Average Error: 32.3 → 0.3
Time: 3.3s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log 2 + \left(\log x - \left(\frac{0.09375}{{x}^{4}} + \frac{0.25}{x \cdot x}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log 2 + \left(\log x - \left(\frac{0.09375}{{x}^{4}} + \frac{0.25}{x \cdot x}\right)\right)
(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (+ (log 2.0) (- (log x) (+ (/ 0.09375 (pow x 4.0)) (/ 0.25 (* x x))))))
double code(double x) {
	return ((double) log(((double) (x + ((double) sqrt(((double) (((double) (x * x)) - 1.0))))))));
}
↓
double code(double x) {
	return ((double) (((double) log(2.0)) + ((double) (((double) log(x)) - ((double) ((0.09375 / ((double) pow(x, 4.0))) + (0.25 / ((double) (x * x)))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program Error: 32.3 bits

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

  2. Taylor expanded around inf Error: 0.3 bits

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

  3. SimplifiedError: 0.3 bits

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

  4. Final simplificationError: 0.3 bits

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

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))