Average Error: 31.4 → 0.2
Time: 23.3s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\left(x + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\left(x + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)
double f(double x) {
        double r4019940 = x;
        double r4019941 = r4019940 * r4019940;
        double r4019942 = 1.0;
        double r4019943 = r4019941 - r4019942;
        double r4019944 = sqrt(r4019943);
        double r4019945 = r4019940 + r4019944;
        double r4019946 = log(r4019945);
        return r4019946;
}
↓
double f(double x) {
        double r4019947 = x;
        double r4019948 = -0.5;
        double r4019949 = r4019948 / r4019947;
        double r4019950 = r4019947 + r4019949;
        double r4019951 = 0.125;
        double r4019952 = r4019951 / r4019947;
        double r4019953 = r4019947 * r4019947;
        double r4019954 = r4019952 / r4019953;
        double r4019955 = r4019950 - r4019954;
        double r4019956 = r4019947 + r4019955;
        double r4019957 = log(r4019956);
        return r4019957;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.4

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

  2. Simplified31.4

    \[\leadsto \color{blue}{\log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)}\]

  3. Taylor expanded around inf 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))