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Derivation

1. Initial program 31.2

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

2. Initial simplification31.2

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

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. Taylor expanded around inf 0.4

   \[\leadsto \color{blue}{\log 2 - \left(\log \left(\frac{1}{x}\right) + \left(\frac{3}{32} \cdot \frac{1}{{x}^{4}} + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right)\right)}\]

6. Simplified0.4

   \[\leadsto \color{blue}{\frac{\frac{-1}{4}}{x \cdot x} - \left(\frac{\frac{3}{32}}{{x}^{4}} - \left(\log x + \log 2\right)\right)}\]
7. Using strategy rm

8. Applied add-log-exp0.5

   \[\leadsto \frac{\frac{-1}{4}}{x \cdot x} - \left(\frac{\frac{3}{32}}{{x}^{4}} - \color{blue}{\log \left(e^{\log x + \log 2}\right)}\right)\]

9. Applied add-log-exp0.5

   \[\leadsto \frac{\frac{-1}{4}}{x \cdot x} - \left(\color{blue}{\log \left(e^{\frac{\frac{3}{32}}{{x}^{4}}}\right)} - \log \left(e^{\log x + \log 2}\right)\right)\]

10. Applied diff-log0.5

    \[\leadsto \frac{\frac{-1}{4}}{x \cdot x} - \color{blue}{\log \left(\frac{e^{\frac{\frac{3}{32}}{{x}^{4}}}}{e^{\log x + \log 2}}\right)}\]

11. Simplified0.2

    \[\leadsto \frac{\frac{-1}{4}}{x \cdot x} - \log \color{blue}{\left(\frac{\frac{1}{2}}{x} \cdot e^{\frac{\frac{3}{32}}{{x}^{4}}}\right)}\]

12. Final simplification0.2

    \[\leadsto \frac{\frac{-1}{4}}{x \cdot x} - \log \left(\frac{\frac{1}{2}}{x} \cdot e^{\frac{\frac{3}{32}}{{x}^{4}}}\right)\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018235 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))