Average Error: 1.0 → 0.0
Time: 1.9m
Precision: 64
Internal Precision: 320
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{\pi \cdot 3}}{1 - v \cdot v}}{\sqrt{\sqrt[3]{2 - \left(v \cdot 6\right) \cdot v}}} \cdot \frac{1}{\left|\sqrt[3]{2 - \left(v \cdot 6\right) \cdot v}\right|}\]

Error

Bits error versus v

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  2. Initial simplification0.0

    \[\leadsto \frac{\frac{\frac{4}{\pi \cdot 3}}{1 - v \cdot v}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.0

    \[\leadsto \frac{\frac{\frac{4}{\pi \cdot 3}}{1 - v \cdot v}}{\sqrt{\color{blue}{\left(\sqrt[3]{2 - v \cdot \left(v \cdot 6\right)} \cdot \sqrt[3]{2 - v \cdot \left(v \cdot 6\right)}\right) \cdot \sqrt[3]{2 - v \cdot \left(v \cdot 6\right)}}}}\]

  5. Applied sqrt-prod0.0

    \[\leadsto \frac{\frac{\frac{4}{\pi \cdot 3}}{1 - v \cdot v}}{\color{blue}{\sqrt{\sqrt[3]{2 - v \cdot \left(v \cdot 6\right)} \cdot \sqrt[3]{2 - v \cdot \left(v \cdot 6\right)}} \cdot \sqrt{\sqrt[3]{2 - v \cdot \left(v \cdot 6\right)}}}}\]

  6. Applied *-un-lft-identity0.0

    \[\leadsto \frac{\frac{\frac{4}{\pi \cdot 3}}{\color{blue}{1 \cdot \left(1 - v \cdot v\right)}}}{\sqrt{\sqrt[3]{2 - v \cdot \left(v \cdot 6\right)} \cdot \sqrt[3]{2 - v \cdot \left(v \cdot 6\right)}} \cdot \sqrt{\sqrt[3]{2 - v \cdot \left(v \cdot 6\right)}}}\]

  7. Applied *-un-lft-identity0.0

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{4}{\pi \cdot 3}}}{1 \cdot \left(1 - v \cdot v\right)}}{\sqrt{\sqrt[3]{2 - v \cdot \left(v \cdot 6\right)} \cdot \sqrt[3]{2 - v \cdot \left(v \cdot 6\right)}} \cdot \sqrt{\sqrt[3]{2 - v \cdot \left(v \cdot 6\right)}}}\]

  8. Applied times-frac0.0

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{\frac{4}{\pi \cdot 3}}{1 - v \cdot v}}}{\sqrt{\sqrt[3]{2 - v \cdot \left(v \cdot 6\right)} \cdot \sqrt[3]{2 - v \cdot \left(v \cdot 6\right)}} \cdot \sqrt{\sqrt[3]{2 - v \cdot \left(v \cdot 6\right)}}}\]

  9. Applied times-frac0.0

    \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\sqrt{\sqrt[3]{2 - v \cdot \left(v \cdot 6\right)} \cdot \sqrt[3]{2 - v \cdot \left(v \cdot 6\right)}}} \cdot \frac{\frac{\frac{4}{\pi \cdot 3}}{1 - v \cdot v}}{\sqrt{\sqrt[3]{2 - v \cdot \left(v \cdot 6\right)}}}}\]

  10. Simplified0.0

    \[\leadsto \color{blue}{\frac{1}{\left|\sqrt[3]{2 - v \cdot \left(v \cdot 6\right)}\right|}} \cdot \frac{\frac{\frac{4}{\pi \cdot 3}}{1 - v \cdot v}}{\sqrt{\sqrt[3]{2 - v \cdot \left(v \cdot 6\right)}}}\]

  11. Final simplification0.0

    \[\leadsto \frac{\frac{\frac{4}{\pi \cdot 3}}{1 - v \cdot v}}{\sqrt{\sqrt[3]{2 - \left(v \cdot 6\right) \cdot v}}} \cdot \frac{1}{\left|\sqrt[3]{2 - \left(v \cdot 6\right) \cdot v}\right|}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018227 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))