Average Error: 19.2 → 8.1
Time: 35.3s
Precision: 64
Internal Precision: 576
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{V \cdot \ell} \le -2.5753218311097967 \cdot 10^{+70}:\\ \;\;\;\;c0 \cdot \frac{\left|\sqrt[3]{\frac{A}{V}}\right|}{\sqrt{\frac{\ell}{\sqrt[3]{\frac{A}{V}}}}}\\ \mathbf{if}\;\frac{1}{V \cdot \ell} \le -7.992760603464358 \cdot 10^{-61}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;\frac{1}{V \cdot \ell} \le 1.1469551436298865 \cdot 10^{-276}:\\ \;\;\;\;c0 \cdot \frac{\left|\sqrt[3]{\frac{A}{V}}\right|}{\sqrt{\frac{\ell}{\sqrt[3]{\frac{A}{V}}}}}\\ \mathbf{if}\;\frac{1}{V \cdot \ell} \le 1.2739427392579735 \cdot 10^{+301}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\frac{\ell}{\sqrt[3]{A}}}} \cdot \sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{V}}}\right)\\ \end{array}\]

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if (/ 1 (* V l)) < -2.5753218311097967e+70 or -7.992760603464358e-61 < (/ 1 (* V l)) < 1.1469551436298865e-276

    1. Initial program 25.8

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

    3. Applied associate-/r*19.9

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

    5. Applied add-cube-cbrt20.2

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

    6. Applied associate-/l*20.2

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

    8. Applied sqrt-div13.6

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

    9. Applied simplify13.6

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

    if -2.5753218311097967e+70 < (/ 1 (* V l)) < -7.992760603464358e-61

    1. Initial program 2.8

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

    if 1.1469551436298865e-276 < (/ 1 (* V l)) < 1.2739427392579735e+301

    1. Initial program 9.8

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

    3. Applied sqrt-div0.4

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

    if 1.2739427392579735e+301 < (/ 1 (* V l))

    1. Initial program 57.5

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

    3. Applied associate-/r*34.5

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

    5. Applied add-cube-cbrt34.7

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

    6. Applied associate-/l*34.7

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

    8. Applied cbrt-div34.6

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

    9. Applied associate-/r/34.6

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

    10. Applied times-frac34.6

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

    11. Applied sqrt-prod22.8

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\frac{\ell}{\sqrt[3]{A}}}} \cdot \sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{V}}}\right)}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 35.3s)Debug logProfile

herbie shell --seed 2018195 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))