Average Error: 19.0 → 13.0
Time: 16.2s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le -6.472040851531678203958266160893295162867 \cdot 10^{-271}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\\ \mathbf{elif}\;V \cdot \ell \le 0.0:\\ \;\;\;\;\left(c0 \cdot \sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right) \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \le 2.700786048354441713096879449910575204342 \cdot 10^{293}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{\sqrt{A}}{V}} \cdot \sqrt{\frac{\sqrt{A}}{\ell}}\right)\\ \end{array}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;V \cdot \ell \le -6.472040851531678203958266160893295162867 \cdot 10^{-271}:\\
\;\;\;\;\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\\

\mathbf{elif}\;V \cdot \ell \le 0.0:\\
\;\;\;\;\left(c0 \cdot \sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right) \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\\

\mathbf{elif}\;V \cdot \ell \le 2.700786048354441713096879449910575204342 \cdot 10^{293}:\\
\;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)\\

\mathbf{else}:\\
\;\;\;\;c0 \cdot \left(\sqrt{\frac{\sqrt{A}}{V}} \cdot \sqrt{\frac{\sqrt{A}}{\ell}}\right)\\

\end{array}
double f(double c0, double A, double V, double l) {
        double r103473 = c0;
        double r103474 = A;
        double r103475 = V;
        double r103476 = l;
        double r103477 = r103475 * r103476;
        double r103478 = r103474 / r103477;
        double r103479 = sqrt(r103478);
        double r103480 = r103473 * r103479;
        return r103480;
}
↓
double f(double c0, double A, double V, double l) {
        double r103481 = V;
        double r103482 = l;
        double r103483 = r103481 * r103482;
        double r103484 = -6.472040851531678e-271;
        bool r103485 = r103483 <= r103484;
        double r103486 = c0;
        double r103487 = A;
        double r103488 = r103487 / r103483;
        double r103489 = sqrt(r103488);
        double r103490 = sqrt(r103489);
        double r103491 = r103486 * r103490;
        double r103492 = r103491 * r103490;
        double r103493 = 0.0;
        bool r103494 = r103483 <= r103493;
        double r103495 = cbrt(r103487);
        double r103496 = r103495 * r103495;
        double r103497 = r103496 / r103481;
        double r103498 = sqrt(r103497);
        double r103499 = r103486 * r103498;
        double r103500 = r103495 / r103482;
        double r103501 = sqrt(r103500);
        double r103502 = r103499 * r103501;
        double r103503 = 2.7007860483544417e+293;
        bool r103504 = r103483 <= r103503;
        double r103505 = sqrt(r103487);
        double r103506 = 1.0;
        double r103507 = r103506 / r103483;
        double r103508 = sqrt(r103507);
        double r103509 = r103505 * r103508;
        double r103510 = r103486 * r103509;
        double r103511 = r103505 / r103481;
        double r103512 = sqrt(r103511);
        double r103513 = r103505 / r103482;
        double r103514 = sqrt(r103513);
        double r103515 = r103512 * r103514;
        double r103516 = r103486 * r103515;
        double r103517 = r103504 ? r103510 : r103516;
        double r103518 = r103494 ? r103502 : r103517;
        double r103519 = r103485 ? r103492 : r103518;
        return r103519;
}

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 (* V l) < -6.472040851531678e-271

    1. Initial program 13.9

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

    3. Applied add-sqr-sqrt13.9

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

    4. Applied sqrt-prod14.1

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

    5. Applied associate-*r*14.1

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

    if -6.472040851531678e-271 < (* V l) < 0.0

    1. Initial program 53.8

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

    3. Applied add-cube-cbrt53.9

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

    4. Applied times-frac34.1

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

    5. Applied sqrt-prod38.4

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

    6. Applied associate-*r*38.5

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

    if 0.0 < (* V l) < 2.7007860483544417e+293

    1. Initial program 16.2

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

    3. Applied div-inv16.5

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

    4. Applied sqrt-prod7.3

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

    if 2.7007860483544417e+293 < (* V l)

    1. Initial program 37.6

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

    3. Applied add-sqr-sqrt37.6

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

    4. Applied times-frac22.8

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

    5. Applied sqrt-prod33.0

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

  4. Final simplification13.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -6.472040851531678203958266160893295162867 \cdot 10^{-271}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\\ \mathbf{elif}\;V \cdot \ell \le 0.0:\\ \;\;\;\;\left(c0 \cdot \sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right) \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \le 2.700786048354441713096879449910575204342 \cdot 10^{293}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{\sqrt{A}}{V}} \cdot \sqrt{\frac{\sqrt{A}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))