Average Error: 31.6 → 0.2
Time: 38.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\mathsf{fma}\left(x, 2, \frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x}\right) - \frac{\frac{1}{2}}{x}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\mathsf{fma}\left(x, 2, \frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x}\right) - \frac{\frac{1}{2}}{x}\right)
double f(double x) {
        double r1869724 = x;
        double r1869725 = r1869724 * r1869724;
        double r1869726 = 1.0;
        double r1869727 = r1869725 - r1869726;
        double r1869728 = sqrt(r1869727);
        double r1869729 = r1869724 + r1869728;
        double r1869730 = log(r1869729);
        return r1869730;
}
↓
double f(double x) {
        double r1869731 = x;
        double r1869732 = 2.0;
        double r1869733 = -0.125;
        double r1869734 = r1869731 * r1869731;
        double r1869735 = r1869734 * r1869731;
        double r1869736 = r1869733 / r1869735;
        double r1869737 = fma(r1869731, r1869732, r1869736);
        double r1869738 = 0.5;
        double r1869739 = r1869738 / r1869731;
        double r1869740 = r1869737 - r1869739;
        double r1869741 = log(r1869740);
        return r1869741;
}

Error

Bits error versus x

Derivation

  1. Initial program 31.6

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

  2. Simplified31.6

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

  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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))