Error

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

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

4. Applied flip--0.0

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

5. Applied associate-/r/0.0

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

6. Applied associate-/l*0.0

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

7. Simplified0.0

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

8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))

Details

Time bar (total: 36.8s)Debug log