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

↓

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

double f(double x) {
        double r11648401 = x;
        double r11648402 = r11648401 * r11648401;
        double r11648403 = 1.0;
        double r11648404 = r11648402 - r11648403;
        double r11648405 = sqrt(r11648404);
        double r11648406 = r11648401 + r11648405;
        double r11648407 = log(r11648406);
        return r11648407;
}

↓

double f(double x) {
        double r11648408 = -0.125;
        double r11648409 = x;
        double r11648410 = r11648408 / r11648409;
        double r11648411 = r11648409 * r11648409;
        double r11648412 = r11648410 / r11648411;
        double r11648413 = 2.0;
        double r11648414 = -0.5;
        double r11648415 = r11648414 / r11648409;
        double r11648416 = fma(r11648413, r11648409, r11648415);
        double r11648417 = r11648412 + r11648416;
        double r11648418 = log(r11648417);
        return r11648418;
}

\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)

Derivation

Initial program 31.5
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
Simplified31.5
\[\leadsto \color{blue}{\log \left(x + \sqrt{(x \cdot x + -1)_*}\right)}\]
Taylor expanded around inf 0.2
\[\leadsto \log \color{blue}{\left(2 \cdot x - \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]
Simplified0.2
\[\leadsto \log \color{blue}{\left((2 \cdot x + \left(\frac{\frac{-1}{2}}{x}\right))_* + \frac{\frac{\frac{-1}{8}}{x}}{x \cdot x}\right)}\]
Final simplification0.2
\[\leadsto \log \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} + (2 \cdot x + \left(\frac{\frac{-1}{2}}{x}\right))_*\right)\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))

Error

Derivation

Reproduce