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

↓

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

double f(double x) {
        double r8603220 = x;
        double r8603221 = r8603220 * r8603220;
        double r8603222 = 1.0;
        double r8603223 = r8603221 - r8603222;
        double r8603224 = sqrt(r8603223);
        double r8603225 = r8603220 + r8603224;
        double r8603226 = log(r8603225);
        return r8603226;
}

↓

double f(double x) {
        double r8603227 = -0.125;
        double r8603228 = x;
        double r8603229 = r8603227 / r8603228;
        double r8603230 = r8603228 * r8603228;
        double r8603231 = r8603229 / r8603230;
        double r8603232 = 2.0;
        double r8603233 = -0.5;
        double r8603234 = r8603233 / r8603228;
        double r8603235 = fma(r8603232, r8603228, r8603234);
        double r8603236 = r8603231 + r8603235;
        double r8603237 = log(r8603236);
        return r8603237;
}

\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 30.5
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
Simplified30.5
\[\leadsto \color{blue}{\log \left(x + \sqrt{(x \cdot x + -1)_*}\right)}\]
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 2019101 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))

Error

Derivation

Reproduce