Average Error: 31.0 → 0.2
Time: 9.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\left(x - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right) - \frac{\frac{1}{2}}{x}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\left(x - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right) - \frac{\frac{1}{2}}{x}\right)\right)
double f(double x) {
        double r1396079 = x;
        double r1396080 = r1396079 * r1396079;
        double r1396081 = 1.0;
        double r1396082 = r1396080 - r1396081;
        double r1396083 = sqrt(r1396082);
        double r1396084 = r1396079 + r1396083;
        double r1396085 = log(r1396084);
        return r1396085;
}
↓
double f(double x) {
        double r1396086 = x;
        double r1396087 = 0.125;
        double r1396088 = r1396086 * r1396086;
        double r1396089 = r1396086 * r1396088;
        double r1396090 = r1396087 / r1396089;
        double r1396091 = r1396086 - r1396090;
        double r1396092 = 0.5;
        double r1396093 = r1396092 / r1396086;
        double r1396094 = r1396091 - r1396093;
        double r1396095 = r1396086 + r1396094;
        double r1396096 = log(r1396095);
        return r1396096;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.0

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

  2. Taylor expanded around inf 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019156 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))