Average Error: 30.8 → 0.2
Time: 10.3s
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 r5713553 = x;
        double r5713554 = r5713553 * r5713553;
        double r5713555 = 1.0;
        double r5713556 = r5713554 - r5713555;
        double r5713557 = sqrt(r5713556);
        double r5713558 = r5713553 + r5713557;
        double r5713559 = log(r5713558);
        return r5713559;
}
↓
double f(double x) {
        double r5713560 = -0.125;
        double r5713561 = x;
        double r5713562 = r5713560 / r5713561;
        double r5713563 = r5713561 * r5713561;
        double r5713564 = r5713562 / r5713563;
        double r5713565 = 2.0;
        double r5713566 = -0.5;
        double r5713567 = r5713566 / r5713561;
        double r5713568 = fma(r5713565, r5713561, r5713567);
        double r5713569 = r5713564 + r5713568;
        double r5713570 = log(r5713569);
        return r5713570;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.8

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

  2. Simplified30.8

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

  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 2019124 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))