Average Error: 32.0 → 0.2
Time: 5.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\left(x - \frac{0.125}{{x}^{3}}\right) - \frac{0.5}{x}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\left(x - \frac{0.125}{{x}^{3}}\right) - \frac{0.5}{x}\right)\right)
double f(double x) {
        double r65683 = x;
        double r65684 = r65683 * r65683;
        double r65685 = 1.0;
        double r65686 = r65684 - r65685;
        double r65687 = sqrt(r65686);
        double r65688 = r65683 + r65687;
        double r65689 = log(r65688);
        return r65689;
}
↓
double f(double x) {
        double r65690 = x;
        double r65691 = 0.125;
        double r65692 = 3.0;
        double r65693 = pow(r65690, r65692);
        double r65694 = r65691 / r65693;
        double r65695 = r65690 - r65694;
        double r65696 = 0.5;
        double r65697 = r65696 / r65690;
        double r65698 = r65695 - r65697;
        double r65699 = r65690 + r65698;
        double r65700 = log(r65699);
        return r65700;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.0

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

  2. Taylor expanded around inf 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2020043 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))