Average Error: 1.6 → 1.3
Time: 15.5s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}} \le 0.999519675329676:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \left(2 \cdot \log \left(\sqrt[3]{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1}}\right)}\\ \end{array}\]
\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(1 + \frac{1}{\sqrt{1}}\right)}\\

\end{array}
double f(double l, double Om, double kx, double ky) {
        double r56436 = 1.0;
        double r56437 = 2.0;
        double r56438 = r56436 / r56437;
        double r56439 = l;
        double r56440 = r56437 * r56439;
        double r56441 = Om;
        double r56442 = r56440 / r56441;
        double r56443 = pow(r56442, r56437);
        double r56444 = kx;
        double r56445 = sin(r56444);
        double r56446 = pow(r56445, r56437);
        double r56447 = ky;
        double r56448 = sin(r56447);
        double r56449 = pow(r56448, r56437);
        double r56450 = r56446 + r56449;
        double r56451 = r56443 * r56450;
        double r56452 = r56436 + r56451;
        double r56453 = sqrt(r56452);
        double r56454 = r56436 / r56453;
        double r56455 = r56436 + r56454;
        double r56456 = r56438 * r56455;
        double r56457 = sqrt(r56456);
        return r56457;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r56458 = 1.0;
        double r56459 = 2.0;
        double r56460 = l;
        double r56461 = r56459 * r56460;
        double r56462 = Om;
        double r56463 = r56461 / r56462;
        double r56464 = pow(r56463, r56459);
        double r56465 = kx;
        double r56466 = sin(r56465);
        double r56467 = pow(r56466, r56459);
        double r56468 = ky;
        double r56469 = sin(r56468);
        double r56470 = pow(r56469, r56459);
        double r56471 = r56467 + r56470;
        double r56472 = r56464 * r56471;
        double r56473 = r56458 + r56472;
        double r56474 = sqrt(r56473);
        double r56475 = r56458 / r56474;
        double r56476 = 0.999519675329676;
        bool r56477 = r56475 <= r56476;
        double r56478 = r56458 / r56459;
        double r56479 = 2.0;
        double r56480 = exp(r56475);
        double r56481 = cbrt(r56480);
        double r56482 = log(r56481);
        double r56483 = r56479 * r56482;
        double r56484 = r56483 + r56482;
        double r56485 = r56458 + r56484;
        double r56486 = r56478 * r56485;
        double r56487 = sqrt(r56486);
        double r56488 = sqrt(r56458);
        double r56489 = r56458 / r56488;
        double r56490 = r56458 + r56489;
        double r56491 = r56478 * r56490;
        double r56492 = sqrt(r56491);
        double r56493 = r56477 ? r56487 : r56492;
        return r56493;
}

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-cbrt-cube2.1

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

    4. Simplified2.1

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt[3]{\color{blue}{{\left(\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}\right)}^{3}}}}\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt2.1

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

    7. Simplified2.1

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

    8. Taylor expanded around 0 1.5

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

  4. Final simplification1.3

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

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))))))))))