Average Error: 1.0 → 0.8
Time: 5.6s
Precision: binary64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\sqrt[3]{\frac{2 \cdot \ell}{Om}} \cdot \sqrt[3]{\frac{2 \cdot \ell}{Om}}\right)}^{2} \cdot \left({\left(\sqrt[3]{\frac{2 \cdot \ell}{Om}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)}}\right)}\]
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}
↓
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\sqrt[3]{\frac{2 \cdot \ell}{Om}} \cdot \sqrt[3]{\frac{2 \cdot \ell}{Om}}\right)}^{2} \cdot \left({\left(\sqrt[3]{\frac{2 \cdot \ell}{Om}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)}}\right)}
double code(double l, double Om, double kx, double ky) {
	return ((double) sqrt(((double) (((double) (1.0 / 2.0)) * ((double) (1.0 + ((double) (1.0 / ((double) sqrt(((double) (1.0 + ((double) (((double) pow(((double) (((double) (2.0 * l)) / Om)), 2.0)) * ((double) (((double) pow(((double) sin(kx)), 2.0)) + ((double) pow(((double) sin(ky)), 2.0))))))))))))))))));
}
↓
double code(double l, double Om, double kx, double ky) {
	return ((double) sqrt(((double) (((double) (1.0 / 2.0)) * ((double) (1.0 + ((double) (1.0 / ((double) sqrt(((double) (1.0 + ((double) (((double) pow(((double) (((double) cbrt(((double) (((double) (2.0 * l)) / Om)))) * ((double) cbrt(((double) (((double) (2.0 * l)) / Om)))))), 2.0)) * ((double) (((double) pow(((double) cbrt(((double) (((double) (2.0 * l)) / Om)))), 2.0)) * ((double) (((double) pow(((double) sin(kx)), 2.0)) + ((double) pow(((double) 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 1.0

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

  3. Applied add-cube-cbrt1.0

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

  4. Applied unpow-prod-down1.0

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

  5. Applied associate-*l*0.8

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

  6. Final simplification0.8

    \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\sqrt[3]{\frac{2 \cdot \ell}{Om}} \cdot \sqrt[3]{\frac{2 \cdot \ell}{Om}}\right)}^{2} \cdot \left({\left(\sqrt[3]{\frac{2 \cdot \ell}{Om}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)\right)}}\right)}\]

Reproduce

herbie shell --seed 2020150 
(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))))))))))