Rust f32::acosh

Average Error: 49.3% → 98.8%

Time: 7.3s

Precision: binary32

Cost: 10144.00

\[x \geq 1\]

Math

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

↓

\[\log \left(x + \left(\left(x + \frac{-0.5}{x}\right) + \left(\frac{-0.125}{{x}^{3}} + \frac{-0.0625}{{x}^{5}}\right)\right)\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary32
 (log
  (+
   x
   (+ (+ x (/ -0.5 x)) (+ (/ -0.125 (pow x 3.0)) (/ -0.0625 (pow x 5.0)))))))

C

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

↓

float code(float x) {
	return logf((x + ((x + (-0.5f / x)) + ((-0.125f / powf(x, 3.0f)) + (-0.0625f / powf(x, 5.0f))))));
}

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 + ((x + ((-0.5e0) / x)) + (((-0.125e0) / (x ** 3.0e0)) + ((-0.0625e0) / (x ** 5.0e0))))))
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(x + Float32(Float32(x + Float32(Float32(-0.5) / x)) + Float32(Float32(Float32(-0.125) / (x ^ Float32(3.0))) + Float32(Float32(-0.0625) / (x ^ Float32(5.0)))))))
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(-0.5) / x)) + ((single(-0.125) / (x ^ single(3.0))) + (single(-0.0625) / (x ^ single(5.0)))))));
end

Target

Original	49.3%
Target	99.3%
Herbie	98.8%

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

Derivation

1. Initial program 49.3

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

2. Taylor expanded in x around inf 98.8

   \[\leadsto \log \left(x + \color{blue}{\left(x - \left(0.5 \cdot \frac{1}{x} + \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right)}\right) \]

3. Simplified 98.8

   \[\leadsto \log \left(x + \color{blue}{\left(\left(x - \frac{0.5}{x}\right) - \left(\frac{0.125}{{x}^{3}} + \frac{0.0625}{{x}^{5}}\right)\right)}\right) \]
   Proof

   [Start] 98.8	\[ \log \left(x + \left(x - \left(0.5 \cdot \frac{1}{x} + \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right) \]
   associate--r+ [=>] 98.8	\[ \log \left(x + \color{blue}{\left(\left(x - 0.5 \cdot \frac{1}{x}\right) - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)}\right) \]
   associate-*r/ [=>] 98.8	\[ \log \left(x + \left(\left(x - \color{blue}{\frac{0.5 \cdot 1}{x}}\right) - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]
   metadata-eval [=>] 98.8	\[ \log \left(x + \left(\left(x - \frac{\color{blue}{0.5}}{x}\right) - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]
   +-commutative [=>] 98.8	\[ \log \left(x + \left(\left(x - \frac{0.5}{x}\right) - \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)}\right)\right) \]
   associate-*r/ [=>] 98.8	\[ \log \left(x + \left(\left(x - \frac{0.5}{x}\right) - \left(\color{blue}{\frac{0.125 \cdot 1}{{x}^{3}}} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)\right) \]
   metadata-eval [=>] 98.8	\[ \log \left(x + \left(\left(x - \frac{0.5}{x}\right) - \left(\frac{\color{blue}{0.125}}{{x}^{3}} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)\right) \]
   associate-*r/ [=>] 98.8	\[ \log \left(x + \left(\left(x - \frac{0.5}{x}\right) - \left(\frac{0.125}{{x}^{3}} + \color{blue}{\frac{0.0625 \cdot 1}{{x}^{5}}}\right)\right)\right) \]
   metadata-eval [=>] 98.8	\[ \log \left(x + \left(\left(x - \frac{0.5}{x}\right) - \left(\frac{0.125}{{x}^{3}} + \frac{\color{blue}{0.0625}}{{x}^{5}}\right)\right)\right) \]

4. Final simplification 98.8

   \[\leadsto \log \left(x + \left(\left(x + \frac{-0.5}{x}\right) + \left(\frac{-0.125}{{x}^{3}} + \frac{-0.0625}{{x}^{5}}\right)\right)\right) \]

Alternatives

Alternative 1
Error	98.7%
Cost	3680.00

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

Alternative 2
Error	98.3%
Cost	3488.00

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

Alternative 3
Error	98.3%
Cost	3424.00

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

Alternative 4
Error	97.1%
Cost	3296.00

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

Alternative 5
Error	44.2%
Cost	3232.00

\[\log x \]

Alternative 6
Error	6.1%
Cost	32.00

\[0 \]

Reproduce

herbie shell --seed 2023097 
(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)))))