Average Error: 31.0 → 0.2
Time: 17.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\mathsf{fma}\left(2, x, \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x \cdot x}}{x}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\mathsf{fma}\left(2, x, \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x \cdot x}}{x}\right)
double f(double x) {
        double r2384608 = x;
        double r2384609 = r2384608 * r2384608;
        double r2384610 = 1.0;
        double r2384611 = r2384609 - r2384610;
        double r2384612 = sqrt(r2384611);
        double r2384613 = r2384608 + r2384612;
        double r2384614 = log(r2384613);
        return r2384614;
}
↓
double f(double x) {
        double r2384615 = 2.0;
        double r2384616 = x;
        double r2384617 = -0.5;
        double r2384618 = r2384617 / r2384616;
        double r2384619 = fma(r2384615, r2384616, r2384618);
        double r2384620 = 0.125;
        double r2384621 = r2384616 * r2384616;
        double r2384622 = r2384620 / r2384621;
        double r2384623 = r2384622 / r2384616;
        double r2384624 = r2384619 - r2384623;
        double r2384625 = log(r2384624);
        return r2384625;
}

Error

Bits error versus x

Derivation

  1. Initial program 31.0

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

  2. Simplified31.0

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

  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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))