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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. Using strategy rm

3. Applied add-cbrt-cube0.0

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

4. Applied add-cbrt-cube0.0

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

5. Applied add-cbrt-cube0.0

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

6. Applied cbrt-unprod0.0

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

7. Applied cbrt-unprod0.0

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

8. Simplified0.0

   \[\leadsto \sqrt[3]{\color{blue}{\left(\left(1 - v \cdot v\right) \cdot \frac{\frac{1}{2} \cdot \sqrt{2}}{16}\right) \cdot \left(\sqrt{1 + v \cdot \left(-3 \cdot v\right)} \cdot \left(\left(1 + v \cdot \left(-3 \cdot v\right)\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right)\right)\right)}}\]
9. Using strategy rm

10. Applied add-sqr-sqrt0.0

    \[\leadsto \sqrt[3]{\left(\left(1 - v \cdot v\right) \cdot \frac{\frac{1}{2} \cdot \sqrt{2}}{16}\right) \cdot \left(\sqrt{\color{blue}{\sqrt{1 + v \cdot \left(-3 \cdot v\right)} \cdot \sqrt{1 + v \cdot \left(-3 \cdot v\right)}}} \cdot \left(\left(1 + v \cdot \left(-3 \cdot v\right)\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right)\right)\right)}\]

11. Applied sqrt-prod0.0

    \[\leadsto \sqrt[3]{\left(\left(1 - v \cdot v\right) \cdot \frac{\frac{1}{2} \cdot \sqrt{2}}{16}\right) \cdot \left(\color{blue}{\left(\sqrt{\sqrt{1 + v \cdot \left(-3 \cdot v\right)}} \cdot \sqrt{\sqrt{1 + v \cdot \left(-3 \cdot v\right)}}\right)} \cdot \left(\left(1 + v \cdot \left(-3 \cdot v\right)\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right)\right)\right)}\]

12. Applied associate-*l*0.0

    \[\leadsto \sqrt[3]{\left(\left(1 - v \cdot v\right) \cdot \frac{\frac{1}{2} \cdot \sqrt{2}}{16}\right) \cdot \color{blue}{\left(\sqrt{\sqrt{1 + v \cdot \left(-3 \cdot v\right)}} \cdot \left(\sqrt{\sqrt{1 + v \cdot \left(-3 \cdot v\right)}} \cdot \left(\left(1 + v \cdot \left(-3 \cdot v\right)\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right)\right)\right)\right)}}\]

13. Final simplification0.0

    \[\leadsto \sqrt[3]{\left(\sqrt{\sqrt{v \cdot \left(-3 \cdot v\right) + 1}} \cdot \left(\sqrt{\sqrt{v \cdot \left(-3 \cdot v\right) + 1}} \cdot \left(\left(v \cdot \left(-3 \cdot v\right) + 1\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right)\right)\right)\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \frac{\sqrt{2} \cdot \frac{1}{2}}{16}\right)}\]

Runtime

Time bar (total: 9.5m)Debug logProfile

herbie shell --seed 2018296 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))