Average Error: 1.6 → 1.5
Time: 8.2s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\sqrt[3]{\frac{2 \cdot \ell}{Om}} \cdot \sqrt[3]{\frac{2 \cdot \ell}{Om}}\right)}^{2} \cdot \left({\left(\sqrt[3]{\frac{2 \cdot \ell}{Om}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)}}\right)}\]
\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]{\frac{2 \cdot \ell}{Om}} \cdot \sqrt[3]{\frac{2 \cdot \ell}{Om}}\right)}^{2} \cdot \left({\left(\sqrt[3]{\frac{2 \cdot \ell}{Om}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)}}\right)}
double f(double l, double Om, double kx, double ky) {
        double r51105 = 1.0;
        double r51106 = 2.0;
        double r51107 = r51105 / r51106;
        double r51108 = l;
        double r51109 = r51106 * r51108;
        double r51110 = Om;
        double r51111 = r51109 / r51110;
        double r51112 = pow(r51111, r51106);
        double r51113 = kx;
        double r51114 = sin(r51113);
        double r51115 = pow(r51114, r51106);
        double r51116 = ky;
        double r51117 = sin(r51116);
        double r51118 = pow(r51117, r51106);
        double r51119 = r51115 + r51118;
        double r51120 = r51112 * r51119;
        double r51121 = r51105 + r51120;
        double r51122 = sqrt(r51121);
        double r51123 = r51105 / r51122;
        double r51124 = r51105 + r51123;
        double r51125 = r51107 * r51124;
        double r51126 = sqrt(r51125);
        return r51126;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r51127 = 1.0;
        double r51128 = 2.0;
        double r51129 = r51127 / r51128;
        double r51130 = l;
        double r51131 = r51128 * r51130;
        double r51132 = Om;
        double r51133 = r51131 / r51132;
        double r51134 = cbrt(r51133);
        double r51135 = r51134 * r51134;
        double r51136 = pow(r51135, r51128);
        double r51137 = pow(r51134, r51128);
        double r51138 = kx;
        double r51139 = sin(r51138);
        double r51140 = pow(r51139, r51128);
        double r51141 = ky;
        double r51142 = sin(r51141);
        double r51143 = pow(r51142, r51128);
        double r51144 = r51140 + r51143;
        double r51145 = r51137 * r51144;
        double r51146 = r51136 * r51145;
        double r51147 = r51127 + r51146;
        double r51148 = sqrt(r51147);
        double r51149 = r51127 / r51148;
        double r51150 = r51127 + r51149;
        double r51151 = r51129 * r51150;
        double r51152 = sqrt(r51151);
        return r51152;
}

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 1.6

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

  3. Applied add-cube-cbrt1.6

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

  4. Applied unpow-prod-down1.6

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

  5. Applied associate-*l*1.5

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

  6. Final simplification1.5

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

Reproduce

herbie shell --seed 2019353 +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))))))))))