Average Error: 1.7 → 1.6
Time: 9.8s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\log \left(e^{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)}}\right)}\right)}\]
\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}{\log \left(e^{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)}}\right)}\right)}
double f(double l, double Om, double kx, double ky) {
        double r57560 = 1.0;
        double r57561 = 2.0;
        double r57562 = r57560 / r57561;
        double r57563 = l;
        double r57564 = r57561 * r57563;
        double r57565 = Om;
        double r57566 = r57564 / r57565;
        double r57567 = pow(r57566, r57561);
        double r57568 = kx;
        double r57569 = sin(r57568);
        double r57570 = pow(r57569, r57561);
        double r57571 = ky;
        double r57572 = sin(r57571);
        double r57573 = pow(r57572, r57561);
        double r57574 = r57570 + r57573;
        double r57575 = r57567 * r57574;
        double r57576 = r57560 + r57575;
        double r57577 = sqrt(r57576);
        double r57578 = r57560 / r57577;
        double r57579 = r57560 + r57578;
        double r57580 = r57562 * r57579;
        double r57581 = sqrt(r57580);
        return r57581;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r57582 = 1.0;
        double r57583 = 2.0;
        double r57584 = r57582 / r57583;
        double r57585 = l;
        double r57586 = r57583 * r57585;
        double r57587 = Om;
        double r57588 = r57586 / r57587;
        double r57589 = 2.0;
        double r57590 = r57583 / r57589;
        double r57591 = pow(r57588, r57590);
        double r57592 = kx;
        double r57593 = sin(r57592);
        double r57594 = pow(r57593, r57583);
        double r57595 = ky;
        double r57596 = sin(r57595);
        double r57597 = pow(r57596, r57583);
        double r57598 = r57594 + r57597;
        double r57599 = r57591 * r57598;
        double r57600 = r57591 * r57599;
        double r57601 = r57582 + r57600;
        double r57602 = sqrt(r57601);
        double r57603 = exp(r57602);
        double r57604 = log(r57603);
        double r57605 = r57582 / r57604;
        double r57606 = r57582 + r57605;
        double r57607 = r57584 * r57606;
        double r57608 = sqrt(r57607);
        return r57608;
}

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. Using strategy rm

  3. Applied sqr-pow1.7

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

  4. Applied associate-*l*1.5

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

  6. Applied add-log-exp1.6

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

  7. Final simplification1.6

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

Reproduce

herbie shell --seed 2020056 
(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))))))))))