Average Error: 18.5 → 11.7
Time: 38.9s
Precision: 64
Internal Precision: 576
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{V \cdot \ell} \le -1.765571419611046 \cdot 10^{+255}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{if}\;\frac{1}{V \cdot \ell} \le -1.9034493027992715 \cdot 10^{-194}:\\ \;\;\;\;\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot c0\right) \cdot \sqrt{\sqrt{A \cdot \frac{1}{V \cdot \ell}}}\\ \mathbf{if}\;\frac{1}{V \cdot \ell} \le 5.6604306642257075 \cdot 10^{-300}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)\\ \end{array}\]

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 3 regimes

  2. if (/ 1 (* V l)) < -1.765571419611046e+255 or -1.9034493027992715e-194 < (/ 1 (* V l)) < 5.6604306642257075e-300

    1. Initial program 37.1

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

    3. Applied associate-/r*24.4

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

    if -1.765571419611046e+255 < (/ 1 (* V l)) < -1.9034493027992715e-194

    1. Initial program 7.7

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

    3. Applied div-inv7.7

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

    5. Applied add-sqr-sqrt8.0

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

    6. Applied associate-*r*7.9

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

    7. Applied simplify7.9

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

    if 5.6604306642257075e-300 < (/ 1 (* V l))

    1. Initial program 14.9

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

    3. Applied div-inv15.1

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

    4. Applied sqrt-prod6.7

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

Runtime

Time bar (total: 38.9s)Debug logProfile

herbie shell --seed 2018167 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))