Average Error: 30.5 → 0.2
Time: 16.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} + \mathsf{fma}\left(2, x, \left(\frac{\frac{-1}{2}}{x}\right)\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} + \mathsf{fma}\left(2, x, \left(\frac{\frac{-1}{2}}{x}\right)\right)\right)
double f(double x) {
        double r10344515 = x;
        double r10344516 = r10344515 * r10344515;
        double r10344517 = 1.0;
        double r10344518 = r10344516 - r10344517;
        double r10344519 = sqrt(r10344518);
        double r10344520 = r10344515 + r10344519;
        double r10344521 = log(r10344520);
        return r10344521;
}
↓
double f(double x) {
        double r10344522 = -0.125;
        double r10344523 = x;
        double r10344524 = r10344522 / r10344523;
        double r10344525 = r10344523 * r10344523;
        double r10344526 = r10344524 / r10344525;
        double r10344527 = 2.0;
        double r10344528 = -0.5;
        double r10344529 = r10344528 / r10344523;
        double r10344530 = fma(r10344527, r10344523, r10344529);
        double r10344531 = r10344526 + r10344530;
        double r10344532 = log(r10344531);
        return r10344532;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.5

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

  2. Simplified30.5

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

  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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))