Average Error: 0.9 → 0.7
Time: 9.7s
Precision: binary64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{2 \cdot \ell}{Om}\\ \sqrt{0.5 + \frac{0.5}{\sqrt{1 + t_0 \cdot \left(t_0 \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)\right)}}} \end{array} \]
\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}
t_0 := \frac{2 \cdot \ell}{Om}\\
\sqrt{0.5 + \frac{0.5}{\sqrt{1 + t_0 \cdot \left(t_0 \cdot \left({\sin kx}^{2} + {\sin ky}^{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
 (let* ((t_0 (/ (* 2.0 l) Om)))
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (+ 1.0 (* t_0 (* t_0 (+ (pow (sin kx) 2.0) (pow (sin ky) 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 t_0 = (2.0 * l) / Om;
	return sqrt(0.5 + (0.5 / sqrt(1.0 + (t_0 * (t_0 * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))))));
}

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 0.9

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

  2. Simplified0.9

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

  4. Applied unpow2_binary640.9

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

  5. Applied associate-*l*_binary640.7

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

  6. Simplified0.7

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

  7. Final simplification0.7

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

Reproduce

herbie shell --seed 2021204 
(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))))))))))