Error

Derivation

1. Initial program 0.0

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

2. Initial simplification0.0

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

4. Applied flip--0.0

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

5. Applied frac-times0.0

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

6. Applied associate-*l/0.0

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

7. Simplified0.0

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

8. Final simplification0.0

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018349 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))