Average Error: 32.6 → 0.3
Time: 14.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\left(x - \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\left(x - \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)
double f(double x) {
        double r64656 = x;
        double r64657 = r64656 * r64656;
        double r64658 = 1.0;
        double r64659 = r64657 - r64658;
        double r64660 = sqrt(r64659);
        double r64661 = r64656 + r64660;
        double r64662 = log(r64661);
        return r64662;
}
↓
double f(double x) {
        double r64663 = x;
        double r64664 = 0.5;
        double r64665 = r64664 / r64663;
        double r64666 = r64663 - r64665;
        double r64667 = 0.125;
        double r64668 = 3.0;
        double r64669 = pow(r64663, r64668);
        double r64670 = r64667 / r64669;
        double r64671 = r64666 - r64670;
        double r64672 = r64663 + r64671;
        double r64673 = log(r64672);
        return r64673;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.6

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

  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

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))