Average Error: 30.9 → 0.2
Time: 14.4s
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 r2254425 = x;
        double r2254426 = r2254425 * r2254425;
        double r2254427 = 1.0;
        double r2254428 = r2254426 - r2254427;
        double r2254429 = sqrt(r2254428);
        double r2254430 = r2254425 + r2254429;
        double r2254431 = log(r2254430);
        return r2254431;
}
↓
double f(double x) {
        double r2254432 = -0.125;
        double r2254433 = x;
        double r2254434 = r2254433 * r2254433;
        double r2254435 = r2254434 * r2254433;
        double r2254436 = r2254432 / r2254435;
        double r2254437 = 2.0;
        double r2254438 = -0.5;
        double r2254439 = r2254438 / r2254433;
        double r2254440 = fma(r2254437, r2254433, r2254439);
        double r2254441 = r2254436 + r2254440;
        double r2254442 = log(r2254441);
        return r2254442;
}

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

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

  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 2019164 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))