Average Error: 18.4 → 14.7
Time: 16.8s
Precision: 64
Internal Precision: 576
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -3.361533605835011 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{\frac{A}{V \cdot \ell}} \cdot c0\\ \mathbf{elif}\;\ell \le -5.604160050265453 \cdot 10^{-65}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\left|\sqrt[3]{\frac{A}{V}} \cdot \sqrt[3]{\frac{1}{\ell}}\right| \cdot \sqrt{\sqrt[3]{\frac{\frac{A}{V}}{\ell}}}}\right) \cdot \sqrt{\sqrt{\frac{\frac{A}{V}}{\ell}}}\\ \mathbf{elif}\;\ell \le 2.533071249827636 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\frac{\ell}{\frac{1}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array}\]

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 4 regimes

  2. if l < -3.361533605835011e+197

    1. Initial program 23.4

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

    2. Initial simplification24.8

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

    4. Applied associate-/l/23.4

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

    if -3.361533605835011e+197 < l < -5.604160050265453e-65

    1. Initial program 13.8

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

    2. Initial simplification13.7

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

    4. Applied add-sqr-sqrt13.9

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

    5. Applied associate-*r*13.9

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

    7. Applied add-cube-cbrt14.0

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

    8. Applied sqrt-prod14.0

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

    9. Simplified14.0

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

    11. Applied div-inv13.9

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

    12. Applied cbrt-prod13.9

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

    if -5.604160050265453e-65 < l < 2.533071249827636e-309

    1. Initial program 21.1

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

    2. Initial simplification21.7

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

    4. Applied div-inv21.8

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

    5. Applied associate-/l*21.1

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

    if 2.533071249827636e-309 < l

    1. Initial program 18.6

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

    2. Initial simplification18.6

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

    4. Applied sqrt-div11.2

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

  4. Final simplification14.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -3.361533605835011 \cdot 10^{+197}:\\ \;\;\;\;\sqrt{\frac{A}{V \cdot \ell}} \cdot c0\\ \mathbf{elif}\;\ell \le -5.604160050265453 \cdot 10^{-65}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\left|\sqrt[3]{\frac{A}{V}} \cdot \sqrt[3]{\frac{1}{\ell}}\right| \cdot \sqrt{\sqrt[3]{\frac{\frac{A}{V}}{\ell}}}}\right) \cdot \sqrt{\sqrt{\frac{\frac{A}{V}}{\ell}}}\\ \mathbf{elif}\;\ell \le 2.533071249827636 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\frac{\ell}{\frac{1}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array}\]

Runtime

Time bar (total: 16.8s)Debug logProfile

herbie shell --seed 2018227 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))