Average Error: 30.7 → 0.3
Time: 15.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\mathsf{fma}\left(2, x, \frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\mathsf{fma}\left(2, x, \frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)
double f(double x) {
        double r2052824 = x;
        double r2052825 = r2052824 * r2052824;
        double r2052826 = 1.0;
        double r2052827 = r2052825 - r2052826;
        double r2052828 = sqrt(r2052827);
        double r2052829 = r2052824 + r2052828;
        double r2052830 = log(r2052829);
        return r2052830;
}
↓
double f(double x) {
        double r2052831 = 2.0;
        double r2052832 = x;
        double r2052833 = -0.125;
        double r2052834 = r2052832 * r2052832;
        double r2052835 = r2052834 * r2052832;
        double r2052836 = r2052833 / r2052835;
        double r2052837 = 0.5;
        double r2052838 = r2052837 / r2052832;
        double r2052839 = r2052836 - r2052838;
        double r2052840 = fma(r2052831, r2052832, r2052839);
        double r2052841 = log(r2052840);
        return r2052841;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.7

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

  2. Simplified30.7

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

  3. Taylor expanded around inf 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))