Average Error: 31.9 → 0.3
Time: 2.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(x - \frac{1}{x} \cdot \left(\frac{0.125}{{x}^{2}} + 0.5\right)\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(x - \frac{1}{x} \cdot \left(\frac{0.125}{{x}^{2}} + 0.5\right)\right)\right)
double code(double x) {
	return log((x + sqrt(((x * x) - 1.0))));
}
↓
double code(double x) {
	return log((x + (x - ((1.0 / x) * ((0.125 / pow(x, 2.0)) + 0.5)))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.9

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

  2. Taylor expanded around inf 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \log \left(x + \left(x - \frac{1}{x} \cdot \left(\frac{0.125}{{x}^{2}} + 0.5\right)\right)\right)\]

Reproduce

herbie shell --seed 2020071 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))