Average Error: 1.1 → 0.7
Time: 6.5s
Precision: binary64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2} \le 3.2541107435 \cdot 10^{-315}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \log \left({\left(e^{{\left(\frac{2}{Om} \cdot \ell\right)}^{2}}\right)}^{\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}}}\right)}\\ \end{array}\]
\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)}
↓
\begin{array}{l}
\mathbf{if}\;{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2} \le 3.2541107435 \cdot 10^{-315}:\\
\;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \log \left({\left(e^{{\left(\frac{2}{Om} \cdot \ell\right)}^{2}}\right)}^{\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}\right)}}\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}}}\right)}\\

\end{array}
double code(double l, double Om, double kx, double ky) {
	return ((double) sqrt(((double) (((double) (1.0 / 2.0)) * ((double) (1.0 + ((double) (1.0 / ((double) sqrt(((double) (1.0 + ((double) (((double) pow(((double) (((double) (2.0 * l)) / Om)), 2.0)) * ((double) (((double) pow(((double) sin(kx)), 2.0)) + ((double) pow(((double) sin(ky)), 2.0))))))))))))))))));
}
↓
double code(double l, double Om, double kx, double ky) {
	double VAR;
	if ((((double) (((double) pow(((double) sin(kx)), 2.0)) + ((double) pow(((double) sin(ky)), 2.0)))) <= 3.2541107435201e-315)) {
		VAR = ((double) sqrt(((double) (((double) (1.0 / 2.0)) * ((double) (1.0 + ((double) (1.0 / ((double) sqrt(((double) (1.0 + ((double) log(((double) pow(((double) exp(((double) pow(((double) (((double) (2.0 / Om)) * l)), 2.0)))), ((double) (((double) pow(((double) sin(kx)), 2.0)) + ((double) pow(((double) sin(ky)), 2.0))))))))))))))))))));
	} else {
		VAR = ((double) sqrt(((double) (((double) (1.0 / 2.0)) * ((double) (1.0 + ((double) (1.0 / ((double) sqrt(((double) (1.0 + ((double) (((double) (((double) pow(((double) sin(kx)), 2.0)) + ((double) pow(((double) sin(ky)), 2.0)))) * ((double) pow(((double) (((double) (2.0 * l)) / Om)), 2.0))))))))))))))));
	}
	return VAR;
}

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. Split input into 2 regimes

  2. if (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)) < 3.2541107435e-315

    1. Initial program 18.1

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

    3. Applied add-log-exp18.1

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

    4. Simplified11.8

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

    if 3.2541107435e-315 < (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))

    1. Initial program 0.0

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2} \le 3.2541107435 \cdot 10^{-315}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \log \left({\left(e^{{\left(\frac{2}{Om} \cdot \ell\right)}^{2}}\right)}^{\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  :precision binary64
  (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))))))))))