Average Error: 1.7 → 1.7
Time: 37.0s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} + \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) + 1}}} \cdot \left(\sqrt[3]{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) + 1}}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) + 1}}}\right)}\]
\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} + \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) + 1}}} \cdot \left(\sqrt[3]{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) + 1}}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) + 1}}}\right)}
double f(double l, double Om, double kx, double ky) {
        double r1272195 = 1.0;
        double r1272196 = 2.0;
        double r1272197 = r1272195 / r1272196;
        double r1272198 = l;
        double r1272199 = r1272196 * r1272198;
        double r1272200 = Om;
        double r1272201 = r1272199 / r1272200;
        double r1272202 = pow(r1272201, r1272196);
        double r1272203 = kx;
        double r1272204 = sin(r1272203);
        double r1272205 = pow(r1272204, r1272196);
        double r1272206 = ky;
        double r1272207 = sin(r1272206);
        double r1272208 = pow(r1272207, r1272196);
        double r1272209 = r1272205 + r1272208;
        double r1272210 = r1272202 * r1272209;
        double r1272211 = r1272195 + r1272210;
        double r1272212 = sqrt(r1272211);
        double r1272213 = r1272195 / r1272212;
        double r1272214 = r1272195 + r1272213;
        double r1272215 = r1272197 * r1272214;
        double r1272216 = sqrt(r1272215);
        return r1272216;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r1272217 = 0.5;
        double r1272218 = ky;
        double r1272219 = sin(r1272218);
        double r1272220 = r1272219 * r1272219;
        double r1272221 = kx;
        double r1272222 = sin(r1272221);
        double r1272223 = r1272222 * r1272222;
        double r1272224 = r1272220 + r1272223;
        double r1272225 = 2.0;
        double r1272226 = l;
        double r1272227 = r1272225 * r1272226;
        double r1272228 = Om;
        double r1272229 = r1272227 / r1272228;
        double r1272230 = r1272229 * r1272229;
        double r1272231 = r1272224 * r1272230;
        double r1272232 = 1.0;
        double r1272233 = r1272231 + r1272232;
        double r1272234 = sqrt(r1272233);
        double r1272235 = r1272217 / r1272234;
        double r1272236 = cbrt(r1272235);
        double r1272237 = r1272236 * r1272236;
        double r1272238 = r1272236 * r1272237;
        double r1272239 = r1272217 + r1272238;
        double r1272240 = sqrt(r1272239);
        return r1272240;
}

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 1.7

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

  2. Simplified1.7

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

  4. Applied add-cube-cbrt1.7

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

  5. Final simplification1.7

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

Reproduce

herbie shell --seed 2019149 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))