Average Error: 31.5 → 0.2
Time: 15.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} + \mathsf{fma}\left(x, 2, \left(\frac{\frac{-1}{2}}{x}\right)\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} + \mathsf{fma}\left(x, 2, \left(\frac{\frac{-1}{2}}{x}\right)\right)\right)
double f(double x) {
        double r2094740 = x;
        double r2094741 = r2094740 * r2094740;
        double r2094742 = 1.0;
        double r2094743 = r2094741 - r2094742;
        double r2094744 = sqrt(r2094743);
        double r2094745 = r2094740 + r2094744;
        double r2094746 = log(r2094745);
        return r2094746;
}
↓
double f(double x) {
        double r2094747 = -0.125;
        double r2094748 = x;
        double r2094749 = r2094747 / r2094748;
        double r2094750 = r2094748 * r2094748;
        double r2094751 = r2094749 / r2094750;
        double r2094752 = 2.0;
        double r2094753 = -0.5;
        double r2094754 = r2094753 / r2094748;
        double r2094755 = fma(r2094748, r2094752, r2094754);
        double r2094756 = r2094751 + r2094755;
        double r2094757 = log(r2094756);
        return r2094757;
}

Error

Bits error versus x

Derivation

  1. Initial program 31.5

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

  2. Simplified31.5

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

  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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))