Average Error: 1.2 → 0.4
Time: 18.3s
Precision: binary64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq -1.4589900259949848 \cdot 10^{+225}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\ell \cdot \sqrt{\frac{4}{Om} \cdot \left(\frac{{\sin ky}^{2}}{Om} + \frac{{\sin kx}^{2}}{Om}\right)} + 0.5 \cdot \frac{\sqrt{\frac{1}{\frac{4}{Om} \cdot \left(\frac{{\sin ky}^{2}}{Om} + \frac{{\sin kx}^{2}}{Om}\right)}}}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{2 \cdot \ell}{Om} \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \left({\sin ky}^{2} + {\sin kx}^{2}\right)\right)}}}\\ \end{array}\]
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq -1.4589900259949848 \cdot 10^{+225}:\\
\;\;\;\;\sqrt{0.5 + \frac{0.5}{\ell \cdot \sqrt{\frac{4}{Om} \cdot \left(\frac{{\sin ky}^{2}}{Om} + \frac{{\sin kx}^{2}}{Om}\right)} + 0.5 \cdot \frac{\sqrt{\frac{1}{\frac{4}{Om} \cdot \left(\frac{{\sin ky}^{2}}{Om} + \frac{{\sin kx}^{2}}{Om}\right)}}}{\ell}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{2 \cdot \ell}{Om} \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \left({\sin ky}^{2} + {\sin kx}^{2}\right)\right)}}}\\

\end{array}
(FPCore (l Om kx ky)
 :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))))))))))
↓
(FPCore (l Om kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l) Om) -1.4589900259949848e+225)
   (sqrt
    (+
     0.5
     (/
      0.5
      (+
       (*
        l
        (sqrt
         (*
          (/ 4.0 Om)
          (+ (/ (pow (sin ky) 2.0) Om) (/ (pow (sin kx) 2.0) Om)))))
       (*
        0.5
        (/
         (sqrt
          (/
           1.0
           (*
            (/ 4.0 Om)
            (+ (/ (pow (sin ky) 2.0) Om) (/ (pow (sin kx) 2.0) Om)))))
         l))))))
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (+
        1.0
        (*
         (/ (* 2.0 l) Om)
         (* (/ (* 2.0 l) Om) (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)))))))))))
double code(double l, double Om, double kx, double ky) {
	return 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)))))));
}
↓
double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((2.0 * l) / Om) <= -1.4589900259949848e+225) {
		tmp = sqrt(0.5 + (0.5 / ((l * sqrt((4.0 / Om) * ((pow(sin(ky), 2.0) / Om) + (pow(sin(kx), 2.0) / Om)))) + (0.5 * (sqrt(1.0 / ((4.0 / Om) * ((pow(sin(ky), 2.0) / Om) + (pow(sin(kx), 2.0) / Om)))) / l)))));
	} else {
		tmp = sqrt(0.5 + (0.5 / sqrt(1.0 + (((2.0 * l) / Om) * (((2.0 * l) / Om) * (pow(sin(ky), 2.0) + pow(sin(kx), 2.0)))))));
	}
	return tmp;
}

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 (/.f64 (*.f64 2 l) Om) < -1.45899002599498478e225

    1. Initial program 4.8

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

    2. Simplified4.8

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}}}\]

    3. Taylor expanded around inf 16.9

      \[\leadsto \sqrt{0.5 + \frac{0.5}{\color{blue}{\sqrt{4 \cdot \frac{{\sin ky}^{2}}{{Om}^{2}} + 4 \cdot \frac{{\sin kx}^{2}}{{Om}^{2}}} \cdot \ell + 0.5 \cdot \left(\sqrt{\frac{1}{4 \cdot \frac{{\sin ky}^{2}}{{Om}^{2}} + 4 \cdot \frac{{\sin kx}^{2}}{{Om}^{2}}}} \cdot \frac{1}{\ell}\right)}}}\]

    4. Simplified0.3

      \[\leadsto \sqrt{0.5 + \frac{0.5}{\color{blue}{\ell \cdot \sqrt{\frac{4}{Om} \cdot \left(\frac{{\sin ky}^{2}}{Om} + \frac{{\sin kx}^{2}}{Om}\right)} + 0.5 \cdot \frac{\sqrt{\frac{1}{\frac{4}{Om} \cdot \left(\frac{{\sin ky}^{2}}{Om} + \frac{{\sin kx}^{2}}{Om}\right)}}}{\ell}}}}\]

    if -1.45899002599498478e225 < (/.f64 (*.f64 2 l) Om)

    1. Initial program 0.8

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

    2. Simplified0.8

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}}}\]
    3. Using strategy rm

    4. Applied unpow2_binary64_1430.8

      \[\leadsto \sqrt{0.5 + \frac{0.5}{\sqrt{1 + \color{blue}{\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right)} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}}\]

    5. Applied associate-*l*_binary64_190.5

      \[\leadsto \sqrt{0.5 + \frac{0.5}{\sqrt{1 + \color{blue}{\frac{2 \cdot \ell}{Om} \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)\right)}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq -1.4589900259949848 \cdot 10^{+225}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\ell \cdot \sqrt{\frac{4}{Om} \cdot \left(\frac{{\sin ky}^{2}}{Om} + \frac{{\sin kx}^{2}}{Om}\right)} + 0.5 \cdot \frac{\sqrt{\frac{1}{\frac{4}{Om} \cdot \left(\frac{{\sin ky}^{2}}{Om} + \frac{{\sin kx}^{2}}{Om}\right)}}}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{2 \cdot \ell}{Om} \cdot \left(\frac{2 \cdot \ell}{Om} \cdot \left({\sin ky}^{2} + {\sin kx}^{2}\right)\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021019 
(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))))))))))