Average Error: 1.6 → 1.5
Time: 6.4s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\sqrt[3]{\frac{2 \cdot \ell}{Om}} \cdot \sqrt[3]{\frac{2 \cdot \ell}{Om}}\right)}^{2} \cdot \left({\left(\sqrt[3]{\frac{2 \cdot \ell}{Om}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)}}\right)}\]
\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}{\sqrt{1 + {\left(\sqrt[3]{\frac{2 \cdot \ell}{Om}} \cdot \sqrt[3]{\frac{2 \cdot \ell}{Om}}\right)}^{2} \cdot \left({\left(\sqrt[3]{\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 r43316 = 1.0;
        double r43317 = 2.0;
        double r43318 = r43316 / r43317;
        double r43319 = l;
        double r43320 = r43317 * r43319;
        double r43321 = Om;
        double r43322 = r43320 / r43321;
        double r43323 = pow(r43322, r43317);
        double r43324 = kx;
        double r43325 = sin(r43324);
        double r43326 = pow(r43325, r43317);
        double r43327 = ky;
        double r43328 = sin(r43327);
        double r43329 = pow(r43328, r43317);
        double r43330 = r43326 + r43329;
        double r43331 = r43323 * r43330;
        double r43332 = r43316 + r43331;
        double r43333 = sqrt(r43332);
        double r43334 = r43316 / r43333;
        double r43335 = r43316 + r43334;
        double r43336 = r43318 * r43335;
        double r43337 = sqrt(r43336);
        return r43337;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r43338 = 1.0;
        double r43339 = 2.0;
        double r43340 = r43338 / r43339;
        double r43341 = l;
        double r43342 = r43339 * r43341;
        double r43343 = Om;
        double r43344 = r43342 / r43343;
        double r43345 = cbrt(r43344);
        double r43346 = r43345 * r43345;
        double r43347 = pow(r43346, r43339);
        double r43348 = pow(r43345, r43339);
        double r43349 = kx;
        double r43350 = sin(r43349);
        double r43351 = pow(r43350, r43339);
        double r43352 = ky;
        double r43353 = sin(r43352);
        double r43354 = pow(r43353, r43339);
        double r43355 = r43351 + r43354;
        double r43356 = r43348 * r43355;
        double r43357 = r43347 * r43356;
        double r43358 = r43338 + r43357;
        double r43359 = sqrt(r43358);
        double r43360 = r43338 / r43359;
        double r43361 = r43338 + r43360;
        double r43362 = r43340 * r43361;
        double r43363 = sqrt(r43362);
        return r43363;
}

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 1.6

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

  3. Applied add-cube-cbrt1.6

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

  4. Applied unpow-prod-down1.6

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

  5. Applied associate-*l*1.5

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

  6. Final simplification1.5

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

Reproduce

herbie shell --seed 2019353 
(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))))))))))