Average Error: 18.9 → 7.6
Time: 22.5s
Precision: 64
Internal Precision: 128
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le -2.388995820113946 \cdot 10^{+278}:\\ \;\;\;\;\frac{\sqrt{\frac{\sqrt[3]{\frac{1}{\ell}}}{\frac{V}{A}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \le -3.063624297741607 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{\frac{A}{V \cdot \ell}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \le 9.9640766643933 \cdot 10^{-315} \lor \neg \left(V \cdot \ell \le 3.7007693884261103 \cdot 10^{+270}\right):\\ \;\;\;\;\frac{\sqrt{\frac{\sqrt[3]{\frac{1}{\ell}}}{\frac{V}{A}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array}\]

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 (* V l) < -2.388995820113946e+278 or -3.063624297741607e-294 < (* V l) < 9.9640766643933e-315 or 3.7007693884261103e+270 < (* V l)

    1. Initial program 45.0

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

    2. Initial simplification27.0

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

    4. Applied div-inv27.0

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt27.2

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

    7. Applied associate-*r*27.2

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{A}{V} \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right)\right) \cdot \sqrt[3]{\frac{1}{\ell}}}}\]
    8. Using strategy rm

    9. Applied cbrt-div27.2

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

    10. Applied cbrt-div27.2

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

    11. Applied associate-*r/27.2

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

    12. Applied associate-*r/27.2

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

    13. Applied frac-times27.2

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

    14. Applied sqrt-div15.4

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

    15. Simplified15.4

      \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{\sqrt[3]{\frac{1}{\ell}}}{\frac{V}{A}}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\]

    if -2.388995820113946e+278 < (* V l) < -3.063624297741607e-294

    1. Initial program 9.1

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

    2. Initial simplification15.6

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

    4. Applied div-inv15.7

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt16.0

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

    7. Applied associate-*r*16.0

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

    8. Taylor expanded around -inf 9.1

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

    if 9.9640766643933e-315 < (* V l) < 3.7007693884261103e+270

    1. Initial program 9.3

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

    2. Initial simplification15.6

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

    4. Applied div-inv15.6

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}}\]
    5. Using strategy rm

    6. Applied frac-times9.3

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A \cdot 1}{V \cdot \ell}}}\]

    7. Applied sqrt-div0.4

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A \cdot 1}}{\sqrt{V \cdot \ell}}}\]

    8. Simplified0.4

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -2.388995820113946 \cdot 10^{+278}:\\ \;\;\;\;\frac{\sqrt{\frac{\sqrt[3]{\frac{1}{\ell}}}{\frac{V}{A}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \le -3.063624297741607 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{\frac{A}{V \cdot \ell}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \le 9.9640766643933 \cdot 10^{-315} \lor \neg \left(V \cdot \ell \le 3.7007693884261103 \cdot 10^{+270}\right):\\ \;\;\;\;\frac{\sqrt{\frac{\sqrt[3]{\frac{1}{\ell}}}{\frac{V}{A}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array}\]

Runtime

Time bar (total: 22.5s)Debug logProfile

herbie shell --seed 2018258 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))