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

↓

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

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Derivation

Initial program 31.1
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
Taylor expanded around inf 0.3
\[\leadsto \log \left(x + \color{blue}{\left(x - \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\]
Simplified0.3
\[\leadsto \log \left(x + \color{blue}{\left(\left(\frac{\frac{-1}{2}}{x} + x\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)}\right)\]
Final simplification0.3
\[\leadsto \log \left(x + \left(\left(x + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\]

herbie shell --seed 2018249 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))