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Derivation

1. Initial program 30.7

   \[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
2. Using strategy rm

3. Applied difference-of-sqr-130.7

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

4. Applied sqrt-prod0.1

   \[\leadsto \log \left(x + \color{blue}{\sqrt{x + 1} \cdot \sqrt{x - 1}}\right)\]
5. Using strategy rm

6. Applied add-sqr-sqrt0.1

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

7. Applied sqrt-prod0.1

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

8. Applied associate-*l*0.1

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

9. Final simplification0.1

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

Runtime

Time bar (total: 31.8s)Debug logProfile

herbie shell --seed 2018263 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))