Average Error: 31.3 → 0.2
Time: 28.3s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\mathsf{fma}\left(2, x, \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\mathsf{fma}\left(2, x, \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)
double f(double x) {
        double r1215332 = x;
        double r1215333 = r1215332 * r1215332;
        double r1215334 = 1.0;
        double r1215335 = r1215333 - r1215334;
        double r1215336 = sqrt(r1215335);
        double r1215337 = r1215332 + r1215336;
        double r1215338 = log(r1215337);
        return r1215338;
}
↓
double f(double x) {
        double r1215339 = 2.0;
        double r1215340 = x;
        double r1215341 = -0.5;
        double r1215342 = r1215341 / r1215340;
        double r1215343 = fma(r1215339, r1215340, r1215342);
        double r1215344 = 0.125;
        double r1215345 = r1215344 / r1215340;
        double r1215346 = r1215340 * r1215340;
        double r1215347 = r1215345 / r1215346;
        double r1215348 = r1215343 - r1215347;
        double r1215349 = log(r1215348);
        return r1215349;
}

Error

Bits error versus x

Derivation

  1. Initial program 31.3

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

  2. Simplified31.3

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

  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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))