Average Error: 0.0 → 0.0
Time: 1.4m
Precision: 64
Internal Precision: 320
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\left(\left(1 - \log \left(e^{{\left(v \cdot v\right)}^{3}}\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{\left(-3 \cdot v\right) \cdot v + 1}}{4 + \left(4 \cdot \left(v \cdot v\right)\right) \cdot \left(1 + v \cdot v\right)}\]

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 flip3--0.0

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

  4. Applied associate-*l/0.0

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

  5. Applied frac-times0.0

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

  6. Simplified0.0

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

  7. Simplified0.0

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

  9. Applied add-log-exp0.0

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

  10. Final simplification0.0

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018250 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))