Arc-hyperbolic-trigonometric functions are very inaccurate

[pavpanchekha]

The functions asinh and acosh for both f32 and f64 are quite inaccurate, both for very large inputs (because x * x overflows) and, for asinh, for very small inputs (due to kinda-complex cancellation).

I tried this code:

fn main() {
    println!("{}", (600.0 as f64).sinh().asinh());
    println!("{}", (1e-16 as f64).sinh().asinh());
    println!("{}", (600.0 as f64).cosh().acosh());
    
    println!("{}", (60.0 as f32).sinh().asinh());
    println!("{}", (1e-8 as f32).sinh().asinh());
    println!("{}", (60.0 as f32).cosh().acosh());
}

This code evaluates asinh(sinh(x)) and acosh(cosh(x)). Naturally I expected the output to be x, but instead the output is inf and 0, indicating overflow and cancellation.

Meta

I ran this in the Rust Playground on both Stable (1.65.0) and Nightly (2022-11-16 e9493d63c2a57b91556d).

[pavpanchekha]

The issue was originally discovered by @finnbear in FPBench/FPBench#118.

I'm testing solutions that don't rely on cmath in herbie-fp/herbie#514.

[pavpanchekha]

In the two PRs above the following implementations are proposed:

fn asinh(x: f64) -> f64 {
	let ax = x.abs();
	let ix = 1.0 / ax;
	((ax + (ax / (f64::hypot(1.0, ix) + ix)))).ln_1p().copysign(x)
}

fn acosh(x: f64) -> f64 {
	if x < 1.0 { f64::NAN } else { (x + ((x - 1.0).sqrt() * (x + 1.0).sqrt())).ln() }
}

I'll file a PR once I get time to figure out how to download + build + test Rust, probably this or next week.

[pavpanchekha]

Here is the accuracy plot from Herbie for f64::asinh, red being the current Rust expression and green being the proposed fix:

And here's f64::acosh:

The f32 versions look similar but the problems occur at different (smaller) inputs.

[finnbear]

I'll file a PR once I get time to figure out how to download + build + test Rust, probably this or next week.

I already have a Rust fork and the required tools to test it so, if it's ok with you, I could write the PR using your proposed implementation.

Looks like #104553 beat me to it 😃