Rust f32::acosh

Percentage Accurate: 53.2% → 98.4%

Time: 14.4s

Precision: binary32

Cost: 6784

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

\[x \geq 1\]

Math

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

↓

\[\log \left(x \cdot 2 - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary32
 (log (- (* x 2.0) (+ (/ 0.5 x) (/ 0.125 (pow x 3.0))))))

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) + (0.125f / powf(x, 3.0f)))));
}

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) + (0.125e0 / (x ** 3.0e0)))))
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(Float32(0.5) / x) + Float32(Float32(0.125) / (x ^ Float32(3.0))))))
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) + (single(0.125) / (x ^ single(3.0))))));
end

Target

Original	53.2%
Target	99.0%
Herbie	98.4%

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

Derivation

1. Initial program 56.1%

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

2. Taylor expanded in x around inf 97.5%

   \[\leadsto \log \color{blue}{\left(2 \cdot x - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)} \]

3. Simplified 97.5%

   \[\leadsto \log \color{blue}{\left(x \cdot 2 - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right)} \]
   Step-by-step derivation

   [Start] 97.5%	\[ \log \left(2 \cdot x - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]
   *-commutative [=>] 97.5%	\[ \log \left(\color{blue}{x \cdot 2} - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]
   associate-*r/ [=>] 97.5%	\[ \log \left(x \cdot 2 - \left(\color{blue}{\frac{0.5 \cdot 1}{x}} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]
   metadata-eval [=>] 97.5%	\[ \log \left(x \cdot 2 - \left(\frac{\color{blue}{0.5}}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]
   associate-*r/ [=>] 97.5%	\[ \log \left(x \cdot 2 - \left(\frac{0.5}{x} + \color{blue}{\frac{0.125 \cdot 1}{{x}^{3}}}\right)\right) \]
   metadata-eval [=>] 97.5%	\[ \log \left(x \cdot 2 - \left(\frac{0.5}{x} + \frac{\color{blue}{0.125}}{{x}^{3}}\right)\right) \]

4. Final simplification 97.5%

   \[\leadsto \log \left(x \cdot 2 - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right) \]

Alternatives

Alternative 1
Accuracy	98.4%
Cost	6784

\[\log \left(x \cdot 2 - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right) \]

Alternative 2
Accuracy	97.9%
Cost	3424

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

Alternative 3
Accuracy	96.7%
Cost	3360

\[\log \left(x \cdot 2 - 0.001953125\right) \]

Alternative 4
Accuracy	44.2%
Cost	3296

\[\log \left(x + 0.875\right) \]

Alternative 5
Accuracy	96.6%
Cost	3296

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

Alternative 6
Accuracy	20.8%
Cost	32

\[0.8333333333333334 \]

Alternative 7
Accuracy	22.2%
Cost	32

\[1.9583333333333333 \]

Alternative 8
Accuracy	22.3%
Cost	32

\[2.125 \]

Reproduce

herbie shell --seed 2023269 
(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)))))