Rust f32::acosh

Average Error: 16.0 → 0.6

Time: 11.2s

Precision: binary32

Cost: 3744

\[x \geq 1\]

Math

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

↓

\[\log \left(\frac{1.5}{x} - \left(-4 \cdot \left(\frac{7.5}{x} - \frac{8}{x}\right) - \left(x + x\right)\right)\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary32
 (log (- (/ 1.5 x) (- (* -4.0 (- (/ 7.5 x) (/ 8.0 x))) (+ x x)))))

C

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

↓

float code(float x) {
	return logf(((1.5f / x) - ((-4.0f * ((7.5f / x) - (8.0f / x))) - (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
    code = log(((1.5e0 / x) - (((-4.0e0) * ((7.5e0 / x) - (8.0e0 / x))) - (x + 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(Float32(Float32(1.5) / x) - Float32(Float32(Float32(-4.0) * Float32(Float32(Float32(7.5) / x) - Float32(Float32(8.0) / x))) - Float32(x + x))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = log(((single(1.5) / x) - ((single(-4.0) * ((single(7.5) / x) - (single(8.0) / x))) - (x + x))));
end

Target

Original	16.0
Target	0.3
Herbie	0.6

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

Derivation

1. Initial program 16.0

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

2. Taylor expanded in x around inf 0.6

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

3. Simplified 0.6

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

   [Start] 0.6	\[ \log \left(x + \left(x - 0.5 \cdot \frac{1}{x}\right)\right) \]
   rational.json-simplify-20 [=>] 0.6	\[ \log \left(x + \left(x - \color{blue}{\frac{1 \cdot 0.5}{x}}\right)\right) \]
   metadata-eval [=>] 0.6	\[ \log \left(x + \left(x - \frac{\color{blue}{0.5}}{x}\right)\right) \]

4. Applied egg-rr 0.6

   \[\leadsto \log \color{blue}{\left(\frac{1.5}{x} - \left(\frac{2}{x} - \left(x + x\right)\right)\right)} \]

5. Applied egg-rr 0.6

   \[\leadsto \log \left(\frac{1.5}{x} - \left(\color{blue}{-4 \cdot \left(\frac{7.5}{x} - \frac{8}{x}\right)} - \left(x + x\right)\right)\right) \]

6. Final simplification 0.6

   \[\leadsto \log \left(\frac{1.5}{x} - \left(-4 \cdot \left(\frac{7.5}{x} - \frac{8}{x}\right) - \left(x + x\right)\right)\right) \]

Alternatives

Alternative 1
Error	0.6
Cost	3552

\[\log \left(\frac{1.5}{x} - \left(\frac{2}{x} - \left(x + x\right)\right)\right) \]

Alternative 2
Error	0.6
Cost	3424

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

Alternative 3
Error	1.0
Cost	3296

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

Reproduce

herbie shell --seed 2023066 
(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)))))