Average Error: 32.0 → 0.2
Time: 7.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(x - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(x - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right)\right)
double f(double x) {
        double r37048 = x;
        double r37049 = r37048 * r37048;
        double r37050 = 1.0;
        double r37051 = r37049 - r37050;
        double r37052 = sqrt(r37051);
        double r37053 = r37048 + r37052;
        double r37054 = log(r37053);
        return r37054;
}
↓
double f(double x) {
        double r37055 = x;
        double r37056 = 0.5;
        double r37057 = r37056 / r37055;
        double r37058 = 0.125;
        double r37059 = 3.0;
        double r37060 = pow(r37055, r37059);
        double r37061 = r37058 / r37060;
        double r37062 = r37057 + r37061;
        double r37063 = r37055 - r37062;
        double r37064 = r37055 + r37063;
        double r37065 = log(r37064);
        return r37065;
}

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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))