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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. Simplified0.0

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

4. Applied add-sqr-sqrt0.0

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

5. Applied *-un-lft-identity0.0

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

6. Applied times-frac0.0

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

7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019005 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))

Details

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