Rust f32::acosh

Average Error: 16.2 → 0.3

Time: 8.8s

Precision: binary32

Cost: 9888

\[x \geq 1\]

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 + (sqrt((x + 1.0e0)) * sqrt((x + (-1.0e0))))))
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(sqrt(Float32(x + Float32(1.0))) * sqrt(Float32(x + Float32(-1.0))))))
end

MATLAB

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

↓

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

Target

Original	16.2
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 16.2

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

2. Applied egg-rr 0.3

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

3. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	3648

\[-\log \left(\frac{\frac{0.125}{x}}{x \cdot x} + 0.5 \cdot \frac{1}{x}\right) \]

Alternative 2
Error	0.4
Cost	3488

\[\frac{-0.25}{x \cdot x} - \log \left(\frac{0.5}{x}\right) \]

Alternative 3
Error	0.6
Cost	3424

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

Alternative 4
Error	0.8
Cost	3328

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

Alternative 5
Error	1.0
Cost	3296

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

Alternative 6
Error	30.0
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2023034 
(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)))))