Rust f32::acosh

Percentage Accurate: 53.7% → 98.3%

Time: 3.6s

Precision: binary32

Cost: 3424

• Unsound rule application detected in e-graph. Results may not be sound.

\[x \geq 1\]

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary32 (log (- (* x 2.0) (/ 0.5 x))))

C

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

↓

float code(float x) {
	return logf(((x * 2.0f) - (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 * 2.0e0) - (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(Float32(x * Float32(2.0)) - 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 * single(2.0)) - (single(0.5) / x)));
end

Target

Original	53.7%
Target	99.3%
Herbie	98.3%

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

Derivation

1. Initial program 58.7%

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

2. Taylor expanded in x around inf 97.5%

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

3. Simplified 97.5%

   \[\leadsto \log \color{blue}{\left(x \cdot 2 - \frac{0.5}{x}\right)} \]
   Step-by-step derivation

   [Start] 97.5%	\[ \log \left(2 \cdot x - 0.5 \cdot \frac{1}{x}\right) \]
   *-commutative [=>] 97.5%	\[ \log \left(\color{blue}{x \cdot 2} - 0.5 \cdot \frac{1}{x}\right) \]
   associate-*r/ [=>] 97.5%	\[ \log \left(x \cdot 2 - \color{blue}{\frac{0.5 \cdot 1}{x}}\right) \]
   metadata-eval [=>] 97.5%	\[ \log \left(x \cdot 2 - \frac{\color{blue}{0.5}}{x}\right) \]

4. Final simplification 97.5%

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

Alternatives

Alternative 1
Accuracy	98.3%
Cost	3424

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

Alternative 2
Accuracy	97.0%
Cost	3296

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

Alternative 3
Accuracy	6.1%
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2023178 
(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)))))