Average Error: 15.3 → 0.9
Time: 16.3s
Precision: 64
Internal Precision: 576
\[\sqrt[3]{\frac{g}{2 \cdot a}}\]
↓
\[\sqrt[3]{\frac{\frac{1}{2}}{\frac{-1}{g}}} \cdot \sqrt[3]{\frac{-1}{a}}\]

Error

Bits error versus g

Bits error versus a

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.3

    \[\sqrt[3]{\frac{g}{2 \cdot a}}\]

  2. Initial simplification15.3

    \[\leadsto \sqrt[3]{\frac{g}{a \cdot 2}}\]
  3. Using strategy rm

  4. Applied cbrt-div0.8

    \[\leadsto \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{a \cdot 2}}}\]

  5. Taylor expanded around -inf 62.8

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

  6. Simplified0.9

    \[\leadsto \color{blue}{\sqrt[3]{\frac{-1}{a}} \cdot \left(\sqrt[3]{\frac{-1}{2}} \cdot \sqrt[3]{g}\right)}\]
  7. Using strategy rm

  8. Applied cbrt-unprod0.8

    \[\leadsto \sqrt[3]{\frac{-1}{a}} \cdot \color{blue}{\sqrt[3]{\frac{-1}{2} \cdot g}}\]

  9. Taylor expanded around -inf 34.0

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

  10. Simplified0.9

    \[\leadsto \sqrt[3]{\frac{-1}{a}} \cdot \color{blue}{\sqrt[3]{\frac{\frac{1}{2}}{\frac{-1}{g}}}}\]

  11. Final simplification0.9

    \[\leadsto \sqrt[3]{\frac{\frac{1}{2}}{\frac{-1}{g}}} \cdot \sqrt[3]{\frac{-1}{a}}\]

Runtime

Time bar (total: 16.3s)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  (cbrt (/ g (* 2 a))))