Average Error: 1.0 → 0.7
Time: 1.3min
Precision: binary64
Cost: 2752
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}\]
↓
\[\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}} \cdot \left|\frac{2 \cdot \ell}{Om}\right|\right) \cdot \left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}} \cdot \left|\frac{2 \cdot \ell}{Om}\right|\right)}}}\]
\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)}
↓
\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}} \cdot \left|\frac{2 \cdot \ell}{Om}\right|\right) \cdot \left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}} \cdot \left|\frac{2 \cdot \ell}{Om}\right|\right)}}}
(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
 (sqrt
  (+
   0.5
   (/
    0.5
    (sqrt
     (+
      1.0
      (*
       (*
        (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
        (fabs (/ (* 2.0 l) Om)))
       (*
        (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
        (fabs (/ (* 2.0 l) 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) {
	return sqrt(0.5 + (0.5 / sqrt(1.0 + ((sqrt(pow(sin(kx), 2.0) + pow(sin(ky), 2.0)) * fabs((2.0 * l) / Om)) * (sqrt(pow(sin(kx), 2.0) + pow(sin(ky), 2.0)) * fabs((2.0 * l) / Om))))));
}

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
Alternative 1
Accuracy0.7
Cost3328
\[\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}} \cdot \left|\frac{2 \cdot \ell}{Om}\right|\right) \cdot \left(\sqrt{\left|\frac{2 \cdot \ell}{Om}\right|} \cdot \left(\sqrt{{\sin kx}^{2} + {\sin ky}^{2}} \cdot \sqrt{\left|\frac{2 \cdot \ell}{Om}\right|}\right)\right)}}}\]

Derivation

  1. Initial program 1.0

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

  2. Simplified1.0

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

  4. Applied add-sqr-sqrt_binary64_1001.0

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

  5. Applied add-sqr-sqrt_binary64_1001.0

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

  6. Applied unswap-sqr_binary64_461.0

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

  7. Simplified1.0

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

  8. Simplified0.7

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

  9. Final simplification0.7

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

Reproduce

herbie shell --seed 2020322 
(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))))))))))