Average Error: 31.9 → 0.3
Time: 7.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(x - \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(x - \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)
double f(double x) {
        double r52618 = x;
        double r52619 = r52618 * r52618;
        double r52620 = 1.0;
        double r52621 = r52619 - r52620;
        double r52622 = sqrt(r52621);
        double r52623 = r52618 + r52622;
        double r52624 = log(r52623);
        return r52624;
}
↓
double f(double x) {
        double r52625 = x;
        double r52626 = 1.0;
        double r52627 = 2.0;
        double r52628 = r52626 / r52627;
        double r52629 = 1.0;
        double r52630 = r52629 / r52625;
        double r52631 = r52628 * r52630;
        double r52632 = 8.0;
        double r52633 = r52626 / r52632;
        double r52634 = 3.0;
        double r52635 = pow(r52625, r52634);
        double r52636 = r52629 / r52635;
        double r52637 = r52633 * r52636;
        double r52638 = r52631 + r52637;
        double r52639 = r52625 - r52638;
        double r52640 = r52625 + r52639;
        double r52641 = log(r52640);
        return r52641;
}

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

    \[\leadsto \log \left(x + \color{blue}{\left(x - \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)}\right)\]

  4. Final simplification0.3

    \[\leadsto \log \left(x + \left(x - \left(\frac{1}{2} \cdot \frac{1}{x} + \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))