Rust f32::acosh

Average Accuracy: 50.1% → 98.6%

Time: 9.0s

Precision: binary32

Cost: 3808

\[x \geq 1\]

Math

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

↓

\[\mathsf{log1p}\left(x \cdot 2 + \left(\left(\frac{-0.5}{x} + -1\right) + \frac{0.25}{x \cdot x} \cdot \frac{-0.5}{x}\right)\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary32
 (log1p (+ (* x 2.0) (+ (+ (/ -0.5 x) -1.0) (* (/ 0.25 (* x x)) (/ -0.5 x))))))

C

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

↓

float code(float x) {
	return log1pf(((x * 2.0f) + (((-0.5f / x) + -1.0f) + ((0.25f / (x * x)) * (-0.5f / x)))));
}

Julia

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

↓

function code(x)
	return log1p(Float32(Float32(x * Float32(2.0)) + Float32(Float32(Float32(Float32(-0.5) / x) + Float32(-1.0)) + Float32(Float32(Float32(0.25) / Float32(x * x)) * Float32(Float32(-0.5) / x)))))
end

Target

Original	50.1%
Target	99.2%
Herbie	98.6%

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

Derivation

1. Initial program 50.1%

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

2. Applied egg-rr 50.1%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(x + \left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} - 1\right)\right)} \]
   Proof

   [Start] 50.1	\[ \log \left(x + \sqrt{x \cdot x - 1}\right) \]
   log1p-expm1-u [=>] 50.1	\[ \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(x + \sqrt{x \cdot x - 1}\right)\right)\right)} \]
   expm1-udef [=>] 50.1	\[ \mathsf{log1p}\left(\color{blue}{e^{\log \left(x + \sqrt{x \cdot x - 1}\right)} - 1}\right) \]
   add-exp-log [<=] 50.1	\[ \mathsf{log1p}\left(\color{blue}{\left(x + \sqrt{x \cdot x - 1}\right)} - 1\right) \]
   associate--l+ [=>] 50.1	\[ \mathsf{log1p}\left(\color{blue}{x + \left(\sqrt{x \cdot x - 1} - 1\right)}\right) \]
   fma-neg [=>] 50.1	\[ \mathsf{log1p}\left(x + \left(\sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} - 1\right)\right) \]
   metadata-eval [=>] 50.1	\[ \mathsf{log1p}\left(x + \left(\sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} - 1\right)\right) \]

3. Simplified 50.1%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} + \left(-1 + x\right)\right)} \]
   Proof

   [Start] 50.1	\[ \mathsf{log1p}\left(x + \left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} - 1\right)\right) \]
   +-commutative [=>] 50.1	\[ \mathsf{log1p}\left(\color{blue}{\left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} - 1\right) + x}\right) \]
   sub-neg [=>] 50.1	\[ \mathsf{log1p}\left(\color{blue}{\left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} + \left(-1\right)\right)} + x\right) \]
   metadata-eval [=>] 50.1	\[ \mathsf{log1p}\left(\left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} + \color{blue}{-1}\right) + x\right) \]
   associate-+l+ [=>] 50.1	\[ \mathsf{log1p}\left(\color{blue}{\sqrt{\mathsf{fma}\left(x, x, -1\right)} + \left(-1 + x\right)}\right) \]

4. Taylor expanded in x around inf 98.6%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{2 \cdot x - \left(0.5 \cdot \frac{1}{x} + \left(1 + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)}\right) \]

5. Simplified 98.6%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(x \cdot 2 - \frac{0.5}{x}\right) - \left(1 + \frac{0.125}{{x}^{3}}\right)}\right) \]
   Proof

   [Start] 98.6	\[ \mathsf{log1p}\left(2 \cdot x - \left(0.5 \cdot \frac{1}{x} + \left(1 + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]
   associate--r+ [=>] 98.6	\[ \mathsf{log1p}\left(\color{blue}{\left(2 \cdot x - 0.5 \cdot \frac{1}{x}\right) - \left(1 + 0.125 \cdot \frac{1}{{x}^{3}}\right)}\right) \]
   *-commutative [=>] 98.6	\[ \mathsf{log1p}\left(\left(\color{blue}{x \cdot 2} - 0.5 \cdot \frac{1}{x}\right) - \left(1 + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]
   associate-*r/ [=>] 98.6	\[ \mathsf{log1p}\left(\left(x \cdot 2 - \color{blue}{\frac{0.5 \cdot 1}{x}}\right) - \left(1 + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]
   metadata-eval [=>] 98.6	\[ \mathsf{log1p}\left(\left(x \cdot 2 - \frac{\color{blue}{0.5}}{x}\right) - \left(1 + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]
   associate-*r/ [=>] 98.6	\[ \mathsf{log1p}\left(\left(x \cdot 2 - \frac{0.5}{x}\right) - \left(1 + \color{blue}{\frac{0.125 \cdot 1}{{x}^{3}}}\right)\right) \]
   metadata-eval [=>] 98.6	\[ \mathsf{log1p}\left(\left(x \cdot 2 - \frac{0.5}{x}\right) - \left(1 + \frac{\color{blue}{0.125}}{{x}^{3}}\right)\right) \]

6. Applied egg-rr 98.6%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot 2 + \left(\left(\frac{-0.5}{x} - 1\right) + \left(-\frac{0.25}{x \cdot x}\right) \cdot \frac{0.5}{x}\right)}\right) \]
   Proof

   [Start] 98.6	\[ \mathsf{log1p}\left(\left(x \cdot 2 - \frac{0.5}{x}\right) - \left(1 + \frac{0.125}{{x}^{3}}\right)\right) \]
   associate--r+ [=>] 98.6	\[ \mathsf{log1p}\left(\color{blue}{\left(\left(x \cdot 2 - \frac{0.5}{x}\right) - 1\right) - \frac{0.125}{{x}^{3}}}\right) \]
   add-cube-cbrt [=>] 98.6	\[ \mathsf{log1p}\left(\left(\left(x \cdot 2 - \frac{0.5}{x}\right) - 1\right) - \color{blue}{\left(\sqrt[3]{\frac{0.125}{{x}^{3}}} \cdot \sqrt[3]{\frac{0.125}{{x}^{3}}}\right) \cdot \sqrt[3]{\frac{0.125}{{x}^{3}}}}\right) \]
   cancel-sign-sub-inv [=>] 98.6	\[ \mathsf{log1p}\left(\color{blue}{\left(\left(x \cdot 2 - \frac{0.5}{x}\right) - 1\right) + \left(-\sqrt[3]{\frac{0.125}{{x}^{3}}} \cdot \sqrt[3]{\frac{0.125}{{x}^{3}}}\right) \cdot \sqrt[3]{\frac{0.125}{{x}^{3}}}}\right) \]
   sub-neg [=>] 98.6	\[ \mathsf{log1p}\left(\left(\color{blue}{\left(x \cdot 2 + \left(-\frac{0.5}{x}\right)\right)} - 1\right) + \left(-\sqrt[3]{\frac{0.125}{{x}^{3}}} \cdot \sqrt[3]{\frac{0.125}{{x}^{3}}}\right) \cdot \sqrt[3]{\frac{0.125}{{x}^{3}}}\right) \]
   associate--l+ [=>] 98.6	\[ \mathsf{log1p}\left(\color{blue}{\left(x \cdot 2 + \left(\left(-\frac{0.5}{x}\right) - 1\right)\right)} + \left(-\sqrt[3]{\frac{0.125}{{x}^{3}}} \cdot \sqrt[3]{\frac{0.125}{{x}^{3}}}\right) \cdot \sqrt[3]{\frac{0.125}{{x}^{3}}}\right) \]
   associate-+l+ [=>] 98.6	\[ \mathsf{log1p}\left(\color{blue}{x \cdot 2 + \left(\left(\left(-\frac{0.5}{x}\right) - 1\right) + \left(-\sqrt[3]{\frac{0.125}{{x}^{3}}} \cdot \sqrt[3]{\frac{0.125}{{x}^{3}}}\right) \cdot \sqrt[3]{\frac{0.125}{{x}^{3}}}\right)}\right) \]
   distribute-neg-frac [=>] 98.6	\[ \mathsf{log1p}\left(x \cdot 2 + \left(\left(\color{blue}{\frac{-0.5}{x}} - 1\right) + \left(-\sqrt[3]{\frac{0.125}{{x}^{3}}} \cdot \sqrt[3]{\frac{0.125}{{x}^{3}}}\right) \cdot \sqrt[3]{\frac{0.125}{{x}^{3}}}\right)\right) \]
   metadata-eval [=>] 98.6	\[ \mathsf{log1p}\left(x \cdot 2 + \left(\left(\frac{\color{blue}{-0.5}}{x} - 1\right) + \left(-\sqrt[3]{\frac{0.125}{{x}^{3}}} \cdot \sqrt[3]{\frac{0.125}{{x}^{3}}}\right) \cdot \sqrt[3]{\frac{0.125}{{x}^{3}}}\right)\right) \]

7. Final simplification 98.6%

   \[\leadsto \mathsf{log1p}\left(x \cdot 2 + \left(\left(\frac{-0.5}{x} + -1\right) + \frac{0.25}{x \cdot x} \cdot \frac{-0.5}{x}\right)\right) \]

Alternatives

Alternative 1
Accuracy	98.6%
Cost	3808

\[\mathsf{log1p}\left(\left(x \cdot 2 + \frac{-0.5}{x}\right) + \left(\frac{0.25}{x \cdot x} \cdot \frac{-0.5}{x} + -1\right)\right) \]

Alternative 2
Accuracy	98.2%
Cost	3424

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

Alternative 3
Accuracy	97.6%
Cost	3328

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

Alternative 4
Accuracy	96.9%
Cost	3296

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

Alternative 5
Accuracy	6.1%
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2023131 
(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)))))