Average Error: 31.1 → 0.3
Time: 22.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\mathsf{fma}\left(x, 2, \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x}\right)\right) - \frac{\frac{1}{2}}{x}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\mathsf{fma}\left(x, 2, \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x}\right)\right) - \frac{\frac{1}{2}}{x}\right)
double f(double x) {
        double r2588342 = x;
        double r2588343 = r2588342 * r2588342;
        double r2588344 = 1.0;
        double r2588345 = r2588343 - r2588344;
        double r2588346 = sqrt(r2588345);
        double r2588347 = r2588342 + r2588346;
        double r2588348 = log(r2588347);
        return r2588348;
}
↓
double f(double x) {
        double r2588349 = x;
        double r2588350 = 2.0;
        double r2588351 = -0.125;
        double r2588352 = r2588349 * r2588349;
        double r2588353 = r2588352 * r2588349;
        double r2588354 = r2588351 / r2588353;
        double r2588355 = fma(r2588349, r2588350, r2588354);
        double r2588356 = 0.5;
        double r2588357 = r2588356 / r2588349;
        double r2588358 = r2588355 - r2588357;
        double r2588359 = log(r2588358);
        return r2588359;
}

Error

Bits error versus x

Derivation

  1. Initial program 31.1

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

  2. Simplified31.1

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

  3. Taylor expanded around inf 0.3

    \[\leadsto \log \color{blue}{\left(2 \cdot x - \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))