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-sqr-sqrt0.0

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

4. Applied sqrt-prod0.0

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

5. Applied associate-*r*0.0

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

6. Simplified0.0

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

8. Applied add-sqr-sqrt0.0

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

9. Applied associate-*r*0.0

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

10. Simplified0.0

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

11. Final simplification0.0

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

Runtime

Time bar (total: 27.8s)Debug logProfile

herbie shell --seed 2018354 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))