Average Error: 1.8 → 1.9
Time: 7.4s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\left(\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}} \cdot \sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right) \cdot \log \left(e^{\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right)}\right)}\]
\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)}
↓
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\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 \log \left(e^{\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)}\right)}
double f(double l, double Om, double kx, double ky) {
        double r42432 = 1.0;
        double r42433 = 2.0;
        double r42434 = r42432 / r42433;
        double r42435 = l;
        double r42436 = r42433 * r42435;
        double r42437 = Om;
        double r42438 = r42436 / r42437;
        double r42439 = pow(r42438, r42433);
        double r42440 = kx;
        double r42441 = sin(r42440);
        double r42442 = pow(r42441, r42433);
        double r42443 = ky;
        double r42444 = sin(r42443);
        double r42445 = pow(r42444, r42433);
        double r42446 = r42442 + r42445;
        double r42447 = r42439 * r42446;
        double r42448 = r42432 + r42447;
        double r42449 = sqrt(r42448);
        double r42450 = r42432 / r42449;
        double r42451 = r42432 + r42450;
        double r42452 = r42434 * r42451;
        double r42453 = sqrt(r42452);
        return r42453;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r42454 = 1.0;
        double r42455 = 2.0;
        double r42456 = r42454 / r42455;
        double r42457 = l;
        double r42458 = r42455 * r42457;
        double r42459 = Om;
        double r42460 = r42458 / r42459;
        double r42461 = pow(r42460, r42455);
        double r42462 = kx;
        double r42463 = sin(r42462);
        double r42464 = pow(r42463, r42455);
        double r42465 = ky;
        double r42466 = sin(r42465);
        double r42467 = pow(r42466, r42455);
        double r42468 = r42464 + r42467;
        double r42469 = r42461 * r42468;
        double r42470 = r42454 + r42469;
        double r42471 = sqrt(r42470);
        double r42472 = cbrt(r42471);
        double r42473 = r42472 * r42472;
        double r42474 = exp(r42472);
        double r42475 = log(r42474);
        double r42476 = r42473 * r42475;
        double r42477 = r42454 / r42476;
        double r42478 = r42454 + r42477;
        double r42479 = r42456 * r42478;
        double r42480 = sqrt(r42479);
        return r42480;
}

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. Initial program 1.8

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

  3. Applied add-cube-cbrt1.8

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

  5. Applied add-log-exp1.9

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

  6. Final simplification1.9

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

Reproduce

herbie shell --seed 2019362 
(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))))))))))