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Derivation

1. Initial program 0.0

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

3. Applied add-cbrt-cube0.0

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

5. Applied pow1/20.0

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

6. Applied pow1/20.0

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

7. Applied pow1/20.0

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

8. Applied pow-prod-up0.0

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

9. Applied pow-prod-up0.0

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

10. Simplified0.0

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

11. Final simplification0.0

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018273 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))