//! Float helpers for a `no_std` environment.
//!
//! 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"), any(feature = "parse-floats", feature = "write-floats")))]
#![cfg_attr(any(), rustfmt::skip)]
/// # Safety
///
/// Safe as long as `e` is properly initialized.
macro_rules! volatile {
($e:expr) => {
// SAFETY: safe as long as `$e` has been properly initialized.
unsafe {
core::ptr::read_volatile(&$e);
}
};
}
/// Floor (f64)
///
/// Finds the nearest integer less than or equal to `x`.
pub(crate) fn floord(x: f64) -> f64 {
const TOINT: f64 = 1. / f64::EPSILON;
let ui = x.to_bits();
let e = ((ui >> 52) & 0x7ff) as i32;
if (e >= 0x3ff + 52) || (x == 0.) {
return x;
}
/* y = int(x) - x, where int(x) is an integer neighbor of x */
let y = if (ui >> 63) != 0 {
x - TOINT + TOINT - x
} else {
x + TOINT - TOINT - x
};
/* special case because of non-nearest rounding modes */
if e < 0x3ff {
volatile!(y);
return if (ui >> 63) != 0 {
-1.
} else {
0.
};
}
if y > 0. {
x + y - 1.
} else {
x + y
}
}
/// Floor (f32)
///
/// Finds the nearest integer less than or equal to `x`.
pub(crate) fn floorf(x: f32) -> f32 {
let mut ui = x.to_bits();
let e = (((ui >> 23) as i32) & 0xff) - 0x7f;
if e >= 23 {
return x;
}
if e >= 0 {
let m: u32 = 0x007fffff >> e;
if (ui & m) == 0 {
return x;
}
volatile!(x + f32::from_bits(0x7b800000));
if ui >> 31 != 0 {
ui += m;
}
ui &= !m;
} else {
volatile!(x + f32::from_bits(0x7b800000));
if ui >> 31 == 0 {
ui = 0;
} else if ui << 1 != 0 {
return -1.0;
}
}
f32::from_bits(ui)
}
/* origin: FreeBSD /usr/src/lib/msun/src/e_log.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.
* ====================================================
*/
/* log(x)
* Return the logarithm of x
*
* Method :
* 1. Argument Reduction: find k and f such that
* x = 2^k * (1+f),
* where sqrt(2)/2 < 1+f < sqrt(2) .
*
* 2. Approximation of log(1+f).
* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
* = 2s + s*R
* We use a special Remez algorithm on [0,0.1716] to generate
* a polynomial of degree 14 to approximate R The maximum error
* of this polynomial approximation is bounded by 2**-58.45. In
* other words,
* 2 4 6 8 10 12 14
* R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
* (the values of Lg1 to Lg7 are listed in the program)
* and
* | 2 14 | -58.45
* | Lg1*s +...+Lg7*s - R(z) | <= 2
* | |
* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
* In order to guarantee error in log below 1ulp, we compute log
* by
* log(1+f) = f - s*(f - R) (if f is not too large)
* log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
*
* 3. Finally, log(x) = k*ln2 + log(1+f).
* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
* Here ln2 is split into two floating point number:
* ln2_hi + ln2_lo,
* where n*ln2_hi is always exact for |n| < 2000.
*
* Special cases:
* log(x) is NaN with signal if x < 0 (including -INF) ;
* log(+INF) is +INF; log(0) is -INF with signal;
* log(NaN) is that NaN with no signal.
*
* Accuracy:
* according to an error analysis, the error is always less than
* 1 ulp (unit in the last place).
*
* 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.
*/
#[allow(clippy::eq_op, clippy::excessive_precision)] // reason="values need to be exact under all conditions"
pub(crate) fn logd(mut x: f64) -> f64 {
const LN2_HI: f64 = 6.93147180369123816490e-01; /* 3fe62e42 fee00000 */
const LN2_LO: f64 = 1.90821492927058770002e-10; /* 3dea39ef 35793c76 */
const LG1: f64 = 6.666666666666735130e-01; /* 3FE55555 55555593 */
const LG2: f64 = 3.999999999940941908e-01; /* 3FD99999 9997FA04 */
const LG3: f64 = 2.857142874366239149e-01; /* 3FD24924 94229359 */
const LG4: f64 = 2.222219843214978396e-01; /* 3FCC71C5 1D8E78AF */
const LG5: f64 = 1.818357216161805012e-01; /* 3FC74664 96CB03DE */
const LG6: f64 = 1.531383769920937332e-01; /* 3FC39A09 D078C69F */
const LG7: f64 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
let x1p54 = f64::from_bits(0x4350000000000000); // 0x1p54 === 2 ^ 54
let mut ui = x.to_bits();
let mut hx: u32 = (ui >> 32) as u32;
let mut k: i32 = 0;
if (hx < 0x00100000) || ((hx >> 31) != 0) {
/* x < 2**-126 */
if ui << 1 == 0 {
return -1. / (x * x); /* log(+-0)=-inf */
}
if hx >> 31 != 0 {
return (x - x) / 0.0; /* log(-#) = NaN */
}
/* subnormal number, scale x up */
k -= 54;
x *= x1p54;
ui = x.to_bits();
hx = (ui >> 32) as u32;
} else if hx >= 0x7ff00000 {
return x;
} else if hx == 0x3ff00000 && ui << 32 == 0 {
return 0.;
}
/* reduce x into [sqrt(2)/2, sqrt(2)] */
hx += 0x3ff00000 - 0x3fe6a09e;
k += ((hx >> 20) as i32) - 0x3ff;
hx = (hx & 0x000fffff) + 0x3fe6a09e;
ui = ((hx as u64) << 32) | (ui & 0xffffffff);
x = f64::from_bits(ui);
let f: f64 = x - 1.0;
let hfsq: f64 = 0.5 * f * f;
let s: f64 = f / (2.0 + f);
let z: f64 = s * s;
let w: f64 = z * z;
let t1: f64 = w * (LG2 + w * (LG4 + w * LG6));
let t2: f64 = z * (LG1 + w * (LG3 + w * (LG5 + w * LG7)));
let r: f64 = t2 + t1;
let dk: f64 = k as f64;
s * (hfsq + r) + dk * LN2_LO - hfsq + f + dk * LN2_HI
}
/* origin: FreeBSD /usr/src/lib/msun/src/e_logf.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.
* ====================================================
*/
#[allow(clippy::eq_op, clippy::excessive_precision)] // reason="values need to be exact under all conditions"
pub(crate) fn logf(mut x: f32) -> f32 {
const LN2_HI: f32 = 6.9313812256e-01; /* 0x3f317180 */
const LN2_LO: f32 = 9.0580006145e-06; /* 0x3717f7d1 */
/* |(log(1+s)-log(1-s))/s - Lg(s)| < 2**-34.24 (~[-4.95e-11, 4.97e-11]). */
const LG1: f32 = 0.66666662693; /* 0xaaaaaa.0p-24 */
const LG2: f32 = 0.40000972152; /* 0xccce13.0p-25 */
const LG3: f32 = 0.28498786688; /* 0x91e9ee.0p-25 */
const LG4: f32 = 0.24279078841; /* 0xf89e26.0p-26 */
let x1p25 = f32::from_bits(0x4c000000); // 0x1p25f === 2 ^ 25
let mut ix = x.to_bits();
let mut k = 0i32;
if (ix < 0x00800000) || ((ix >> 31) != 0) {
/* x < 2**-126 */
if ix << 1 == 0 {
return -1. / (x * x); /* log(+-0)=-inf */
}
if (ix >> 31) != 0 {
return (x - x) / 0.; /* log(-#) = NaN */
}
/* subnormal number, scale up x */
k -= 25;
x *= x1p25;
ix = x.to_bits();
} else if ix >= 0x7f800000 {
return x;
} else if ix == 0x3f800000 {
return 0.;
}
/* reduce x into [sqrt(2)/2, sqrt(2)] */
ix += 0x3f800000 - 0x3f3504f3;
k += ((ix >> 23) as i32) - 0x7f;
ix = (ix & 0x007fffff) + 0x3f3504f3;
x = f32::from_bits(ix);
let f = x - 1.;
let s = f / (2. + f);
let z = s * s;
let w = z * z;
let t1 = w * (LG2 + w * LG4);
let t2 = z * (LG1 + w * LG3);
let r = t2 + t1;
let hfsq = 0.5 * f * f;
let dk = k as f32;
s * (hfsq + r) + dk * LN2_LO - hfsq + f + dk * LN2_HI
}
