Average Error: 31.4 → 0.4
Time: 4.7s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log 2 - \left(\frac{0.09375}{{x}^{4}} + \left(\frac{0.25}{x \cdot x} - \log x\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log 2 - \left(\frac{0.09375}{{x}^{4}} + \left(\frac{0.25}{x \cdot x} - \log x\right)\right)
(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (- (log 2.0) (+ (/ 0.09375 (pow x 4.0)) (- (/ 0.25 (* x x)) (log x)))))
double code(double x) {
	return log(x + sqrt((x * x) - 1.0));
}
↓
double code(double x) {
	return log(2.0) - ((0.09375 / pow(x, 4.0)) + ((0.25 / (x * x)) - log(x)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.4

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

  2. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{\log 2 - \left(\log \left(\frac{1}{x}\right) + \left(0.25 \cdot \frac{1}{{x}^{2}} + 0.09375 \cdot \frac{1}{{x}^{4}}\right)\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{\log 2 - \left(\frac{0.09375}{{x}^{4}} + \left(\frac{0.25}{x \cdot x} - \log x\right)\right)}\]

  4. Final simplification0.4

    \[\leadsto \log 2 - \left(\frac{0.09375}{{x}^{4}} + \left(\frac{0.25}{x \cdot x} - \log x\right)\right)\]

Reproduce

herbie shell --seed 2020308 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))