Rust f32::acosh

Average Accuracy: 50.9% → 98.7%

Time: 11.0s

Precision: binary32

Cost: 6784

\[x \geq 1\]

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

float code(float x) {
	return logf((((x + x) - (0.125f * powf(x, -3.0f))) + (-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 + x) - (0.125e0 * (x ** (-3.0e0)))) + ((-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(Float32(x + x) - Float32(Float32(0.125) * (x ^ Float32(-3.0)))) + 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 + x) - (single(0.125) * (x ^ single(-3.0)))) + (single(-0.5) / x)));
end

Target

Original	50.9%
Target	99.3%
Herbie	98.7%

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

Derivation

1. Initial program 50.9%

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

2. Applied egg-rr 50.9%

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

   [Start] 50.9	\[ \log \left(x + \sqrt{x \cdot x - 1}\right) \]
   flip-+ [=>] 7.3	\[ \log \color{blue}{\left(\frac{x \cdot x - \sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}}{x - \sqrt{x \cdot x - 1}}\right)} \]
   clear-num [=>] 7.3	\[ \log \color{blue}{\left(\frac{1}{\frac{x - \sqrt{x \cdot x - 1}}{x \cdot x - \sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}}}\right)} \]
   *-un-lft-identity [=>] 7.3	\[ \log \left(\frac{1}{\frac{\color{blue}{1 \cdot \left(x - \sqrt{x \cdot x - 1}\right)}}{x \cdot x - \sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}}}\right) \]
   associate-/l* [=>] 7.3	\[ \log \left(\frac{1}{\color{blue}{\frac{1}{\frac{x \cdot x - \sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}}{x - \sqrt{x \cdot x - 1}}}}}\right) \]
   flip-+ [<=] 50.9	\[ \log \left(\frac{1}{\frac{1}{\color{blue}{x + \sqrt{x \cdot x - 1}}}}\right) \]
   fma-neg [=>] 50.9	\[ \log \left(\frac{1}{\frac{1}{x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}}}\right) \]
   metadata-eval [=>] 50.9	\[ \log \left(\frac{1}{\frac{1}{x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}}}\right) \]

3. Taylor expanded in x around inf 98.7%

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

4. Simplified 98.7%

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

   [Start] 98.7	\[ \log \left(\frac{1}{\frac{1}{x + \left(x - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)}}\right) \]
   associate--r+ [=>] 98.7	\[ \log \left(\frac{1}{\frac{1}{x + \color{blue}{\left(\left(x - 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)}}}\right) \]
   sub-neg [=>] 98.7	\[ \log \left(\frac{1}{\frac{1}{x + \color{blue}{\left(\left(x - 0.5 \cdot \frac{1}{x}\right) + \left(-0.125 \cdot \frac{1}{{x}^{3}}\right)\right)}}}\right) \]
   associate-+l- [=>] 98.7	\[ \log \left(\frac{1}{\frac{1}{x + \color{blue}{\left(x - \left(0.5 \cdot \frac{1}{x} - \left(-0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right)}}}\right) \]
   associate-*r/ [=>] 98.7	\[ \log \left(\frac{1}{\frac{1}{x + \left(x - \left(\color{blue}{\frac{0.5 \cdot 1}{x}} - \left(-0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right)}}\right) \]
   metadata-eval [=>] 98.7	\[ \log \left(\frac{1}{\frac{1}{x + \left(x - \left(\frac{\color{blue}{0.5}}{x} - \left(-0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right)}}\right) \]
   distribute-lft-neg-in [=>] 98.7	\[ \log \left(\frac{1}{\frac{1}{x + \left(x - \left(\frac{0.5}{x} - \color{blue}{\left(-0.125\right) \cdot \frac{1}{{x}^{3}}}\right)\right)}}\right) \]
   metadata-eval [=>] 98.7	\[ \log \left(\frac{1}{\frac{1}{x + \left(x - \left(\frac{0.5}{x} - \color{blue}{-0.125} \cdot \frac{1}{{x}^{3}}\right)\right)}}\right) \]
   associate-*r/ [=>] 98.7	\[ \log \left(\frac{1}{\frac{1}{x + \left(x - \left(\frac{0.5}{x} - \color{blue}{\frac{-0.125 \cdot 1}{{x}^{3}}}\right)\right)}}\right) \]
   metadata-eval [=>] 98.7	\[ \log \left(\frac{1}{\frac{1}{x + \left(x - \left(\frac{0.5}{x} - \frac{\color{blue}{-0.125}}{{x}^{3}}\right)\right)}}\right) \]

5. Applied egg-rr 98.7%

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

   [Start] 98.7	\[ \log \left(\frac{1}{\frac{1}{x + \left(x - \left(\frac{0.5}{x} - \frac{-0.125}{{x}^{3}}\right)\right)}}\right) \]
   remove-double-div [=>] 98.7	\[ \log \color{blue}{\left(x + \left(x - \left(\frac{0.5}{x} - \frac{-0.125}{{x}^{3}}\right)\right)\right)} \]
   associate-+r- [=>] 98.7	\[ \log \color{blue}{\left(\left(x + x\right) - \left(\frac{0.5}{x} - \frac{-0.125}{{x}^{3}}\right)\right)} \]
   *-un-lft-identity [=>] 98.7	\[ \log \left(\left(\color{blue}{1 \cdot x} + x\right) - \left(\frac{0.5}{x} - \frac{-0.125}{{x}^{3}}\right)\right) \]
   *-un-lft-identity [=>] 98.7	\[ \log \left(\left(1 \cdot x + \color{blue}{1 \cdot x}\right) - \left(\frac{0.5}{x} - \frac{-0.125}{{x}^{3}}\right)\right) \]
   distribute-rgt-out [=>] 98.7	\[ \log \left(\color{blue}{x \cdot \left(1 + 1\right)} - \left(\frac{0.5}{x} - \frac{-0.125}{{x}^{3}}\right)\right) \]
   metadata-eval [=>] 98.7	\[ \log \left(x \cdot \color{blue}{2} - \left(\frac{0.5}{x} - \frac{-0.125}{{x}^{3}}\right)\right) \]
   metadata-eval [<=] 98.7	\[ \log \left(x \cdot \color{blue}{\frac{1}{0.5}} - \left(\frac{0.5}{x} - \frac{-0.125}{{x}^{3}}\right)\right) \]
   div-inv [<=] 98.7	\[ \log \left(\color{blue}{\frac{x}{0.5}} - \left(\frac{0.5}{x} - \frac{-0.125}{{x}^{3}}\right)\right) \]
   sub-neg [=>] 98.7	\[ \log \left(\frac{x}{0.5} - \color{blue}{\left(\frac{0.5}{x} + \left(-\frac{-0.125}{{x}^{3}}\right)\right)}\right) \]
   +-commutative [=>] 98.7	\[ \log \left(\frac{x}{0.5} - \color{blue}{\left(\left(-\frac{-0.125}{{x}^{3}}\right) + \frac{0.5}{x}\right)}\right) \]
   distribute-neg-frac [=>] 98.7	\[ \log \left(\frac{x}{0.5} - \left(\color{blue}{\frac{--0.125}{{x}^{3}}} + \frac{0.5}{x}\right)\right) \]
   metadata-eval [=>] 98.7	\[ \log \left(\frac{x}{0.5} - \left(\frac{\color{blue}{0.125}}{{x}^{3}} + \frac{0.5}{x}\right)\right) \]
   metadata-eval [<=] 98.7	\[ \log \left(\frac{x}{0.5} - \left(\frac{\color{blue}{{0.5}^{3}}}{{x}^{3}} + \frac{0.5}{x}\right)\right) \]
   cube-div [<=] 98.7	\[ \log \left(\frac{x}{0.5} - \left(\color{blue}{{\left(\frac{0.5}{x}\right)}^{3}} + \frac{0.5}{x}\right)\right) \]
   associate--r+ [=>] 98.7	\[ \log \color{blue}{\left(\left(\frac{x}{0.5} - {\left(\frac{0.5}{x}\right)}^{3}\right) - \frac{0.5}{x}\right)} \]

6. Final simplification 98.7%

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

Alternatives

Alternative 1
Accuracy	97.5%
Cost	3392

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

Alternative 2
Accuracy	97.5%
Cost	3328

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

Alternative 3
Accuracy	96.9%
Cost	3296

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

Alternative 4
Accuracy	6.1%
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2023133 
(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)))))