Average Error: 30.9 → 0.3
Time: 21.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} + (2 \cdot x + \left(\frac{\frac{-1}{2}}{x}\right))_*\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} + (2 \cdot x + \left(\frac{\frac{-1}{2}}{x}\right))_*\right)
double f(double x) {
        double r5491332 = x;
        double r5491333 = r5491332 * r5491332;
        double r5491334 = 1.0;
        double r5491335 = r5491333 - r5491334;
        double r5491336 = sqrt(r5491335);
        double r5491337 = r5491332 + r5491336;
        double r5491338 = log(r5491337);
        return r5491338;
}
↓
double f(double x) {
        double r5491339 = -0.125;
        double r5491340 = x;
        double r5491341 = r5491339 / r5491340;
        double r5491342 = r5491340 * r5491340;
        double r5491343 = r5491341 / r5491342;
        double r5491344 = 2.0;
        double r5491345 = -0.5;
        double r5491346 = r5491345 / r5491340;
        double r5491347 = fma(r5491344, r5491340, r5491346);
        double r5491348 = r5491343 + r5491347;
        double r5491349 = log(r5491348);
        return r5491349;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.9

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

  2. Simplified30.9

    \[\leadsto \color{blue}{\log \left(x + \sqrt{(x \cdot x + -1)_*}\right)}\]

  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

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

  5. Final simplification0.3

    \[\leadsto \log \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} + (2 \cdot x + \left(\frac{\frac{-1}{2}}{x}\right))_*\right)\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))