Average Error: 1.2 → 1.2
Time: 7.1s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \sqrt[3]{{\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}^{3}}\right)}\]
\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 + \sqrt[3]{{\left(\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)}^{3}}\right)}
double f(double l, double Om, double kx, double ky) {
        double r35201 = 1.0;
        double r35202 = 2.0;
        double r35203 = r35201 / r35202;
        double r35204 = l;
        double r35205 = r35202 * r35204;
        double r35206 = Om;
        double r35207 = r35205 / r35206;
        double r35208 = pow(r35207, r35202);
        double r35209 = kx;
        double r35210 = sin(r35209);
        double r35211 = pow(r35210, r35202);
        double r35212 = ky;
        double r35213 = sin(r35212);
        double r35214 = pow(r35213, r35202);
        double r35215 = r35211 + r35214;
        double r35216 = r35208 * r35215;
        double r35217 = r35201 + r35216;
        double r35218 = sqrt(r35217);
        double r35219 = r35201 / r35218;
        double r35220 = r35201 + r35219;
        double r35221 = r35203 * r35220;
        double r35222 = sqrt(r35221);
        return r35222;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r35223 = 1.0;
        double r35224 = 2.0;
        double r35225 = r35223 / r35224;
        double r35226 = l;
        double r35227 = r35224 * r35226;
        double r35228 = Om;
        double r35229 = r35227 / r35228;
        double r35230 = pow(r35229, r35224);
        double r35231 = kx;
        double r35232 = sin(r35231);
        double r35233 = pow(r35232, r35224);
        double r35234 = ky;
        double r35235 = sin(r35234);
        double r35236 = pow(r35235, r35224);
        double r35237 = r35233 + r35236;
        double r35238 = r35230 * r35237;
        double r35239 = r35223 + r35238;
        double r35240 = sqrt(r35239);
        double r35241 = r35223 / r35240;
        double r35242 = 3.0;
        double r35243 = pow(r35241, r35242);
        double r35244 = cbrt(r35243);
        double r35245 = r35223 + r35244;
        double r35246 = r35225 * r35245;
        double r35247 = sqrt(r35246);
        return r35247;
}

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 1.2

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

  3. Applied add-cbrt-cube1.2

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

  4. Applied add-cbrt-cube1.2

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

  5. Applied cbrt-undiv1.2

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

  6. Simplified1.2

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

  7. Final simplification1.2

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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))))))))))