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Derivation

1. Initial program 31.0

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

2. Initial simplification31.0

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

3. Taylor expanded around inf 0.2

   \[\leadsto \log \left(x + \color{blue}{\left(x - \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\]

4. Simplified0.2

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

5. Final simplification0.2

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

Runtime

Time bar (total: 10.5s)Debug logProfile

herbie shell --seed 2018355 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))