Average Error: 1.0 → 0.7
Time: 10.0s
Precision: binary64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;{\sin kx}^{2} + {\sin ky}^{2} \leq 1.976262583365 \cdot 10^{-323}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\sqrt{1 + \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}}}}\\ \end{array}\]
\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}\;{\sin kx}^{2} + {\sin ky}^{2} \leq 1.976262583365 \cdot 10^{-323}:\\
\;\;\;\;1\\

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

\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 (sin kx) 2.0) (pow (sin ky) 2.0)) 1.976262583365e-323)
   1.0
   (sqrt
    (+
     0.5
     (*
      (sqrt 0.5)
      (/
       (sqrt 0.5)
       (sqrt
        (+
         1.0
         (*
          (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))
          (pow (/ (* 2.0 l) Om) 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 ((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)) <= 1.976262583365e-323) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5 + (sqrt(0.5) * (sqrt(0.5) / sqrt(1.0 + ((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)) * pow(((2.0 * l) / Om), 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 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2)) < 1.97626e-323

    1. Initial program 18.5

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

    2. Simplified18.5

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

    3. Taylor expanded around 0 12.6

      \[\leadsto \sqrt{0.5 + \frac{0.5}{\color{blue}{1}}}\]

    if 1.97626e-323 < (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2))

    1. Initial program 0.0

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

    2. Simplified0.0

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

    4. Applied *-un-lft-identity_binary640.0

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

    5. Applied sqrt-prod_binary640.0

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

    6. Applied add-sqr-sqrt_binary640.0

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

    7. Applied times-frac_binary640.0

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

    8. Simplified0.0

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} + {\sin ky}^{2} \leq 1.976262583365 \cdot 10^{-323}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \sqrt{0.5} \cdot \frac{\sqrt{0.5}}{\sqrt{1 + \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020253 
(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))))))))))