|x|) */
|
|
z = 1.0 / z;
|
|
}
|
|
|
|
if hx < 0 {
|
|
if ((ix - 0x3ff00000) | yisint) == 0 {
|
|
z = (z - z) / (z - z); /* (-1)**non-int is NaN */
|
|
} else if yisint == 1 {
|
|
z = -z; /* (x<0)**odd = -(|x|**odd) */
|
|
}
|
|
}
|
|
|
|
return z;
|
|
}
|
|
}
|
|
|
|
let mut s: f64 = 1.0; /* sign of result */
|
|
if hx < 0 {
|
|
if yisint == 0 {
|
|
/* (x<0)**(non-int) is NaN */
|
|
return (x - x) / (x - x);
|
|
}
|
|
|
|
if yisint == 1 {
|
|
/* (x<0)**(odd int) */
|
|
s = -1.0;
|
|
}
|
|
}
|
|
|
|
/* |y| is HUGE */
|
|
if iy > 0x41e00000 {
|
|
/* if |y| > 2**31 */
|
|
if iy > 0x43f00000 {
|
|
/* if |y| > 2**64, must o/uflow */
|
|
if ix <= 0x3fefffff {
|
|
return if hy < 0 {
|
|
HUGE * HUGE
|
|
} else {
|
|
TINY * TINY
|
|
};
|
|
}
|
|
|
|
if ix >= 0x3ff00000 {
|
|
return if hy > 0 {
|
|
HUGE * HUGE
|
|
} else {
|
|
TINY * TINY
|
|
};
|
|
}
|
|
}
|
|
|
|
/* over/underflow if x is not close to one */
|
|
if ix < 0x3fefffff {
|
|
return if hy < 0 {
|
|
s * HUGE * HUGE
|
|
} else {
|
|
s * TINY * TINY
|
|
};
|
|
}
|
|
if ix > 0x3ff00000 {
|
|
return if hy > 0 {
|
|
s * HUGE * HUGE
|
|
} else {
|
|
s * TINY * TINY
|
|
};
|
|
}
|
|
|
|
/* now |1-x| is TINY <= 2**-20, suffice to compute
|
|
log(x) by x-x^2/2+x^3/3-x^4/4 */
|
|
let t: f64 = ax - 1.0; /* t has 20 trailing zeros */
|
|
let w: f64 = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
|
|
let u: f64 = IVLN2_H * t; /* ivln2_h has 21 sig. bits */
|
|
let v: f64 = t * IVLN2_L - w * IVLN2;
|
|
t1 = with_set_low_word(u + v, 0);
|
|
t2 = v - (t1 - u);
|
|
} else {
|
|
// double ss,s2,s_h,s_l,t_h,t_l;
|
|
let mut n: i32 = 0;
|
|
|
|
if ix < 0x00100000 {
|
|
/* take care subnormal number */
|
|
ax *= TWO53;
|
|
n -= 53;
|
|
ix = get_high_word(ax) as i32;
|
|
}
|
|
|
|
n += (ix >> 20) - 0x3ff;
|
|
j = ix & 0x000fffff;
|
|
|
|
/* determine interval */
|
|
let k: i32;
|
|
ix = j | 0x3ff00000; /* normalize ix */
|
|
if j <= 0x3988E {
|
|
/* |x|<sqrt(3/2) */
|
|
k = 0;
|
|
} else if j < 0xBB67A {
|
|
/* |x|<sqrt(3) */
|
|
k = 1;
|
|
} else {
|
|
k = 0;
|
|
n += 1;
|
|
ix -= 0x00100000;
|
|
}
|
|
ax = with_set_high_word(ax, ix as u32);
|
|
|
|
/* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
|
|
let u: f64 = ax - i!(BP, k as usize); /* bp[0]=1.0, bp[1]=1.5 */
|
|
let v: f64 = 1.0 / (ax + i!(BP, k as usize));
|
|
let ss: f64 = u * v;
|
|
let s_h = with_set_low_word(ss, 0);
|
|
|
|
/* t_h=ax+bp[k] High */
|
|
let t_h: f64 = with_set_high_word(
|
|
0.0,
|
|
((ix as u32 >> 1) | 0x20000000) + 0x00080000 + ((k as u32) << 18),
|
|
);
|
|
let t_l: f64 = ax - (t_h - i!(BP, k as usize));
|
|
let s_l: f64 = v * ((u - s_h * t_h) - s_h * t_l);
|
|
|
|
/* compute log(ax) */
|
|
let s2: f64 = ss * ss;
|
|
let mut r: f64 = s2 * s2 * (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
|
|
r += s_l * (s_h + ss);
|
|
let s2: f64 = s_h * s_h;
|
|
let t_h: f64 = with_set_low_word(3.0 + s2 + r, 0);
|
|
let t_l: f64 = r - ((t_h - 3.0) - s2);
|
|
|
|
/* u+v = ss*(1+...) */
|
|
let u: f64 = s_h * t_h;
|
|
let v: f64 = s_l * t_h + t_l * ss;
|
|
|
|
/* 2/(3log2)*(ss+...) */
|
|
let p_h: f64 = with_set_low_word(u + v, 0);
|
|
let p_l = v - (p_h - u);
|
|
let z_h: f64 = CP_H * p_h; /* cp_h+cp_l = 2/(3*log2) */
|
|
let z_l: f64 = CP_L * p_h + p_l * CP + i!(DP_L, k as usize);
|
|
|
|
/* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
|
|
let t: f64 = n as f64;
|
|
t1 = with_set_low_word(((z_h + z_l) + i!(DP_H, k as usize)) + t, 0);
|
|
t2 = z_l - (((t1 - t) - i!(DP_H, k as usize)) - z_h);
|
|
}
|
|
|
|
/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
|
|
let y1: f64 = with_set_low_word(y, 0);
|
|
let p_l: f64 = (y - y1) * t1 + y * t2;
|
|
let mut p_h: f64 = y1 * t1;
|
|
let z: f64 = p_l + p_h;
|
|
let mut j: i32 = (z.to_bits() >> 32) as i32;
|
|
let i: i32 = z.to_bits() as i32;
|
|
// let (j, i): (i32, i32) = ((z.to_bits() >> 32) as i32, z.to_bits() as i32);
|
|
|
|
if j >= 0x40900000 {
|
|
/* z >= 1024 */
|
|
if (j - 0x40900000) | i != 0 {
|
|
/* if z > 1024 */
|
|
return s * HUGE * HUGE; /* overflow */
|
|
}
|
|
|
|
if p_l + OVT > z - p_h {
|
|
return s * HUGE * HUGE; /* overflow */
|
|
}
|
|
} else if (j & 0x7fffffff) >= 0x4090cc00 {
|
|
/* z <= -1075 */
|
|
// FIXME: instead of abs(j) use unsigned j
|
|
|
|
if (((j as u32) - 0xc090cc00) | (i as u32)) != 0 {
|
|
/* z < -1075 */
|
|
return s * TINY * TINY; /* underflow */
|
|
}
|
|
|
|
if p_l <= z - p_h {
|
|
return s * TINY * TINY; /* underflow */
|
|
}
|
|
}
|
|
|
|
/* compute 2**(p_h+p_l) */
|
|
let i: i32 = j & (0x7fffffff as i32);
|
|
k = (i >> 20) - 0x3ff;
|
|
let mut n: i32 = 0;
|
|
|
|
if i > 0x3fe00000 {
|
|
/* if |z| > 0.5, set n = [z+0.5] */
|
|
n = j + (0x00100000 >> (k + 1));
|
|
k = ((n & 0x7fffffff) >> 20) - 0x3ff; /* new k for n */
|
|
let t: f64 = with_set_high_word(0.0, (n & !(0x000fffff >> k)) as u32);
|
|
n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
|
|
if j < 0 {
|
|
n = -n;
|
|
}
|
|
p_h -= t;
|
|
}
|
|
|
|
let t: f64 = with_set_low_word(p_l + p_h, 0);
|
|
let u: f64 = t * LG2_H;
|
|
let v: f64 = (p_l - (t - p_h)) * LG2 + t * LG2_L;
|
|
let mut z: f64 = u + v;
|
|
let w: f64 = v - (z - u);
|
|
let t: f64 = z * z;
|
|
let t1: f64 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
|
|
let r: f64 = (z * t1) / (t1 - 2.0) - (w + z * w);
|
|
z = 1.0 - (r - z);
|
|
j = get_high_word(z) as i32;
|
|
j += n << 20;
|
|
|
|
if (j >> 20) <= 0 {
|
|
/* subnormal output */
|
|
z = scalbnd(z, n);
|
|
} else {
|
|
z = with_set_high_word(z, j as u32);
|
|
}
|
|
|
|
s * z
|
|
}
|
|
|
|
/// Absolute value (magnitude) (f64)
|
|
/// Calculates the absolute value (magnitude) of the argument `x`,
|
|
/// by direct manipulation of the bit representation of `x`.
|
|
pub fn fabsd(x: f64) -> f64 {
|
|
f64::from_bits(x.to_bits() & (u64::MAX / 2))
|
|
}
|
|
|
|
pub fn scalbnd(x: f64, mut n: i32) -> f64 {
|
|
let x1p1023 = f64::from_bits(0x7fe0000000000000); // 0x1p1023 === 2 ^ 1023
|
|
let x1p53 = f64::from_bits(0x4340000000000000); // 0x1p53 === 2 ^ 53
|
|
let x1p_1022 = f64::from_bits(0x0010000000000000); // 0x1p-1022 === 2 ^ (-1022)
|
|
|
|
let mut y = x;
|
|
|
|
if n > 1023 {
|
|
y *= x1p1023;
|
|
n -= 1023;
|
|
if n > 1023 {
|
|
y *= x1p1023;
|
|
n -= 1023;
|
|
if n > 1023 {
|
|
n = 1023;
|
|
}
|
|
}
|
|
} else if n < -1022 {
|
|
/* make sure final n < -53 to avoid double
|
|
rounding in the subnormal range */
|
|
y *= x1p_1022 * x1p53;
|
|
n += 1022 - 53;
|
|
if n < -1022 {
|
|
y *= x1p_1022 * x1p53;
|
|
n += 1022 - 53;
|
|
if n < -1022 {
|
|
n = -1022;
|
|
}
|
|
}
|
|
}
|
|
y * f64::from_bits(((0x3ff + n) as u64) << 52)
|
|
}
|
|
|
|
/* 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
|
|
*/
|
|
|
|
pub fn sqrtd(x: f64) -> f64 {
|
|
#[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::*;
|
|
// SAFETY: safe, since `_mm_set_sd` takes a 64-bit float, and returns
|
|
// a 128-bit type with the lowest 64-bits as `x`, `_mm_sqrt_ss` calculates
|
|
// the sqrt of this 128-bit vector, and `_mm_cvtss_f64` extracts the lower
|
|
// 64-bits as a 64-bit float.
|
|
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)
|
|
}
|
|
}
|
|
|
|
#[inline]
|
|
fn get_high_word(x: f64) -> u32 {
|
|
(x.to_bits() >> 32) as u32
|
|
}
|
|
|
|
#[inline]
|
|
fn with_set_high_word(f: f64, hi: u32) -> f64 {
|
|
let mut tmp = f.to_bits();
|
|
tmp &= 0x00000000_ffffffff;
|
|
tmp |= (hi as u64) << 32;
|
|
f64::from_bits(tmp)
|
|
}
|
|
|
|
#[inline]
|
|
fn with_set_low_word(f: f64, lo: u32) -> f64 {
|
|
let mut tmp = f.to_bits();
|
|
tmp &= 0xffffffff_00000000;
|
|
tmp |= lo as u64;
|
|
f64::from_bits(tmp)
|
|
}
|