Average Error: 0.0 → 0.0
Time: 25.7s
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{\left(\left(\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \frac{1 - v \cdot v}{\frac{4}{\sqrt{2}}}\right) \cdot \left(\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \frac{1 - v \cdot v}{\frac{4}{\sqrt{2}}}\right)\right) \cdot \left(\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \frac{1 - v \cdot v}{\frac{4}{\sqrt{2}}}\right)}\]

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Initial simplification0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Final simplification0.0

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

Runtime

Time bar (total: 25.7s)Debug logProfile

herbie shell --seed 2018277 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))