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/*
* // Copyright (c) Radzivon Bartoshyk 4/2025. All rights reserved.
* //
* // Redistribution and use in source and binary forms, with or without modification,
* // are permitted provided that the following conditions are met:
* //
* // 1. Redistributions of source code must retain the above copyright notice, this
* // list of conditions and the following disclaimer.
* //
* // 2. Redistributions in binary form must reproduce the above copyright notice,
* // this list of conditions and the following disclaimer in the documentation
* // and/or other materials provided with the distribution.
* //
* // 3. Neither the name of the copyright holder nor the names of its
* // contributors may be used to endorse or promote products derived from
* // this software without specific prior written permission.
* //
* // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
* // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
* // DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
* // FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* // DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
* // SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
* // CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
* // OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
use crate::common::{f_fmla, f_fmlaf};
/// Computes atan using FMA
///
/// Max found ULP 0.49999973
#[inline]
pub fn f_atanf(x: f32) -> f32 {
const PI2: f64 = f64::from_bits(0x3ff921fb54442d18);
let t = x.to_bits();
let e = (t >> 23) & 0xff;
let gt = e >= 127;
let ta = t & 0x7fffffff;
if ta >= 0x4c700518u32 {
// |x| >= 6.29198e+07
if ta > 0x7f800000u32 {
return x + x;
} // nan
return f32::copysign(PI2 as f32, x); // inf or |x| >= 6.29198e+07
}
if e < 127 - 13 {
// |x| < 2^-13
if e < 127 - 25 {
// |x| < 2^-25
if t << 1 == 0 {
return x;
}
let res = f_fmlaf(-x, x.abs(), x);
return res;
}
return f_fmlaf(-f64::from_bits(0x3fd5555560000000) as f32 * x, x * x, x);
}
/* now |x| >= 0.00012207 */
let mut z = x as f64;
if gt {
z = 1.0 / z;
} /* gt is non-zero for |x| >= 1 */
let z2 = z * z;
let z4 = z2 * z2;
let z8 = z4 * z4;
/* polynomials generated using rminimax
(https://gitlab.inria.fr/sfilip/rminimax) with the following command:
./ratapprox --function="atan(x)" --dom=[0.000122070,1] --num=[x,x^3,x^5,x^7,x^9,x^11,x^13] --den=[1,x^2,x^4,x^6,x^8,x^10,x^12] --output=atanf.sollya --log
(see output atanf.sollya)
The coefficient cd[0] was slightly reduced from the original value
0.330005 to avoid an exceptional case for |x| = 0.069052
and rounding to nearest.
*/
const CN: [u64; 7] = [
0x3fd51eccde075d67,
0x3fea76bb5637f2f2,
0x3fe81e0eed20de88,
0x3fd376c8ca67d11d,
0x3faaec7b69202ac6,
0x3f69561899acc73e,
0x3efbf9fa5b67e600,
];
const CD: [u64; 7] = [
0x3fd51eccde075d66,
0x3fedfbdd7b392d28,
0x3ff0000000000000,
0x3fdfd22bf0e89b54,
0x3fbd91ff8b576282,
0x3f8653ea99fc9bb0,
0x3f31e7fcc202340a,
];
let mut cn0 = f_fmla(z2, f64::from_bits(CN[1]), f64::from_bits(CN[0]));
let cn2 = f_fmla(z2, f64::from_bits(CN[3]), f64::from_bits(CN[2]));
let mut cn4 = f_fmla(z2, f64::from_bits(CN[5]), f64::from_bits(CN[4]));
let cn6 = f64::from_bits(CN[6]);
cn0 = f_fmla(z4, cn2, cn0);
cn4 = f_fmla(z4, cn6, cn4);
cn0 = f_fmla(z8, cn4, cn0);
cn0 *= z;
let mut cd0 = f_fmla(z2, f64::from_bits(CD[1]), f64::from_bits(CD[0]));
let cd2 = f_fmla(z2, f64::from_bits(CD[3]), f64::from_bits(CD[2]));
let mut cd4 = f_fmla(z2, f64::from_bits(CD[5]), f64::from_bits(CD[4]));
let cd6 = f64::from_bits(CD[6]);
cd0 = f_fmla(z4, cd2, cd0);
cd4 = f_fmla(z4, cd6, cd4);
cd0 = f_fmla(z8, cd4, cd0);
let r = cn0 / cd0;
if !gt {
return r as f32;
} /* for |x| < 1, (float) r is correctly rounded */
const PI_OVER2_H: f64 = f64::from_bits(0x3ff9000000000000);
const PI_OVER2_L: f64 = f64::from_bits(0x3f80fdaa22168c23);
/* now r approximates atan(1/x), we use atan(x) + atan(1/x) = sign(x)*pi/2,
where PI_OVER2_H + PI_OVER2_L approximates pi/2.
With sign(z)*L + (-r + sign(z)*H), it fails for x=0x1.98c252p+12 and
rounding upward.
With sign(z)*PI - r, where PI is a double approximation of pi to nearest,
it fails for x=0x1.ddf9f6p+0 and rounding upward. */
((f64::copysign(PI_OVER2_L, z) - r) + f64::copysign(PI_OVER2_H, z)) as f32
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn f_atan_test() {
assert!(
(f_atanf(1.0) - std::f32::consts::PI / 4f32).abs() < 1e-6,
"Invalid result {}",
f_atanf(1f32)
);
assert!(
(f_atanf(2f32) - 1.107148717794090503017065f32).abs() < 1e-6,
"Invalid result {}",
f_atanf(2f32)
);
assert!(
(f_atanf(5f32) - 1.3734007669450158608612719264f32).abs() < 1e-6,
"Invalid result {}",
f_atanf(5f32)
);
}
}