Average Error: 18.5 → 14.4
Time: 35.4s
Precision: 64
Internal Precision: 576
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{A}{V} \le -1.6532709800542353 \cdot 10^{+300}:\\ \;\;\;\;\sqrt{\left|\sqrt[3]{\frac{A}{\ell \cdot V}}\right| \cdot \sqrt{\sqrt[3]{\frac{A}{\ell \cdot V}}}} \cdot \left(\sqrt{\sqrt{\frac{A}{\ell \cdot V}}} \cdot c0\right)\\ \mathbf{elif}\;\frac{A}{V} \le 3.1729883907217 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\ell} \cdot \frac{A}{V}}\\ \mathbf{elif}\;\frac{A}{V} \le 2.0190676349963465 \cdot 10^{+267}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot \sqrt{\frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A} \cdot c0\right) \cdot \sqrt{\frac{1}{\ell \cdot V}}\\ \end{array}\]

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 (/ A V) < -1.6532709800542353e+300

    1. Initial program 36.2

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

    3. Applied add-sqr-sqrt36.2

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

    4. Applied sqrt-prod36.3

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

    5. Applied associate-*r*36.3

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

    7. Applied add-cube-cbrt36.3

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

    8. Applied sqrt-prod36.3

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

    9. Simplified36.3

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

    if -1.6532709800542353e+300 < (/ A V) < 3.1729883907217e-319

    1. Initial program 17.9

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

    2. Initial simplification17.8

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

    4. Applied div-inv17.8

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

    if 3.1729883907217e-319 < (/ A V) < 2.0190676349963465e+267

    1. Initial program 14.9

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

    2. Initial simplification9.4

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

    4. Applied div-inv9.4

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

    5. Applied sqrt-prod0.5

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

    6. Applied associate-*r*2.8

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

    if 2.0190676349963465e+267 < (/ A V)

    1. Initial program 32.4

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

    3. Applied div-inv32.4

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

    4. Applied sqrt-prod38.0

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

    5. Applied associate-*r*38.4

      \[\leadsto \color{blue}{\left(c0 \cdot \sqrt{A}\right) \cdot \sqrt{\frac{1}{V \cdot \ell}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification14.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V} \le -1.6532709800542353 \cdot 10^{+300}:\\ \;\;\;\;\sqrt{\left|\sqrt[3]{\frac{A}{\ell \cdot V}}\right| \cdot \sqrt{\sqrt[3]{\frac{A}{\ell \cdot V}}}} \cdot \left(\sqrt{\sqrt{\frac{A}{\ell \cdot V}}} \cdot c0\right)\\ \mathbf{elif}\;\frac{A}{V} \le 3.1729883907217 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\ell} \cdot \frac{A}{V}}\\ \mathbf{elif}\;\frac{A}{V} \le 2.0190676349963465 \cdot 10^{+267}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot \sqrt{\frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A} \cdot c0\right) \cdot \sqrt{\frac{1}{\ell \cdot V}}\\ \end{array}\]

Runtime

Time bar (total: 35.4s)Debug logProfile

herbie shell --seed 2018215 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))