//! A small number of math routines for floats and doubles.
//!
//! These are adapted from libm, a port of musl libc's libm to Rust.
//! libm can be found online [here](
https://github.com/rust-lang/libm),
//! and is similarly licensed under an Apache2.0/MIT license
#![cfg(all(not(feature = "std"), feature = "compact"))]
#![doc(hidden)]
/* origin: FreeBSD /usr/src/lib/msun/src/e_powf.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.
* ====================================================
*/
/// # Safety
///
/// Safe if `index < array.len()`.
macro_rules! i {
($array:ident, $index:expr) => {
// SAFETY: safe if `index < array.len()`.
unsafe { *$array.get_unchecked($index) }
};
}
pub fn powf(x: f32, y: f32) -> f32 {
const BP: [f32; 2] = [1.0, 1.5];
const DP_H: [f32; 2] = [0.0, 5.84960938e-01]; /* 0x3f15c000 */
const DP_L: [f32; 2] = [0.0, 1.56322085e-06]; /* 0x35d1cfdc */
const TWO24: f32 = 16777216.0; /* 0x4b800000 */
const HUGE: f32 = 1.0e30;
const TINY: f32 = 1.0e-30;
const L1: f32 = 6.0000002384e-01; /* 0x3f19999a */
const L2: f32 = 4.2857143283e-01; /* 0x3edb6db7 */
const L3: f32 = 3.3333334327e-01; /* 0x3eaaaaab */
const L4: f32 = 2.7272811532e-01; /* 0x3e8ba305 */
const L5: f32 = 2.3066075146e-01; /* 0x3e6c3255 */
const L6: f32 = 2.0697501302e-01; /* 0x3e53f142 */
const P1: f32 = 1.6666667163e-01; /* 0x3e2aaaab */
const P2: f32 = -2.7777778450e-03; /* 0xbb360b61 */
const P3: f32 = 6.6137559770e-05; /* 0x388ab355 */
const P4: f32 = -1.6533901999e-06; /* 0xb5ddea0e */
const P5: f32 = 4.1381369442e-08; /* 0x3331bb4c */
const LG2: f32 = 6.9314718246e-01; /* 0x3f317218 */
const LG2_H: f32 = 6.93145752e-01; /* 0x3f317200 */
const LG2_L: f32 = 1.42860654e-06; /* 0x35bfbe8c */
const OVT: f32 = 4.2995665694e-08; /* -(128-log2(ovfl+.5ulp)) */
const CP: f32 = 9.6179670095e-01; /* 0x3f76384f =2/(3ln2) */
const CP_H: f32 = 9.6191406250e-01; /* 0x3f764000 =12b cp */
const CP_L: f32 = -1.1736857402e-04; /* 0xb8f623c6 =tail of cp_h */
const IVLN2: f32 = 1.4426950216e+00;
const IVLN2_H: f32 = 1.4426879883e+00;
const IVLN2_L: f32 = 7.0526075433e-06;
let mut z: f32;
let mut ax: f32;
let z_h: f32;
let z_l: f32;
let mut p_h: f32;
let mut p_l: f32;
let y1: f32;
let mut t1: f32;
let t2: f32;
let mut r: f32;
let s: f32;
let mut sn: f32;
let mut t: f32;
let mut u: f32;
let mut v: f32;
let mut w: f32;
let i: i32;
let mut j: i32;
let mut k: i32;
let mut yisint: i32;
let mut n: i32;
let hx: i32;
let hy: i32;
let mut ix: i32;
let iy: i32;
let mut is: i32;
hx = x.to_bits() as i32;
hy = y.to_bits() as i32;
ix = hx & 0x7fffffff;
iy = hy & 0x7fffffff;
/* x**0 = 1, even if x is NaN */
if iy == 0 {
return 1.0;
}
/* 1**y = 1, even if y is NaN */
if hx == 0x3f800000 {
return 1.0;
}
/* NaN if either arg is NaN */
if ix > 0x7f800000 || iy > 0x7f800000 {
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
*/
yisint = 0;
if hx < 0 {
if iy >= 0x4b800000 {
yisint = 2; /* even integer y */
} else if iy >= 0x3f800000 {
k = (iy >> 23) - 0x7f; /* exponent */
j = iy >> (23 - k);
if (j << (23 - k)) == iy {
yisint = 2 - (j & 1);
}
}
}
/* special value of y */
if iy == 0x7f800000 {
/* y is +-inf */
if ix == 0x3f800000 {
/* (-1)**+-inf is 1 */
return 1.0;
} else if ix > 0x3f800000 {
/* (|x|>1)**+-inf = inf,0 */
return if hy >= 0 {
y
} else {
0.0
};
} else {
/* (|x|<1)**+-inf = 0,inf */
return if hy >= 0 {
0.0
} else {
-y
};
}
}
if iy == 0x3f800000 {
/* y is +-1 */
return if hy >= 0 {
x
} else {
1.0 / x
};
}
if hy == 0x40000000 {
/* y is 2 */
return x * x;
}
if hy == 0x3f000000
/* y is 0.5 */
&& hx >= 0
{
/* x >= +0 */
return sqrtf(x);
}
ax = fabsf(x);
/* special value of x */
if ix == 0x7f800000 || ix == 0 || ix == 0x3f800000 {
/* x is +-0,+-inf,+-1 */
z = ax;
if hy < 0 {
/* z = (1/|x|) */
z = 1.0 / z;
}
if hx < 0 {
if ((ix - 0x3f800000) | 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;
}
sn = 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) */
sn = -1.0;
}
}
/* |y| is HUGE */
if iy > 0x4d000000 {
/* if |y| > 2**27 */
/* over/underflow if x is not close to one */
if ix < 0x3f7ffff8 {
return if hy < 0 {
sn * HUGE * HUGE
} else {
sn * TINY * TINY
};
}
if ix > 0x3f800007 {
return if hy > 0 {
sn * HUGE * HUGE
} else {
sn * 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 */
t = ax - 1.; /* t has 20 trailing zeros */
w = (t * t) * (0.5 - t * (0.333333333333 - t * 0.25));
u = IVLN2_H * t; /* IVLN2_H has 16 sig. bits */
v = t * IVLN2_L - w * IVLN2;
t1 = u + v;
is = t1.to_bits() as i32;
t1 = f32::from_bits(is as u32 & 0xfffff000);
t2 = v - (t1 - u);
} else {
let mut s2: f32;
let mut s_h: f32;
let s_l: f32;
let mut t_h: f32;
let mut t_l: f32;
n = 0;
/* take care subnormal number */
if ix < 0x00800000 {
ax *= TWO24;
n -= 24;
ix = ax.to_bits() as i32;
}
n += ((ix) >> 23) - 0x7f;
j = ix & 0x007fffff;
/* determine interval */
ix = j | 0x3f800000; /* normalize ix */
if j <= 0x1cc471 {
/* |x|<sqrt(3/2) */
k = 0;
} else if j < 0x5db3d7 {
/* |x|<sqrt(3) */
k = 1;
} else {
k = 0;
n += 1;
ix -= 0x00800000;
}
ax = f32::from_bits(ix as u32);
/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
u = ax - i!(BP, k as usize); /* bp[0]=1.0, bp[1]=1.5 */
v = 1.0 / (ax + i!(BP, k as usize));
s = u * v;
s_h = s;
is = s_h.to_bits() as i32;
s_h = f32::from_bits(is as u32 & 0xfffff000);
/* t_h=ax+bp[k] High */
is = (((ix as u32 >> 1) & 0xfffff000) | 0x20000000) as i32;
t_h = f32::from_bits(is as u32 + 0x00400000 + ((k as u32) << 21));
t_l = ax - (t_h - i!(BP, k as usize));
s_l = v * ((u - s_h * t_h) - s_h * t_l);
/* compute log(ax) */
s2 = s * s;
r = s2 * s2 * (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
r += s_l * (s_h + s);
s2 = s_h * s_h;
t_h = 3.0 + s2 + r;
is = t_h.to_bits() as i32;
t_h = f32::from_bits(is as u32 & 0xfffff000);
t_l = r - ((t_h - 3.0) - s2);
/* u+v = s*(1+...) */
u = s_h * t_h;
v = s_l * t_h + t_l * s;
/* 2/(3log2)*(s+...) */
p_h = u + v;
is = p_h.to_bits() as i32;
p_h = f32::from_bits(is as u32 & 0xfffff000);
p_l = v - (p_h - u);
z_h = CP_H * p_h; /* cp_h+cp_l = 2/(3*log2) */
z_l = CP_L * p_h + p_l * CP + i!(DP_L, k as usize);
/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
t = n as f32;
t1 = ((z_h + z_l) + i!(DP_H, k as usize)) + t;
is = t1.to_bits() as i32;
t1 = f32::from_bits(is as u32 & 0xfffff000);
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) */
is = y.to_bits() as i32;
y1 = f32::from_bits(is as u32 & 0xfffff000);
p_l = (y - y1) * t1 + y * t2;
p_h = y1 * t1;
z = p_l + p_h;
j = z.to_bits() as i32;
if j > 0x43000000 {
/* if z > 128 */
return sn * HUGE * HUGE; /* overflow */
} else if j == 0x43000000 {
/* if z == 128 */
if p_l + OVT > z - p_h {
return sn * HUGE * HUGE; /* overflow */
}
} else if (j & 0x7fffffff) > 0x43160000 {
/* z < -150 */
// FIXME: check should be (uint32_t)j > 0xc3160000
return sn * TINY * TINY; /* underflow */
} else if j as u32 == 0xc3160000
/* z == -150 */
&& p_l <= z - p_h
{
return sn * TINY * TINY; /* underflow */
}
/*
* compute 2**(p_h+p_l)
*/
i = j & 0x7fffffff;
k = (i >> 23) - 0x7f;
n = 0;
if i > 0x3f000000 {
/* if |z| > 0.5, set n = [z+0.5] */
n = j + (0x00800000 >> (k + 1));
k = ((n & 0x7fffffff) >> 23) - 0x7f; /* new k for n */
t = f32::from_bits(n as u32 & !(0x007fffff >> k));
n = ((n & 0x007fffff) | 0x00800000) >> (23 - k);
if j < 0 {
n = -n;
}
p_h -= t;
}
t = p_l + p_h;
is = t.to_bits() as i32;
t = f32::from_bits(is as u32 & 0xffff8000);
u = t * LG2_H;
v = (p_l - (t - p_h)) * LG2 + t * LG2_L;
z = u + v;
w = v - (z - u);
t = z * z;
t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
r = (z * t1) / (t1 - 2.0) - (w + z * w);
z = 1.0 - (r - z);
j = z.to_bits() as i32;
j += n << 23;
if (j >> 23) <= 0 {
/* subnormal output */
z = scalbnf(z, n);
} else {
z = f32::from_bits(j as u32);
}
sn * z
}
/* 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 {
#[cfg(target_feature = "sse")]
{
// Note: This path is unlikely since LLVM will usually have already
// optimized sqrt calls into hardware instructions if sse 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_ss` takes a 32-bit float, and returns
// 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
// 32-bits as a 32-bit float.
unsafe {
let m = _mm_set_ss(x);
let m_sqrt = _mm_sqrt_ss(m);
_mm_cvtss_f32(m_sqrt)
}
}
#[cfg(not(target_feature = "sse"))]
{
const TINY: f32 = 1.0e-30;
let mut z: f32;
let sign: i32 = 0x80000000u32 as i32;
let mut ix: i32;
let mut s: i32;
let mut q: i32;
let mut m: i32;
let mut t: i32;
let mut i: i32;
let mut r: u32;
ix = x.to_bits() as i32;
/* take care of Inf and NaN */
if (ix as u32 & 0x7f800000) == 0x7f800000 {
return x * x + x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf, sqrt(-inf)=sNaN */
}
/* take care of zero */
if ix <= 0 {
if (ix & !sign) == 0 {
return x; /* sqrt(+-0) = +-0 */
}
if ix < 0 {
return (x - x) / (x - x); /* sqrt(-ve) = sNaN */
}
}
/* normalize x */
m = ix >> 23;
if m == 0 {
/* subnormal x */
i = 0;
while ix & 0x00800000 == 0 {
ix <<= 1;
i = i + 1;
}
m -= i - 1;
}
m -= 127; /* unbias exponent */
ix = (ix & 0x007fffff) | 0x00800000;
if m & 1 == 1 {
/* odd m, double x to make it even */
ix += ix;
}
m >>= 1; /* m = [m/2] */
/* generate sqrt(x) bit by bit */
ix += ix;
q = 0;
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)
}
