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

↓

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

Derivation

Initial program 0.0
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
Initial simplification0.0
\[\leadsto \left(\left(1 - v \cdot v\right) \cdot \frac{\sqrt{2}}{4}\right) \cdot \sqrt{(\left(-3 \cdot v\right) \cdot v + 1)_*}\]
Using strategy rm
Applied add-cbrt-cube0.0
\[\leadsto \color{blue}{\sqrt[3]{\left(\left(\left(\left(1 - v \cdot v\right) \cdot \frac{\sqrt{2}}{4}\right) \cdot \sqrt{(\left(-3 \cdot v\right) \cdot v + 1)_*}\right) \cdot \left(\left(\left(1 - v \cdot v\right) \cdot \frac{\sqrt{2}}{4}\right) \cdot \sqrt{(\left(-3 \cdot v\right) \cdot v + 1)_*}\right)\right) \cdot \left(\left(\left(1 - v \cdot v\right) \cdot \frac{\sqrt{2}}{4}\right) \cdot \sqrt{(\left(-3 \cdot v\right) \cdot v + 1)_*}\right)}}\]
Final simplification0.0
\[\leadsto \sqrt[3]{\left(\left(\sqrt{(\left(v \cdot -3\right) \cdot v + 1)_*} \cdot \left(\left(1 - v \cdot v\right) \cdot \frac{\sqrt{2}}{4}\right)\right) \cdot \left(\sqrt{(\left(v \cdot -3\right) \cdot v + 1)_*} \cdot \left(\left(1 - v \cdot v\right) \cdot \frac{\sqrt{2}}{4}\right)\right)\right) \cdot \left(\sqrt{(\left(v \cdot -3\right) \cdot v + 1)_*} \cdot \left(\left(1 - v \cdot v\right) \cdot \frac{\sqrt{2}}{4}\right)\right)}\]

Runtime

herbie shell --seed 2018251 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))

Error

Derivation

Runtime