Rust f32::acosh

Average Error: 16.0 → 0.2

Time: 2.5s

Precision: binary32

Cost: 13024

\[x \geq 1\]

Math

\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]

↓

\[\log \left(\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x + -1}, x\right)\right) \]

FPCore

(FPCore (x) :precision binary32 (log (+ x (sqrt (- (* x x) 1.0)))))

↓

(FPCore (x)
 :precision binary32
 (log (fma (sqrt (+ x 1.0)) (sqrt (+ x -1.0)) x)))

C

float code(float x) {
	return logf((x + sqrtf(((x * x) - 1.0f))));
}

↓

float code(float x) {
	return logf(fmaf(sqrtf((x + 1.0f)), sqrtf((x + -1.0f)), x));
}

Julia

function code(x)
	return log(Float32(x + sqrt(Float32(Float32(x * x) - Float32(1.0)))))
end

↓

function code(x)
	return log(fma(sqrt(Float32(x + Float32(1.0))), sqrt(Float32(x + Float32(-1.0))), x))
end

Target

Original	16.0
Target	0.3
Herbie	0.2

\[\log \left(x + \sqrt{x - 1} \cdot \sqrt{x + 1}\right) \]

Derivation

1. Initial program 16.0

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]

2. Applied egg-rr 0.3

   \[\leadsto \log \left(x + \color{blue}{\sqrt{x + 1} \cdot \sqrt{x + -1}}\right) \]

3. Applied egg-rr 0.2

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x + -1}, x\right)\right)} \]

4. Final simplification 0.2

   \[\leadsto \log \left(\mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x + -1}, x\right)\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	9888

\[\log \left(x + \sqrt{x + 1} \cdot \sqrt{x + -1}\right) \]

Alternative 2
Error	16.0
Cost	6624

\[\log \left(x + \sqrt{-1 + x \cdot x}\right) \]

Reproduce

herbie shell --seed 2022334 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

  :herbie-target
  (log (+ x (* (sqrt (- x 1.0)) (sqrt (+ x 1.0)))))

  (log (+ x (sqrt (- (* x x) 1.0)))))