Average Error: 19.2 → 8.9
Time: 4.8s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le -4.2577037844572332 \cdot 10^{280}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\left(\sqrt[3]{1} \cdot \sqrt[3]{\frac{1}{V}}\right) \cdot \left(\sqrt[3]{1} \cdot \frac{A}{\ell}\right)}}{\left|\sqrt[3]{V}\right|}\\ \mathbf{elif}\;V \cdot \ell \le -2.1210801829800363 \cdot 10^{-101}:\\ \;\;\;\;\left(\left|\sqrt[3]{\frac{A}{V \cdot \ell}}\right| \cdot c0\right) \cdot \sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}}}\\ \mathbf{elif}\;V \cdot \ell \le 5.13828 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\left(\sqrt[3]{1} \cdot \sqrt[3]{\frac{1}{V}}\right) \cdot \left(\sqrt[3]{1} \cdot \frac{A}{\ell}\right)}}{\left|\sqrt[3]{V}\right|}\\ \mathbf{elif}\;V \cdot \ell \le 7.96962524666391019 \cdot 10^{302}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\left(\sqrt[3]{1} \cdot \sqrt[3]{\frac{1}{V}}\right) \cdot \left(\sqrt[3]{1} \cdot \frac{A}{\ell}\right)}}{\left|\sqrt[3]{V}\right|}\\ \end{array}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;V \cdot \ell \le -4.2577037844572332 \cdot 10^{280}:\\
\;\;\;\;c0 \cdot \frac{\sqrt{\left(\sqrt[3]{1} \cdot \sqrt[3]{\frac{1}{V}}\right) \cdot \left(\sqrt[3]{1} \cdot \frac{A}{\ell}\right)}}{\left|\sqrt[3]{V}\right|}\\

\mathbf{elif}\;V \cdot \ell \le -2.1210801829800363 \cdot 10^{-101}:\\
\;\;\;\;\left(\left|\sqrt[3]{\frac{A}{V \cdot \ell}}\right| \cdot c0\right) \cdot \sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}}}\\

\mathbf{elif}\;V \cdot \ell \le 5.13828 \cdot 10^{-322}:\\
\;\;\;\;c0 \cdot \frac{\sqrt{\left(\sqrt[3]{1} \cdot \sqrt[3]{\frac{1}{V}}\right) \cdot \left(\sqrt[3]{1} \cdot \frac{A}{\ell}\right)}}{\left|\sqrt[3]{V}\right|}\\

\mathbf{elif}\;V \cdot \ell \le 7.96962524666391019 \cdot 10^{302}:\\
\;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\

\mathbf{else}:\\
\;\;\;\;c0 \cdot \frac{\sqrt{\left(\sqrt[3]{1} \cdot \sqrt[3]{\frac{1}{V}}\right) \cdot \left(\sqrt[3]{1} \cdot \frac{A}{\ell}\right)}}{\left|\sqrt[3]{V}\right|}\\

\end{array}
double f(double c0, double A, double V, double l) {
        double r230197 = c0;
        double r230198 = A;
        double r230199 = V;
        double r230200 = l;
        double r230201 = r230199 * r230200;
        double r230202 = r230198 / r230201;
        double r230203 = sqrt(r230202);
        double r230204 = r230197 * r230203;
        return r230204;
}
↓
double f(double c0, double A, double V, double l) {
        double r230205 = V;
        double r230206 = l;
        double r230207 = r230205 * r230206;
        double r230208 = -4.257703784457233e+280;
        bool r230209 = r230207 <= r230208;
        double r230210 = c0;
        double r230211 = 1.0;
        double r230212 = cbrt(r230211);
        double r230213 = r230211 / r230205;
        double r230214 = cbrt(r230213);
        double r230215 = r230212 * r230214;
        double r230216 = A;
        double r230217 = r230216 / r230206;
        double r230218 = r230212 * r230217;
        double r230219 = r230215 * r230218;
        double r230220 = sqrt(r230219);
        double r230221 = cbrt(r230205);
        double r230222 = fabs(r230221);
        double r230223 = r230220 / r230222;
        double r230224 = r230210 * r230223;
        double r230225 = -2.1210801829800363e-101;
        bool r230226 = r230207 <= r230225;
        double r230227 = r230216 / r230207;
        double r230228 = cbrt(r230227);
        double r230229 = fabs(r230228);
        double r230230 = r230229 * r230210;
        double r230231 = sqrt(r230228);
        double r230232 = r230230 * r230231;
        double r230233 = 5.138282716749e-322;
        bool r230234 = r230207 <= r230233;
        double r230235 = 7.96962524666391e+302;
        bool r230236 = r230207 <= r230235;
        double r230237 = sqrt(r230216);
        double r230238 = r230210 * r230237;
        double r230239 = sqrt(r230207);
        double r230240 = r230238 / r230239;
        double r230241 = r230236 ? r230240 : r230224;
        double r230242 = r230234 ? r230224 : r230241;
        double r230243 = r230226 ? r230232 : r230242;
        double r230244 = r230209 ? r230224 : r230243;
        return r230244;
}

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 (* V l) < -4.257703784457233e+280 or -2.1210801829800363e-101 < (* V l) < 5.138282716749e-322 or 7.96962524666391e+302 < (* V l)

    1. Initial program 38.3

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

    3. Applied *-un-lft-identity38.3

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

    4. Applied times-frac25.7

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

    6. Applied add-cube-cbrt25.9

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

    7. Applied associate-*l*25.9

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{\frac{1}{V}}\right) \cdot \left(\sqrt[3]{\frac{1}{V}} \cdot \frac{A}{\ell}\right)}}\]
    8. Using strategy rm

    9. Applied cbrt-div25.9

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

    10. Applied associate-*l/25.9

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

    11. Applied cbrt-div25.9

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

    12. Applied associate-*r/25.9

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{1}}{\sqrt[3]{V}}} \cdot \frac{\sqrt[3]{1} \cdot \frac{A}{\ell}}{\sqrt[3]{V}}}\]

    13. Applied frac-times25.9

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

    14. Applied sqrt-div16.5

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

    15. Simplified16.5

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

    16. Simplified16.5

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

    if -4.257703784457233e+280 < (* V l) < -2.1210801829800363e-101

    1. Initial program 7.6

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

    3. Applied add-cube-cbrt8.0

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

    4. Applied sqrt-prod8.0

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

    5. Applied associate-*r*8.0

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

    6. Simplified8.0

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

    if 5.138282716749e-322 < (* V l) < 7.96962524666391e+302

    1. Initial program 9.9

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

    3. Applied sqrt-div0.6

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

    4. Applied associate-*r/2.5

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -4.2577037844572332 \cdot 10^{280}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\left(\sqrt[3]{1} \cdot \sqrt[3]{\frac{1}{V}}\right) \cdot \left(\sqrt[3]{1} \cdot \frac{A}{\ell}\right)}}{\left|\sqrt[3]{V}\right|}\\ \mathbf{elif}\;V \cdot \ell \le -2.1210801829800363 \cdot 10^{-101}:\\ \;\;\;\;\left(\left|\sqrt[3]{\frac{A}{V \cdot \ell}}\right| \cdot c0\right) \cdot \sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}}}\\ \mathbf{elif}\;V \cdot \ell \le 5.13828 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\left(\sqrt[3]{1} \cdot \sqrt[3]{\frac{1}{V}}\right) \cdot \left(\sqrt[3]{1} \cdot \frac{A}{\ell}\right)}}{\left|\sqrt[3]{V}\right|}\\ \mathbf{elif}\;V \cdot \ell \le 7.96962524666391019 \cdot 10^{302}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\left(\sqrt[3]{1} \cdot \sqrt[3]{\frac{1}{V}}\right) \cdot \left(\sqrt[3]{1} \cdot \frac{A}{\ell}\right)}}{\left|\sqrt[3]{V}\right|}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))