Average Error: 0.0 → 0.0
Time: 57.2s
Precision: 64
Internal Precision: 576
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{2} \cdot \sqrt{1 - {\left(v \cdot \left(v \cdot 3\right)\right)}^{3}}}{\sqrt{\left(\left(v \cdot v\right) \cdot 3 + 1\right) + \left(\left(v \cdot v\right) \cdot 3\right) \cdot \left(\left(v \cdot v\right) \cdot 3\right)} \cdot 4} \cdot \left(1 - v \cdot v\right)\]

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

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

  4. Applied sqrt-div0.0

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

  5. Applied frac-times0.0

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

  6. Simplified0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Runtime

Time bar (total: 57.2s)Debug logProfile

herbie shell --seed 2018221 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))