Rust f32::acosh

Average Accuracy: 49.9% → 98.9%

Time: 6.2s

Precision: binary32

Cost: 3488

\[x \geq 1\]

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

float code(float x) {
	return (-0.25f / (x * x)) - logf((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 = ((-0.25e0) / (x * x)) - log((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 Float32(Float32(Float32(-0.25) / Float32(x * x)) - log(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 = (single(-0.25) / (x * x)) - log((single(0.5) / x));
end

Target

Original	49.9%
Target	99.2%
Herbie	98.9%

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

Derivation

1. Initial program 49.9%

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

2. Taylor expanded in x around inf 98.3%

   \[\leadsto \log \color{blue}{\left(2 \cdot x - 0.5 \cdot \frac{1}{x}\right)} \]

3. Simplified 98.3%

   \[\leadsto \log \color{blue}{\left(x \cdot 2 - \frac{0.5}{x}\right)} \]
   Proof

   [Start] 98.3	\[ \log \left(2 \cdot x - 0.5 \cdot \frac{1}{x}\right) \]
   *-commutative [=>] 98.3	\[ \log \left(\color{blue}{x \cdot 2} - 0.5 \cdot \frac{1}{x}\right) \]
   associate-*r/ [=>] 98.3	\[ \log \left(x \cdot 2 - \color{blue}{\frac{0.5 \cdot 1}{x}}\right) \]
   metadata-eval [=>] 98.3	\[ \log \left(x \cdot 2 - \frac{\color{blue}{0.5}}{x}\right) \]

4. Taylor expanded in x around inf 98.1%

   \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]

5. Simplified 98.9%

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

   [Start] 98.1	\[ \left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right) - 0.25 \cdot \frac{1}{{x}^{2}} \]
   cancel-sign-sub-inv [=>] 98.1	\[ \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right) + \left(-0.25\right) \cdot \frac{1}{{x}^{2}}} \]
   +-commutative [=>] 98.1	\[ \color{blue}{\left(-0.25\right) \cdot \frac{1}{{x}^{2}} + \left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right)} \]
   log-rec [=>] 98.1	\[ \left(-0.25\right) \cdot \frac{1}{{x}^{2}} + \left(-1 \cdot \color{blue}{\left(-\log x\right)} + \log 2\right) \]
   distribute-rgt-neg-in [<=] 98.1	\[ \left(-0.25\right) \cdot \frac{1}{{x}^{2}} + \left(\color{blue}{\left(--1 \cdot \log x\right)} + \log 2\right) \]
   neg-sub0 [=>] 98.1	\[ \left(-0.25\right) \cdot \frac{1}{{x}^{2}} + \left(\color{blue}{\left(0 - -1 \cdot \log x\right)} + \log 2\right) \]
   associate-+l- [=>] 98.1	\[ \left(-0.25\right) \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(0 - \left(-1 \cdot \log x - \log 2\right)\right)} \]
   mul-1-neg [=>] 98.1	\[ \left(-0.25\right) \cdot \frac{1}{{x}^{2}} + \left(0 - \left(\color{blue}{\left(-\log x\right)} - \log 2\right)\right) \]
   log-rec [<=] 98.1	\[ \left(-0.25\right) \cdot \frac{1}{{x}^{2}} + \left(0 - \left(\color{blue}{\log \left(\frac{1}{x}\right)} - \log 2\right)\right) \]
   log-div [<=] 98.9	\[ \left(-0.25\right) \cdot \frac{1}{{x}^{2}} + \left(0 - \color{blue}{\log \left(\frac{\frac{1}{x}}{2}\right)}\right) \]
   associate-/l/ [=>] 98.2	\[ \left(-0.25\right) \cdot \frac{1}{{x}^{2}} + \left(0 - \log \color{blue}{\left(\frac{1}{2 \cdot x}\right)}\right) \]
   associate-/r* [=>] 98.9	\[ \left(-0.25\right) \cdot \frac{1}{{x}^{2}} + \left(0 - \log \color{blue}{\left(\frac{\frac{1}{2}}{x}\right)}\right) \]
   metadata-eval [=>] 98.9	\[ \left(-0.25\right) \cdot \frac{1}{{x}^{2}} + \left(0 - \log \left(\frac{\color{blue}{0.5}}{x}\right)\right) \]
   associate-+r- [=>] 98.9	\[ \color{blue}{\left(\left(-0.25\right) \cdot \frac{1}{{x}^{2}} + 0\right) - \log \left(\frac{0.5}{x}\right)} \]

6. Final simplification 98.9%

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

Alternatives

Alternative 1
Accuracy	98.3%
Cost	3424

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

Alternative 2
Accuracy	98.3%
Cost	3424

\[\log \left(x \cdot 2 + \frac{-0.5}{x}\right) \]

Alternative 3
Accuracy	97.7%
Cost	3328

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

Alternative 4
Accuracy	97.0%
Cost	3296

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

Alternative 5
Accuracy	6.1%
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2023141 
(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)))))