Rust f32::acosh

Average Error: 50.37% → 1.59%

Time: 6.9s

Precision: binary32

Cost: 3424

\[x \geq 1\]

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

float code(float x) {
	return logf((x + (x + (-0.5f / 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) / 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) / 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) / x))));
end

Target

Original	50.37%
Target	0.61%
Herbie	1.59%

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

Derivation

1. Initial program 50.37

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

2. Taylor expanded in x around inf 1.59

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

3. Simplified 1.59

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

   [Start] 1.59	\[ \log \left(x + \left(x - 0.5 \cdot \frac{1}{x}\right)\right) \]
   associate-*r/ [=>] 1.59	\[ \log \left(x + \left(x - \color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right) \]
   metadata-eval [=>] 1.59	\[ \log \left(x + \left(x - \frac{\color{blue}{0.5}}{x}\right)\right) \]

4. Final simplification 1.59

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

Alternatives

Alternative 1
Error	2.36%
Cost	3328

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

Alternative 2
Error	2.91%
Cost	3296

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

Alternative 3
Error	55.81%
Cost	3232

\[\log x \]

Alternative 4
Error	93.89%
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2023121 
(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)))))