Average Error: 30.9 → 0.2
Time: 35.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\mathsf{fma}\left(2, x, \frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x}\right) - \frac{\frac{1}{2}}{x}\right)\]
\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}\right) - \frac{\frac{1}{2}}{x}\right)
double f(double x) {
        double r2003334 = x;
        double r2003335 = r2003334 * r2003334;
        double r2003336 = 1.0;
        double r2003337 = r2003335 - r2003336;
        double r2003338 = sqrt(r2003337);
        double r2003339 = r2003334 + r2003338;
        double r2003340 = log(r2003339);
        return r2003340;
}
↓
double f(double x) {
        double r2003341 = 2.0;
        double r2003342 = x;
        double r2003343 = -0.125;
        double r2003344 = r2003342 * r2003342;
        double r2003345 = r2003344 * r2003342;
        double r2003346 = r2003343 / r2003345;
        double r2003347 = fma(r2003341, r2003342, r2003346);
        double r2003348 = 0.5;
        double r2003349 = r2003348 / r2003342;
        double r2003350 = r2003347 - r2003349;
        double r2003351 = log(r2003350);
        return r2003351;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.9

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

  2. Simplified30.9

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

  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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))