Average Error: 32.2 → 0.4
Time: 3.6s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\left(\log 2 + 2 \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(\log \left(\sqrt[3]{x}\right) - \left(\frac{0.09375}{{x}^{4}} + \frac{0.25}{x \cdot x}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\left(\log 2 + 2 \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(\log \left(\sqrt[3]{x}\right) - \left(\frac{0.09375}{{x}^{4}} + \frac{0.25}{x \cdot x}\right)\right)
double code(double x) {
	return ((double) log(((double) (x + ((double) sqrt(((double) (((double) (x * x)) - 1.0))))))));
}
↓
double code(double x) {
	return ((double) (((double) (((double) log(2.0)) + ((double) (2.0 * ((double) log(((double) cbrt(x)))))))) + ((double) (((double) log(((double) cbrt(x)))) - ((double) (((double) (0.09375 / ((double) pow(x, 4.0)))) + ((double) (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 32.2

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

  2. Taylor expanded around inf 0.3

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

  3. Simplified0.3

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

  5. Applied add-cube-cbrt0.3

    \[\leadsto \log 2 + \left(\log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - \left(\frac{0.09375}{{x}^{4}} + \frac{0.25}{x \cdot x}\right)\right)\]

  6. Applied log-prod0.5

    \[\leadsto \log 2 + \left(\color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - \left(\frac{0.09375}{{x}^{4}} + \frac{0.25}{x \cdot x}\right)\right)\]

  7. Applied associate--l+0.5

    \[\leadsto \log 2 + \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) - \left(\frac{0.09375}{{x}^{4}} + \frac{0.25}{x \cdot x}\right)\right)\right)}\]

  8. Applied associate-+r+0.4

    \[\leadsto \color{blue}{\left(\log 2 + \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + \left(\log \left(\sqrt[3]{x}\right) - \left(\frac{0.09375}{{x}^{4}} + \frac{0.25}{x \cdot x}\right)\right)}\]

  9. Simplified0.4

    \[\leadsto \color{blue}{\left(\log 2 + 2 \cdot \log \left(\sqrt[3]{x}\right)\right)} + \left(\log \left(\sqrt[3]{x}\right) - \left(\frac{0.09375}{{x}^{4}} + \frac{0.25}{x \cdot x}\right)\right)\]

  10. Final simplification0.4

    \[\leadsto \left(\log 2 + 2 \cdot \log \left(\sqrt[3]{x}\right)\right) + \left(\log \left(\sqrt[3]{x}\right) - \left(\frac{0.09375}{{x}^{4}} + \frac{0.25}{x \cdot x}\right)\right)\]

Reproduce

herbie shell --seed 2020185 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))