Average Error: 1.6 → 0.9
Time: 31.3s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{2 \cdot \left(\frac{\frac{\ell \cdot \sin kx}{Om} \cdot \frac{\ell \cdot \sin kx}{Om}}{\sqrt{1}} + \frac{\frac{\sin ky \cdot \ell}{Om} \cdot \frac{\sin ky \cdot \ell}{Om}}{\sqrt{1}}\right) + \sqrt{1}}\right)}\]
\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} \cdot \left(1 + \frac{1}{2 \cdot \left(\frac{\frac{\ell \cdot \sin kx}{Om} \cdot \frac{\ell \cdot \sin kx}{Om}}{\sqrt{1}} + \frac{\frac{\sin ky \cdot \ell}{Om} \cdot \frac{\sin ky \cdot \ell}{Om}}{\sqrt{1}}\right) + \sqrt{1}}\right)}
double f(double l, double Om, double kx, double ky) {
        double r1929629 = 1.0;
        double r1929630 = 2.0;
        double r1929631 = r1929629 / r1929630;
        double r1929632 = l;
        double r1929633 = r1929630 * r1929632;
        double r1929634 = Om;
        double r1929635 = r1929633 / r1929634;
        double r1929636 = pow(r1929635, r1929630);
        double r1929637 = kx;
        double r1929638 = sin(r1929637);
        double r1929639 = pow(r1929638, r1929630);
        double r1929640 = ky;
        double r1929641 = sin(r1929640);
        double r1929642 = pow(r1929641, r1929630);
        double r1929643 = r1929639 + r1929642;
        double r1929644 = r1929636 * r1929643;
        double r1929645 = r1929629 + r1929644;
        double r1929646 = sqrt(r1929645);
        double r1929647 = r1929629 / r1929646;
        double r1929648 = r1929629 + r1929647;
        double r1929649 = r1929631 * r1929648;
        double r1929650 = sqrt(r1929649);
        return r1929650;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r1929651 = 1.0;
        double r1929652 = 2.0;
        double r1929653 = r1929651 / r1929652;
        double r1929654 = l;
        double r1929655 = kx;
        double r1929656 = sin(r1929655);
        double r1929657 = r1929654 * r1929656;
        double r1929658 = Om;
        double r1929659 = r1929657 / r1929658;
        double r1929660 = r1929659 * r1929659;
        double r1929661 = sqrt(r1929651);
        double r1929662 = r1929660 / r1929661;
        double r1929663 = ky;
        double r1929664 = sin(r1929663);
        double r1929665 = r1929664 * r1929654;
        double r1929666 = r1929665 / r1929658;
        double r1929667 = r1929666 * r1929666;
        double r1929668 = r1929667 / r1929661;
        double r1929669 = r1929662 + r1929668;
        double r1929670 = r1929652 * r1929669;
        double r1929671 = r1929670 + r1929661;
        double r1929672 = r1929651 / r1929671;
        double r1929673 = r1929651 + r1929672;
        double r1929674 = r1929653 * r1929673;
        double r1929675 = sqrt(r1929674);
        return r1929675;
}

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

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

  4. Applied times-frac1.6

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

  5. Applied unpow-prod-down3.6

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

  6. Applied associate-*l*3.5

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

  7. Taylor expanded around inf 17.1

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

  8. Simplified0.9

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

  9. Final simplification0.9

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

Reproduce

herbie shell --seed 2019169 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (sqrt (+ 1.0 (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))