Rust f32::acosh

Average Error: 50.71% → 1.89%

Time: 10.7s

Precision: binary32

Cost: 3936

\[x \geq 1\]

Math

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

↓

\[\begin{array}{l} t_0 := x + \frac{0.5}{x}\\ \log \left(\left(x + \frac{x}{\frac{t_0}{x}}\right) - \frac{0.25}{t_0 \cdot \left(x \cdot x\right)}\right) \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary32
 (let* ((t_0 (+ x (/ 0.5 x))))
   (log (- (+ x (/ x (/ t_0 x))) (/ 0.25 (* t_0 (* x x)))))))

C

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

↓

float code(float x) {
	float t_0 = x + (0.5f / x);
	return logf(((x + (x / (t_0 / x))) - (0.25f / (t_0 * (x * 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
    real(4) :: t_0
    t_0 = x + (0.5e0 / x)
    code = log(((x + (x / (t_0 / x))) - (0.25e0 / (t_0 * (x * x)))))
end function

Julia

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

↓

function code(x)
	t_0 = Float32(x + Float32(Float32(0.5) / x))
	return log(Float32(Float32(x + Float32(x / Float32(t_0 / x))) - Float32(Float32(0.25) / Float32(t_0 * Float32(x * x)))))
end

MATLAB

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

↓

function tmp = code(x)
	t_0 = x + (single(0.5) / x);
	tmp = log(((x + (x / (t_0 / x))) - (single(0.25) / (t_0 * (x * x)))));
end

Target

Original	50.71%
Target	0.81%
Herbie	1.89%

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

Derivation

1. Initial program 50.71

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

2. Taylor expanded in x around inf 1.88

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

3. Simplified 1.88

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

   [Start] 1.88	\[ \log \left(x + \left(x - 0.5 \cdot \frac{1}{x}\right)\right) \]
   associate-*r/ [=>] 1.88	\[ \log \left(x + \left(x - \color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right) \]
   metadata-eval [=>] 1.88	\[ \log \left(x + \left(x - \frac{\color{blue}{0.5}}{x}\right)\right) \]

4. Applied egg-rr 1.89

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

5. Final simplification 1.89

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

Alternatives

Alternative 1
Error	1.88%
Cost	3552

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

Alternative 2
Error	1.89%
Cost	3552

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

Alternative 3
Error	1.88%
Cost	3424

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

Alternative 4
Error	2.53%
Cost	3328

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

Alternative 5
Error	3.26%
Cost	3296

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

Alternative 6
Error	93.89%
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2023090 
(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)))))