Average Error: 1.8 → 1.8
Time: 38.4s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \left(\log \left(\sqrt{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right) + \log \left(\sqrt{e^{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)\right)\right)}\]
\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 + \left(\log \left(\sqrt{e^{\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) + \log \left(\sqrt{e^{\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)\right)\right)}
double f(double l, double Om, double kx, double ky) {
        double r58986 = 1.0;
        double r58987 = 2.0;
        double r58988 = r58986 / r58987;
        double r58989 = l;
        double r58990 = r58987 * r58989;
        double r58991 = Om;
        double r58992 = r58990 / r58991;
        double r58993 = pow(r58992, r58987);
        double r58994 = kx;
        double r58995 = sin(r58994);
        double r58996 = pow(r58995, r58987);
        double r58997 = ky;
        double r58998 = sin(r58997);
        double r58999 = pow(r58998, r58987);
        double r59000 = r58996 + r58999;
        double r59001 = r58993 * r59000;
        double r59002 = r58986 + r59001;
        double r59003 = sqrt(r59002);
        double r59004 = r58986 / r59003;
        double r59005 = r58986 + r59004;
        double r59006 = r58988 * r59005;
        double r59007 = sqrt(r59006);
        return r59007;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r59008 = 1.0;
        double r59009 = 2.0;
        double r59010 = r59008 / r59009;
        double r59011 = l;
        double r59012 = r59009 * r59011;
        double r59013 = Om;
        double r59014 = r59012 / r59013;
        double r59015 = pow(r59014, r59009);
        double r59016 = kx;
        double r59017 = sin(r59016);
        double r59018 = pow(r59017, r59009);
        double r59019 = ky;
        double r59020 = sin(r59019);
        double r59021 = pow(r59020, r59009);
        double r59022 = r59018 + r59021;
        double r59023 = r59015 * r59022;
        double r59024 = r59008 + r59023;
        double r59025 = sqrt(r59024);
        double r59026 = r59008 / r59025;
        double r59027 = exp(r59026);
        double r59028 = sqrt(r59027);
        double r59029 = log(r59028);
        double r59030 = r59029 + r59029;
        double r59031 = r59008 + r59030;
        double r59032 = r59010 * r59031;
        double r59033 = sqrt(r59032);
        return r59033;
}

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 1.8

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

  3. Applied add-log-exp1.8

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

  5. Applied add-sqr-sqrt1.8

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

  6. Applied log-prod1.8

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

  7. Final simplification1.8

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

Reproduce

herbie shell --seed 2019199 
(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))))))))))