Average Error: 0.8 → 0.8
Time: 6.3s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \left(\sqrt[3]{{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)} \cdot \sqrt[3]{{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}\right) \cdot \sqrt[3]{{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right)}\]
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\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]{{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)} \cdot \sqrt[3]{{\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]{{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right)}
double f(double l, double Om, double kx, double ky) {
        double r38254 = 1.0;
        double r38255 = 2.0;
        double r38256 = r38254 / r38255;
        double r38257 = l;
        double r38258 = r38255 * r38257;
        double r38259 = Om;
        double r38260 = r38258 / r38259;
        double r38261 = pow(r38260, r38255);
        double r38262 = kx;
        double r38263 = sin(r38262);
        double r38264 = pow(r38263, r38255);
        double r38265 = ky;
        double r38266 = sin(r38265);
        double r38267 = pow(r38266, r38255);
        double r38268 = r38264 + r38267;
        double r38269 = r38261 * r38268;
        double r38270 = r38254 + r38269;
        double r38271 = sqrt(r38270);
        double r38272 = r38254 / r38271;
        double r38273 = r38254 + r38272;
        double r38274 = r38256 * r38273;
        double r38275 = sqrt(r38274);
        return r38275;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r38276 = 1.0;
        double r38277 = 2.0;
        double r38278 = r38276 / r38277;
        double r38279 = l;
        double r38280 = r38277 * r38279;
        double r38281 = Om;
        double r38282 = r38280 / r38281;
        double r38283 = pow(r38282, r38277);
        double r38284 = kx;
        double r38285 = sin(r38284);
        double r38286 = pow(r38285, r38277);
        double r38287 = ky;
        double r38288 = sin(r38287);
        double r38289 = pow(r38288, r38277);
        double r38290 = r38286 + r38289;
        double r38291 = r38283 * r38290;
        double r38292 = cbrt(r38291);
        double r38293 = r38292 * r38292;
        double r38294 = r38293 * r38292;
        double r38295 = r38276 + r38294;
        double r38296 = sqrt(r38295);
        double r38297 = r38276 / r38296;
        double r38298 = r38276 + r38297;
        double r38299 = r38278 * r38298;
        double r38300 = sqrt(r38299);
        return r38300;
}

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 0.8

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

  3. Applied add-cube-cbrt0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2020065 
(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))))))))))