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 r34130 = 1.0;
        double r34131 = 2.0;
        double r34132 = r34130 / r34131;
        double r34133 = l;
        double r34134 = r34131 * r34133;
        double r34135 = Om;
        double r34136 = r34134 / r34135;
        double r34137 = pow(r34136, r34131);
        double r34138 = kx;
        double r34139 = sin(r34138);
        double r34140 = pow(r34139, r34131);
        double r34141 = ky;
        double r34142 = sin(r34141);
        double r34143 = pow(r34142, r34131);
        double r34144 = r34140 + r34143;
        double r34145 = r34137 * r34144;
        double r34146 = r34130 + r34145;
        double r34147 = sqrt(r34146);
        double r34148 = r34130 / r34147;
        double r34149 = r34130 + r34148;
        double r34150 = r34132 * r34149;
        double r34151 = sqrt(r34150);
        return r34151;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r34152 = 1.0;
        double r34153 = 2.0;
        double r34154 = r34152 / r34153;
        double r34155 = l;
        double r34156 = r34153 * r34155;
        double r34157 = Om;
        double r34158 = r34156 / r34157;
        double r34159 = pow(r34158, r34153);
        double r34160 = kx;
        double r34161 = sin(r34160);
        double r34162 = pow(r34161, r34153);
        double r34163 = ky;
        double r34164 = sin(r34163);
        double r34165 = pow(r34164, r34153);
        double r34166 = r34162 + r34165;
        double r34167 = r34159 * r34166;
        double r34168 = r34152 + r34167;
        double r34169 = sqrt(r34168);
        double r34170 = r34152 / r34169;
        double r34171 = 3.0;
        double r34172 = pow(r34170, r34171);
        double r34173 = cbrt(r34172);
        double r34174 = r34152 + r34173;
        double r34175 = r34154 * r34174;
        double r34176 = sqrt(r34175);
        return r34176;
}

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 
(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))))))))))