Average Error: 18.2 → 12.3
Time: 36.6s
Precision: 64
Internal Precision: 320
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{V \cdot \ell} \le -4.977339900586398 \cdot 10^{-297}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\left|\sqrt[3]{A} \cdot \sqrt[3]{\frac{1}{\ell \cdot V}}\right| \cdot \sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}}}}\\ \mathbf{if}\;\frac{1}{V \cdot \ell} \le 4.8822813688907365 \cdot 10^{-287}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array}\]

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (/ 1 (* V l)) < -4.977339900586398e-297

    1. Initial program 14.2

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt14.2

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\sqrt{\frac{A}{V \cdot \ell}} \cdot \sqrt{\frac{A}{V \cdot \ell}}}}\]

    4. Applied sqrt-prod14.5

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}\]

    5. Applied associate-*r*14.5

      \[\leadsto \color{blue}{\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt14.5

      \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\color{blue}{\left(\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right) \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}}}}\]

    8. Applied sqrt-prod14.5

      \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\color{blue}{\sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}}}}}\]

    9. Applied simplify14.5

      \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\color{blue}{\left|\sqrt[3]{\frac{A}{\ell \cdot V}}\right|} \cdot \sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}}}}\]
    10. Using strategy rm

    11. Applied div-inv14.7

      \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\left|\sqrt[3]{\color{blue}{A \cdot \frac{1}{\ell \cdot V}}}\right| \cdot \sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}}}}\]

    12. Applied cbrt-prod14.7

      \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\left|\color{blue}{\sqrt[3]{A} \cdot \sqrt[3]{\frac{1}{\ell \cdot V}}}\right| \cdot \sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}}}}\]

    if -4.977339900586398e-297 < (/ 1 (* V l)) < 4.8822813688907365e-287

    1. Initial program 36.9

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt36.9

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\sqrt{\frac{A}{V \cdot \ell}} \cdot \sqrt{\frac{A}{V \cdot \ell}}}}\]

    4. Applied sqrt-prod36.9

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}\]

    5. Applied associate-*r*36.9

      \[\leadsto \color{blue}{\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}}\]

    6. Taylor expanded around 0 36.9

      \[\leadsto \left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\color{blue}{\frac{A}{\ell \cdot V}}}}\]

    7. Applied simplify21.1

      \[\leadsto \color{blue}{c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}}\]

    if 4.8822813688907365e-287 < (/ 1 (* V l))

    1. Initial program 15.7

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
    2. Using strategy rm

    3. Applied sqrt-div6.6

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 36.6s)Debug logProfile

herbie shell --seed '#(1072936661 1621281212 3440817831 3219514234 460296804 1258167384)' 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))