Average Error: 32.2 → 0.3
Time: 6.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log 2 - \left(\left(\frac{0.09375}{{x}^{4}} + \frac{0.25}{x \cdot x}\right) - \log x\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log 2 - \left(\left(\frac{0.09375}{{x}^{4}} + \frac{0.25}{x \cdot x}\right) - \log x\right)
double f(double x) {
        double r58511 = x;
        double r58512 = r58511 * r58511;
        double r58513 = 1.0;
        double r58514 = r58512 - r58513;
        double r58515 = sqrt(r58514);
        double r58516 = r58511 + r58515;
        double r58517 = log(r58516);
        return r58517;
}
↓
double f(double x) {
        double r58518 = 2.0;
        double r58519 = log(r58518);
        double r58520 = 0.09375;
        double r58521 = x;
        double r58522 = 4.0;
        double r58523 = pow(r58521, r58522);
        double r58524 = r58520 / r58523;
        double r58525 = 0.25;
        double r58526 = r58521 * r58521;
        double r58527 = r58525 / r58526;
        double r58528 = r58524 + r58527;
        double r58529 = log(r58521);
        double r58530 = r58528 - r58529;
        double r58531 = r58519 - r58530;
        return r58531;
}

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

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2020046 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))