Average Error: 32.2 → 0.3
Time: 4.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log 2 + \left(\left(\log x - \frac{\frac{0.25}{x}}{x}\right) - \frac{0.09375}{{x}^{4}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log 2 + \left(\left(\log x - \frac{\frac{0.25}{x}}{x}\right) - \frac{0.09375}{{x}^{4}}\right)
double f(double x) {
        double r74009 = x;
        double r74010 = r74009 * r74009;
        double r74011 = 1.0;
        double r74012 = r74010 - r74011;
        double r74013 = sqrt(r74012);
        double r74014 = r74009 + r74013;
        double r74015 = log(r74014);
        return r74015;
}
↓
double f(double x) {
        double r74016 = 2.0;
        double r74017 = log(r74016);
        double r74018 = x;
        double r74019 = log(r74018);
        double r74020 = 0.25;
        double r74021 = r74020 / r74018;
        double r74022 = r74021 / r74018;
        double r74023 = r74019 - r74022;
        double r74024 = 0.09375;
        double r74025 = 4.0;
        double r74026 = pow(r74018, r74025);
        double r74027 = r74024 / r74026;
        double r74028 = r74023 - r74027;
        double r74029 = r74017 + r74028;
        return r74029;
}

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

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

  4. Final simplification0.3

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

Reproduce

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