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Derivation

1. Initial program 0.1

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

2. Initial simplification0.1

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

4. Applied div-inv0.1

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

5. Applied *-un-lft-identity0.1

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

6. Applied distribute-rgt-out0.1

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

7. Final simplification0.1

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

Runtime

Time bar (total: 20.9s)Debug logProfile

herbie shell --seed 2018252 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))