Average Error: 1.8 → 1.8
Time: 20.5s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}\right)}^{2} \cdot {\left(\sqrt[3]{\sin ky}\right)}^{2}\right)}}\right)}\]
\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(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}\right)}^{2} \cdot {\left(\sqrt[3]{\sin ky}\right)}^{2}\right)}}\right)}
double f(double l, double Om, double kx, double ky) {
        double r36154 = 1.0;
        double r36155 = 2.0;
        double r36156 = r36154 / r36155;
        double r36157 = l;
        double r36158 = r36155 * r36157;
        double r36159 = Om;
        double r36160 = r36158 / r36159;
        double r36161 = pow(r36160, r36155);
        double r36162 = kx;
        double r36163 = sin(r36162);
        double r36164 = pow(r36163, r36155);
        double r36165 = ky;
        double r36166 = sin(r36165);
        double r36167 = pow(r36166, r36155);
        double r36168 = r36164 + r36167;
        double r36169 = r36161 * r36168;
        double r36170 = r36154 + r36169;
        double r36171 = sqrt(r36170);
        double r36172 = r36154 / r36171;
        double r36173 = r36154 + r36172;
        double r36174 = r36156 * r36173;
        double r36175 = sqrt(r36174);
        return r36175;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r36176 = 1.0;
        double r36177 = 2.0;
        double r36178 = r36176 / r36177;
        double r36179 = l;
        double r36180 = r36177 * r36179;
        double r36181 = Om;
        double r36182 = r36180 / r36181;
        double r36183 = pow(r36182, r36177);
        double r36184 = kx;
        double r36185 = sin(r36184);
        double r36186 = pow(r36185, r36177);
        double r36187 = ky;
        double r36188 = sin(r36187);
        double r36189 = cbrt(r36188);
        double r36190 = r36189 * r36189;
        double r36191 = pow(r36190, r36177);
        double r36192 = pow(r36189, r36177);
        double r36193 = r36191 * r36192;
        double r36194 = r36186 + r36193;
        double r36195 = r36183 * r36194;
        double r36196 = r36176 + r36195;
        double r36197 = sqrt(r36196);
        double r36198 = r36176 / r36197;
        double r36199 = r36176 + r36198;
        double r36200 = r36178 * r36199;
        double r36201 = sqrt(r36200);
        return r36201;
}

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

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

  4. Applied unpow-prod-down1.8

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

  5. Final simplification1.8

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

Reproduce

herbie shell --seed 2019304 
(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))))))))))