Average Error: 31.2 → 0.2
Time: 12.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} + \mathsf{fma}\left(2, x, \left(\frac{\frac{-1}{2}}{x}\right)\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} + \mathsf{fma}\left(2, x, \left(\frac{\frac{-1}{2}}{x}\right)\right)\right)
double f(double x) {
        double r3152881 = x;
        double r3152882 = r3152881 * r3152881;
        double r3152883 = 1.0;
        double r3152884 = r3152882 - r3152883;
        double r3152885 = sqrt(r3152884);
        double r3152886 = r3152881 + r3152885;
        double r3152887 = log(r3152886);
        return r3152887;
}
↓
double f(double x) {
        double r3152888 = -0.125;
        double r3152889 = x;
        double r3152890 = r3152888 / r3152889;
        double r3152891 = r3152889 * r3152889;
        double r3152892 = r3152890 / r3152891;
        double r3152893 = 2.0;
        double r3152894 = -0.5;
        double r3152895 = r3152894 / r3152889;
        double r3152896 = fma(r3152893, r3152889, r3152895);
        double r3152897 = r3152892 + r3152896;
        double r3152898 = log(r3152897);
        return r3152898;
}

Error

Bits error versus x

Derivation

  1. Initial program 31.2

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

  2. Simplified31.2

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

  3. Taylor expanded around inf 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))