Average Error: 19.3 → 12.0
Time: 43.7s
Precision: 64
Internal Precision: 576
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{V \cdot \ell} \le -5.107158068521334 \cdot 10^{+307}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;\frac{1}{V \cdot \ell} \le -7.306433839188941 \cdot 10^{-275}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\frac{V \cdot \ell}{A}}}\\ \mathbf{if}\;\frac{1}{V \cdot \ell} \le 1.9932829043408157 \cdot 10^{-292}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;\frac{1}{V \cdot \ell} \le 5.491448021325038 \cdot 10^{+278}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array}\]

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 (/ 1 (* V l)) < -5.107158068521334e+307 or -7.306433839188941e-275 < (/ 1 (* V l)) < 1.9932829043408157e-292

    1. Initial program 43.3

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

    3. Applied associate-/r*26.7

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

    if -5.107158068521334e+307 < (/ 1 (* V l)) < -7.306433839188941e-275

    1. Initial program 9.9

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

    3. Applied clear-num10.3

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

    if 1.9932829043408157e-292 < (/ 1 (* V l)) < 5.491448021325038e+278

    1. Initial program 9.2

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

    3. Applied div-inv9.2

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

    4. Applied sqrt-prod0.4

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

    if 5.491448021325038e+278 < (/ 1 (* V l))

    1. Initial program 51.9

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

    3. Applied add-sqr-sqrt51.9

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

    4. Applied sqrt-prod51.9

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

    5. Applied associate-*r*51.9

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

    6. Taylor expanded around 0 51.9

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

    7. Applied simplify34.0

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

    9. Applied div-inv34.0

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

    11. Applied associate-*l/34.0

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

    12. Applied sqrt-div38.6

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

    13. Applied simplify38.6

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

Runtime

Time bar (total: 43.7s)Debug logProfile

herbie shell --seed 2018214 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))