Average Error: 18.5 → 16.0
Time: 21.0s
Precision: 64
Internal Precision: 576
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le 4.652115632423765 \cdot 10^{-288}:\\ \;\;\;\;\left(\sqrt{\left|\sqrt[3]{\frac{\frac{A}{V}}{\ell}}\right| \cdot \sqrt{\sqrt[3]{\frac{\frac{A}{V}}{\ell}}}} \cdot c0\right) \cdot \sqrt{\sqrt{\frac{\frac{A}{V}}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot \sqrt{\frac{1}{\ell}}\\ \end{array}\]

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 2 regimes

  2. if l < 4.652115632423765e-288

    1. Initial program 18.8

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

    2. Initial simplification18.7

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

    4. Applied add-sqr-sqrt18.9

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

    5. Applied associate-*r*18.9

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

    7. Applied add-cube-cbrt18.9

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

    8. Applied sqrt-prod18.9

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

    9. Simplified18.9

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

    if 4.652115632423765e-288 < l

    1. Initial program 18.2

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

    2. Initial simplification18.1

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

    4. Applied div-inv18.1

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

    5. Applied sqrt-prod10.9

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

    6. Applied associate-*r*12.8

      \[\leadsto \color{blue}{\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot \sqrt{\frac{1}{\ell}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le 4.652115632423765 \cdot 10^{-288}:\\ \;\;\;\;\left(\sqrt{\left|\sqrt[3]{\frac{\frac{A}{V}}{\ell}}\right| \cdot \sqrt{\sqrt[3]{\frac{\frac{A}{V}}{\ell}}}} \cdot c0\right) \cdot \sqrt{\sqrt{\frac{\frac{A}{V}}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot \sqrt{\frac{1}{\ell}}\\ \end{array}\]

Runtime

Time bar (total: 21.0s)Debug logProfile

herbie shell --seed 2018248 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))