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Derivation

1. Initial program 1.0

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

2. Initial simplification0.0

   \[\leadsto \frac{\frac{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}\]
3. Using strategy rm

4. Applied flip-+0.0

   \[\leadsto \frac{\frac{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}{\sqrt{\color{blue}{\frac{2 \cdot 2 - \left(\left(v \cdot v\right) \cdot -6\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)}{2 - \left(v \cdot v\right) \cdot -6}}}}\]

5. Applied sqrt-div1.0

   \[\leadsto \frac{\frac{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}{\color{blue}{\frac{\sqrt{2 \cdot 2 - \left(\left(v \cdot v\right) \cdot -6\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)}}{\sqrt{2 - \left(v \cdot v\right) \cdot -6}}}}\]

6. Applied associate-/r/0.0

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}{\sqrt{2 \cdot 2 - \left(\left(v \cdot v\right) \cdot -6\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)}} \cdot \sqrt{2 - \left(v \cdot v\right) \cdot -6}}\]

7. Final simplification0.0

   \[\leadsto \frac{\frac{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}{\sqrt{4 - \left(\left(v \cdot v\right) \cdot -6\right) \cdot \left(\left(v \cdot v\right) \cdot -6\right)}} \cdot \sqrt{2 - \left(v \cdot v\right) \cdot -6}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018263 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))