Average Error: 31.1 → 0.2
Time: 20.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \mathsf{fma}\left(2, x, \frac{\frac{-1}{2}}{x}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \mathsf{fma}\left(2, x, \frac{\frac{-1}{2}}{x}\right)\right)
double f(double x) {
        double r2588927 = x;
        double r2588928 = r2588927 * r2588927;
        double r2588929 = 1.0;
        double r2588930 = r2588928 - r2588929;
        double r2588931 = sqrt(r2588930);
        double r2588932 = r2588927 + r2588931;
        double r2588933 = log(r2588932);
        return r2588933;
}
↓
double f(double x) {
        double r2588934 = -0.125;
        double r2588935 = x;
        double r2588936 = r2588935 * r2588935;
        double r2588937 = r2588936 * r2588935;
        double r2588938 = r2588934 / r2588937;
        double r2588939 = 2.0;
        double r2588940 = -0.5;
        double r2588941 = r2588940 / r2588935;
        double r2588942 = fma(r2588939, r2588935, r2588941);
        double r2588943 = r2588938 + r2588942;
        double r2588944 = log(r2588943);
        return r2588944;
}

Error

Bits error versus x

Derivation

  1. Initial program 31.1

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

  2. Simplified31.1

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

  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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))