Average Error: 30.7 → 0.2
Time: 13.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} + \mathsf{fma}\left(x, 2, \left(\frac{\frac{-1}{2}}{x}\right)\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} + \mathsf{fma}\left(x, 2, \left(\frac{\frac{-1}{2}}{x}\right)\right)\right)
double f(double x) {
        double r1250540 = x;
        double r1250541 = r1250540 * r1250540;
        double r1250542 = 1.0;
        double r1250543 = r1250541 - r1250542;
        double r1250544 = sqrt(r1250543);
        double r1250545 = r1250540 + r1250544;
        double r1250546 = log(r1250545);
        return r1250546;
}
↓
double f(double x) {
        double r1250547 = -0.125;
        double r1250548 = x;
        double r1250549 = r1250547 / r1250548;
        double r1250550 = r1250548 * r1250548;
        double r1250551 = r1250549 / r1250550;
        double r1250552 = 2.0;
        double r1250553 = -0.5;
        double r1250554 = r1250553 / r1250548;
        double r1250555 = fma(r1250548, r1250552, r1250554);
        double r1250556 = r1250551 + r1250555;
        double r1250557 = log(r1250556);
        return r1250557;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.7

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

  2. Simplified30.7

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

  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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))