Average Error: 1.1 → 1.0
Time: 9.1s
Precision: binary64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \leq 3.543223448247394 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + 2 \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \left(\frac{ky \cdot ky}{Om} + \sin kx \cdot \frac{\sin kx}{Om}\right)\right)}}\\ \end{array}\]
\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}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \leq 3.543223448247394 \cdot 10^{+157}:\\
\;\;\;\;\sqrt{0.5 \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)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + 2 \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \left(\frac{ky \cdot ky}{Om} + \sin kx \cdot \frac{\sin kx}{Om}\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 (<= (pow (/ (* 2.0 l) Om) 2.0) 3.543223448247394e+157)
   (sqrt
    (*
     0.5
     (+
      1.0
      (/
       1.0
       (sqrt
        (+
         1.0
         (*
          (pow (/ (* 2.0 l) Om) 2.0)
          (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))))))
   (sqrt
    (+
     0.5
     (/
      0.5
      (+
       1.0
       (*
        2.0
        (*
         (/ (* l l) Om)
         (+ (/ (* ky ky) Om) (* (sin kx) (/ (sin kx) Om)))))))))))
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 (pow(((2.0 * l) / Om), 2.0) <= 3.543223448247394e+157) {
		tmp = sqrt(0.5 * (1.0 + (1.0 / sqrt(1.0 + (pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))))));
	} else {
		tmp = sqrt(0.5 + (0.5 / (1.0 + (2.0 * (((l * l) / Om) * (((ky * ky) / Om) + (sin(kx) * (sin(kx) / Om))))))));
	}
	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 (pow.f64 (/.f64 (*.f64 2 l) Om) 2) < 3.5432234482473943e157

    1. Initial program 0.0

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

    if 3.5432234482473943e157 < (pow.f64 (/.f64 (*.f64 2 l) Om) 2)

    1. Initial program 2.9

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

    2. Simplified2.9

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

    3. Taylor expanded around 0 24.3

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

    4. Simplified3.9

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

    6. Applied *-un-lft-identity_binary643.9

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

    7. Applied add-sqr-sqrt_binary6433.7

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

    8. Applied unpow-prod-down_binary6433.7

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

    9. Applied times-frac_binary6433.0

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

    10. Simplified33.0

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

    11. Simplified2.6

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

    12. Taylor expanded around 0 2.7

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

    13. Simplified2.7

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \leq 3.543223448247394 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + 2 \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \left(\frac{ky \cdot ky}{Om} + \sin kx \cdot \frac{\sin kx}{Om}\right)\right)}}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(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))))))))))