Average Error: 18.6 → 7.8
Time: 17.8s
Precision: 64
Internal Precision: 576
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le -2.7879072072422225 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left|\sqrt[3]{\frac{A}{V}}\right|}{\sqrt{\frac{\ell}{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \le -6.001573491311489 \cdot 10^{-196}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \le 5.297604086874348 \cdot 10^{-308}:\\ \;\;\;\;\frac{\left|\sqrt[3]{\frac{A}{V}}\right|}{\sqrt{\frac{\ell}{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \le 4.335909566247426 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{V}}} \cdot \sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\frac{\ell}{\sqrt[3]{A}}}}\right) \cdot c0\\ \end{array}\]

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 4 regimes

  2. if (* V l) < -2.7879072072422225e+124 or -6.001573491311489e-196 < (* V l) < 5.297604086874348e-308

    1. Initial program 33.8

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

    2. Initial simplification24.0

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

    4. Applied add-cube-cbrt24.3

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

    5. Applied associate-/l*24.3

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

    7. Applied sqrt-div16.2

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

    8. Simplified16.2

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

    10. Applied cbrt-div16.2

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

    if -2.7879072072422225e+124 < (* V l) < -6.001573491311489e-196

    1. Initial program 5.6

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

    if 5.297604086874348e-308 < (* V l) < 4.335909566247426e+290

    1. Initial program 9.6

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

    2. Initial simplification16.3

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

    4. Applied div-inv16.3

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

    6. Applied frac-times9.6

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

    7. Applied sqrt-div0.4

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

    8. Simplified0.4

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

    if 4.335909566247426e+290 < (* V l)

    1. Initial program 37.8

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

    2. Initial simplification23.8

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

    4. Applied add-cube-cbrt24.0

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

    5. Applied associate-/l*24.0

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

    7. Applied cbrt-div24.0

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

    8. Applied associate-/r/24.0

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

    9. Applied times-frac24.0

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

    10. Applied sqrt-prod16.7

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

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -2.7879072072422225 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left|\sqrt[3]{\frac{A}{V}}\right|}{\sqrt{\frac{\ell}{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \le -6.001573491311489 \cdot 10^{-196}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \le 5.297604086874348 \cdot 10^{-308}:\\ \;\;\;\;\frac{\left|\sqrt[3]{\frac{A}{V}}\right|}{\sqrt{\frac{\ell}{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \le 4.335909566247426 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{V}}} \cdot \sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\frac{\ell}{\sqrt[3]{A}}}}\right) \cdot c0\\ \end{array}\]

Runtime

Time bar (total: 17.8s)Debug logProfile

herbie shell --seed 2018225 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))