Average Error: 1.6 → 1.3
Time: 15.2s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}} \le 0.999519675329676:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \left(2 \cdot \log \left(\sqrt[3]{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(\log \left(e^{\frac{1}{\sqrt{1}}}\right) + 1\right)}\\ \end{array}\]
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}} \le 0.999519675329676:\\
\;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \left(2 \cdot \log \left(\sqrt[3]{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{1}{2} \cdot \left(\log \left(e^{\frac{1}{\sqrt{1}}}\right) + 1\right)}\\

\end{array}
double f(double l, double Om, double kx, double ky) {
        double r53449 = 1.0;
        double r53450 = 2.0;
        double r53451 = r53449 / r53450;
        double r53452 = l;
        double r53453 = r53450 * r53452;
        double r53454 = Om;
        double r53455 = r53453 / r53454;
        double r53456 = pow(r53455, r53450);
        double r53457 = kx;
        double r53458 = sin(r53457);
        double r53459 = pow(r53458, r53450);
        double r53460 = ky;
        double r53461 = sin(r53460);
        double r53462 = pow(r53461, r53450);
        double r53463 = r53459 + r53462;
        double r53464 = r53456 * r53463;
        double r53465 = r53449 + r53464;
        double r53466 = sqrt(r53465);
        double r53467 = r53449 / r53466;
        double r53468 = r53449 + r53467;
        double r53469 = r53451 * r53468;
        double r53470 = sqrt(r53469);
        return r53470;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r53471 = 1.0;
        double r53472 = 2.0;
        double r53473 = l;
        double r53474 = r53472 * r53473;
        double r53475 = Om;
        double r53476 = r53474 / r53475;
        double r53477 = pow(r53476, r53472);
        double r53478 = kx;
        double r53479 = sin(r53478);
        double r53480 = pow(r53479, r53472);
        double r53481 = ky;
        double r53482 = sin(r53481);
        double r53483 = pow(r53482, r53472);
        double r53484 = r53480 + r53483;
        double r53485 = r53477 * r53484;
        double r53486 = r53471 + r53485;
        double r53487 = sqrt(r53486);
        double r53488 = r53471 / r53487;
        double r53489 = 0.999519675329676;
        bool r53490 = r53488 <= r53489;
        double r53491 = r53471 / r53472;
        double r53492 = 2.0;
        double r53493 = exp(r53488);
        double r53494 = cbrt(r53493);
        double r53495 = log(r53494);
        double r53496 = r53492 * r53495;
        double r53497 = r53496 + r53495;
        double r53498 = r53471 + r53497;
        double r53499 = r53491 * r53498;
        double r53500 = sqrt(r53499);
        double r53501 = sqrt(r53471);
        double r53502 = r53471 / r53501;
        double r53503 = exp(r53502);
        double r53504 = log(r53503);
        double r53505 = r53504 + r53471;
        double r53506 = r53491 * r53505;
        double r53507 = sqrt(r53506);
        double r53508 = r53490 ? r53500 : r53507;
        return r53508;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ 1.0 (sqrt (+ 1.0 (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))) < 0.999519675329676

    1. Initial program 1.0

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
    2. Using strategy rm

    3. Applied add-log-exp1.0

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \color{blue}{\log \left(e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right)}\right)}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt1.0

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \log \color{blue}{\left(\left(\sqrt[3]{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}} \cdot \sqrt[3]{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right) \cdot \sqrt[3]{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)}\right)}\]

    6. Applied log-prod1.0

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \color{blue}{\left(\log \left(\sqrt[3]{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}} \cdot \sqrt[3]{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)\right)}\right)}\]

    7. Simplified1.0

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)} + \log \left(\sqrt[3]{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)\right)\right)}\]

    if 0.999519675329676 < (/ 1.0 (sqrt (+ 1.0 (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))

    1. Initial program 2.1

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
    2. Using strategy rm

    3. Applied add-log-exp2.1

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \color{blue}{\log \left(e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right)}\right)}\]

    4. Taylor expanded around 0 1.5

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \log \left(e^{\frac{1}{\sqrt{1 + \color{blue}{0}}}}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}} \le 0.999519675329676:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \left(2 \cdot \log \left(\sqrt[3]{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(\log \left(e^{\frac{1}{\sqrt{1}}}\right) + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  :precision binary64
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))