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Derivation

1. Initial program 63.0

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

2. Initial simplification63.0

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

4. Applied add-cube-cbrt63.0

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

6. Applied pow1/363.0

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

7. Applied pow1/363.0

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

8. Applied pow-prod-up63.0

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

9. Simplified63.0

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

10. Final simplification63.0

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

Runtime

Time bar (total: 42.4s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))