Average Error: 0.0 → 0.0
Time: 1.5m
Precision: 64
Internal Precision: 320
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\left(\frac{1}{4} - v \cdot \left(v \cdot \frac{1}{4}\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}\]

Error

Bits error versus v

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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. Using strategy rm

  3. Applied add-log-exp0.0

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

  4. Taylor expanded around -inf 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018252 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))