Average Error: 54.0 → 54.0
Time: 20.5s
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\left(\sqrt[3]{\frac{\sqrt{2}}{4}} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(\left(\left(\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\frac{\sqrt{2}}{4}}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\frac{\sqrt{2}}{4}}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\frac{\sqrt{2}}{4}}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{\sqrt{2}}{4}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt{2}}{4}} \cdot \sqrt[3]{\sqrt[3]{\frac{\sqrt{2}}{4}}}\right)\right)\right) \cdot \left(1 - v \cdot v\right)\]

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 54.0

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

  3. Applied add-cube-cbrt54.0

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

  4. Applied associate-*l*54.0

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

  6. Applied add-cube-cbrt54.0

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

  7. Applied associate-*l*54.0

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

  9. Applied add-cube-cbrt54.0

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

  10. Final simplification54.0

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

Runtime

Time bar (total: 20.5s)Debug logProfile

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