Average Error: 19.1 → 7.7
Time: 43.6s
Precision: 64
Internal Precision: 576
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le -4.1903899166213757 \cdot 10^{+164}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\ell}} \cdot \left|\sqrt[3]{A} \cdot \sqrt[3]{\frac{1}{V}}\right|\right)\\ \mathbf{if}\;V \cdot \ell \le -6.215981705788424 \cdot 10^{-137}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\\ \mathbf{if}\;V \cdot \ell \le 3.6988896375228 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\left|\sqrt[3]{\frac{A}{V}}\right|}{\sqrt{\frac{\ell}{\sqrt[3]{\frac{A}{V}}}}}\\ \mathbf{if}\;V \cdot \ell \le 6.5042641181115005 \cdot 10^{+258}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\frac{\ell}{\sqrt[3]{A}}}} \cdot \sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{V}}}\right)\\ \end{array}\]

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 5 regimes

  2. if (* V l) < -4.1903899166213757e+164

    1. Initial program 27.5

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

    3. Applied associate-/r*19.6

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

    5. Applied add-cube-cbrt19.8

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

    6. Applied associate-/l*19.8

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

    8. Applied div-inv19.8

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

    9. Applied times-frac19.8

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

    10. Applied sqrt-prod12.6

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

    11. Applied simplify12.6

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

    13. Applied div-inv12.6

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

    14. Applied cbrt-prod12.7

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

    if -4.1903899166213757e+164 < (* V l) < -6.215981705788424e-137

    1. Initial program 5.2

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

    3. Applied add-sqr-sqrt5.2

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

    4. Applied sqrt-prod5.4

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

    5. Applied associate-*r*5.4

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

    if -6.215981705788424e-137 < (* V l) < 3.6988896375228e-310

    1. Initial program 40.4

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

    3. Applied associate-/r*27.9

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

    5. Applied add-cube-cbrt28.3

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

    6. Applied associate-/l*28.3

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

    8. Applied sqrt-div18.1

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

    9. Applied simplify18.1

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

    if 3.6988896375228e-310 < (* V l) < 6.5042641181115005e+258

    1. Initial program 9.5

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

    3. Applied sqrt-div0.4

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

    if 6.5042641181115005e+258 < (* V l)

    1. Initial program 34.8

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

    3. Applied associate-/r*20.3

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

    5. Applied add-cube-cbrt20.5

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

    6. Applied associate-/l*20.5

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

    8. Applied cbrt-div20.5

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

    9. Applied associate-/r/20.5

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

    10. Applied times-frac20.5

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

    11. Applied sqrt-prod13.2

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\frac{\ell}{\sqrt[3]{A}}}} \cdot \sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{V}}}\right)}\]
  3. Recombined 5 regimes into one program.

Runtime

Time bar (total: 43.6s)Debug logProfile

herbie shell --seed 2018206 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))