Average Error: 1.7 → 1.4
Time: 32.2s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{\frac{1}{2}}{\sqrt{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}}}{\sqrt{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}} + \frac{1}{2}}\]
\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)}
↓
\sqrt{\frac{\frac{\frac{1}{2}}{\sqrt{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}}}{\sqrt{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r786607 = 1.0;
        double r786608 = 2.0;
        double r786609 = r786607 / r786608;
        double r786610 = l;
        double r786611 = r786608 * r786610;
        double r786612 = Om;
        double r786613 = r786611 / r786612;
        double r786614 = pow(r786613, r786608);
        double r786615 = kx;
        double r786616 = sin(r786615);
        double r786617 = pow(r786616, r786608);
        double r786618 = ky;
        double r786619 = sin(r786618);
        double r786620 = pow(r786619, r786608);
        double r786621 = r786617 + r786620;
        double r786622 = r786614 * r786621;
        double r786623 = r786607 + r786622;
        double r786624 = sqrt(r786623);
        double r786625 = r786607 / r786624;
        double r786626 = r786607 + r786625;
        double r786627 = r786609 * r786626;
        double r786628 = sqrt(r786627);
        return r786628;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r786629 = 0.5;
        double r786630 = 2.0;
        double r786631 = l;
        double r786632 = r786630 * r786631;
        double r786633 = Om;
        double r786634 = r786632 / r786633;
        double r786635 = ky;
        double r786636 = sin(r786635);
        double r786637 = r786636 * r786636;
        double r786638 = kx;
        double r786639 = sin(r786638);
        double r786640 = r786639 * r786639;
        double r786641 = r786637 + r786640;
        double r786642 = r786634 * r786641;
        double r786643 = r786642 * r786634;
        double r786644 = 1.0;
        double r786645 = r786643 + r786644;
        double r786646 = sqrt(r786645);
        double r786647 = sqrt(r786646);
        double r786648 = r786629 / r786647;
        double r786649 = r786648 / r786647;
        double r786650 = r786649 + r786629;
        double r786651 = sqrt(r786650);
        return r786651;
}

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 associate-*r*1.4

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

  6. Applied add-sqr-sqrt1.4

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

  7. Applied associate-/r*1.4

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

  8. Final simplification1.4

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

Reproduce

herbie shell --seed 2019152 
(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))))))))))