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-exp-log0.0

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

9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))

Details

Time bar (total: 40.4s)Debug log