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. Using strategy rm

3. Applied add-cbrt-cube0.0

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

4. Applied add-cbrt-cube0.0

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

5. Applied cbrt-unprod0.0

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

6. Simplified0.0

   \[\leadsto \sqrt[3]{\color{blue}{(\left(\frac{1}{32} \cdot \sqrt{2}\right) \cdot \left(-3 \cdot \left(v \cdot v\right)\right) + \left(\frac{1}{32} \cdot \sqrt{2}\right))_* \cdot \sqrt{(-3 \cdot \left(v \cdot v\right) + 1)_*}}} \cdot \left(1 - v \cdot v\right)\]
7. Using strategy rm

8. Applied add-cube-cbrt0.0

   \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt[3]{(\left(\frac{1}{32} \cdot \sqrt{2}\right) \cdot \left(-3 \cdot \left(v \cdot v\right)\right) + \left(\frac{1}{32} \cdot \sqrt{2}\right))_* \cdot \sqrt{(-3 \cdot \left(v \cdot v\right) + 1)_*}} \cdot \sqrt[3]{(\left(\frac{1}{32} \cdot \sqrt{2}\right) \cdot \left(-3 \cdot \left(v \cdot v\right)\right) + \left(\frac{1}{32} \cdot \sqrt{2}\right))_* \cdot \sqrt{(-3 \cdot \left(v \cdot v\right) + 1)_*}}\right) \cdot \sqrt[3]{(\left(\frac{1}{32} \cdot \sqrt{2}\right) \cdot \left(-3 \cdot \left(v \cdot v\right)\right) + \left(\frac{1}{32} \cdot \sqrt{2}\right))_* \cdot \sqrt{(-3 \cdot \left(v \cdot v\right) + 1)_*}}}} \cdot \left(1 - v \cdot v\right)\]

9. Final simplification0.0

   \[\leadsto \sqrt[3]{\left(\sqrt[3]{\sqrt{(-3 \cdot \left(v \cdot v\right) + 1)_*} \cdot (\left(\frac{1}{32} \cdot \sqrt{2}\right) \cdot \left(\left(v \cdot v\right) \cdot -3\right) + \left(\frac{1}{32} \cdot \sqrt{2}\right))_*} \cdot \sqrt[3]{\sqrt{(-3 \cdot \left(v \cdot v\right) + 1)_*} \cdot (\left(\frac{1}{32} \cdot \sqrt{2}\right) \cdot \left(\left(v \cdot v\right) \cdot -3\right) + \left(\frac{1}{32} \cdot \sqrt{2}\right))_*}\right) \cdot \sqrt[3]{\sqrt{(-3 \cdot \left(v \cdot v\right) + 1)_*} \cdot (\left(\frac{1}{32} \cdot \sqrt{2}\right) \cdot \left(\left(v \cdot v\right) \cdot -3\right) + \left(\frac{1}{32} \cdot \sqrt{2}\right))_*}} \cdot \left(1 - v \cdot v\right)\]

Runtime

Time bar (total: 33.2s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))