Average Error: 30.7 → 0.3
Time: 11.4s
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 r4721279 = x;
        double r4721280 = r4721279 * r4721279;
        double r4721281 = 1.0;
        double r4721282 = r4721280 - r4721281;
        double r4721283 = sqrt(r4721282);
        double r4721284 = r4721279 + r4721283;
        double r4721285 = log(r4721284);
        return r4721285;
}
↓
double f(double x) {
        double r4721286 = -0.125;
        double r4721287 = x;
        double r4721288 = r4721286 / r4721287;
        double r4721289 = r4721287 * r4721287;
        double r4721290 = r4721288 / r4721289;
        double r4721291 = 2.0;
        double r4721292 = -0.5;
        double r4721293 = r4721292 / r4721287;
        double r4721294 = fma(r4721291, r4721287, r4721293);
        double r4721295 = r4721290 + r4721294;
        double r4721296 = log(r4721295);
        return r4721296;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.7

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

  2. Simplified30.7

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

  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 2019107 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))