Rust f32::acosh

Average Error: 15.9 → 0.5

Time: 1.8s

Precision: binary32

Cost: 3488

\[x \geq 1\]

Math

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

↓

\[\log \left(x + \left(x - 0.5 \cdot \frac{1}{x}\right)\right) \]

FPCore

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

↓

(FPCore (x) :precision binary32 (log (+ x (- x (* 0.5 (/ 1.0 x))))))

C

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

↓

float code(float x) {
	return logf((x + (x - (0.5f * (1.0f / x)))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + sqrt(((x * x) - 1.0e0))))
end function

↓

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + (x - (0.5e0 * (1.0e0 / x)))))
end function

Julia

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

↓

function code(x)
	return log(Float32(x + Float32(x - Float32(Float32(0.5) * Float32(Float32(1.0) / x)))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) - single(1.0)))));
end

↓

function tmp = code(x)
	tmp = log((x + (x - (single(0.5) * (single(1.0) / x)))));
end

Target

Original	15.9
Target	0.2
Herbie	0.5

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

Derivation

1. Initial program 15.9

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

2. Taylor expanded in x around inf 0.5

   \[\leadsto \log \left(x + \color{blue}{\left(x - 0.5 \cdot \frac{1}{x}\right)}\right) \]

3. Final simplification 0.5

   \[\leadsto \log \left(x + \left(x - 0.5 \cdot \frac{1}{x}\right)\right) \]

Alternatives

Alternative 1
Error	0.9
Cost	3296

\[\log \left(x + x\right) \]

Reproduce

herbie shell --seed 2023053 
(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)))))