Average Error: 31.3 → 0.2
Time: 11.0s
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 r977750 = x;
        double r977751 = r977750 * r977750;
        double r977752 = 1.0;
        double r977753 = r977751 - r977752;
        double r977754 = sqrt(r977753);
        double r977755 = r977750 + r977754;
        double r977756 = log(r977755);
        return r977756;
}
↓
double f(double x) {
        double r977757 = -0.125;
        double r977758 = x;
        double r977759 = r977758 * r977758;
        double r977760 = r977759 * r977758;
        double r977761 = r977757 / r977760;
        double r977762 = 2.0;
        double r977763 = -0.5;
        double r977764 = r977763 / r977758;
        double r977765 = fma(r977762, r977758, r977764);
        double r977766 = r977761 + r977765;
        double r977767 = log(r977766);
        return r977767;
}

Error

Bits error versus x

Derivation

  1. Initial program 31.3

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

  2. Simplified31.3

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

  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

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

  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 2019151 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))