Average Error: 1.1 → 0.7
Time: 16.6s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{\sqrt{2} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{2} \cdot \sqrt{{\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}\right)}\right)}\]
\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}{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{\sqrt{2} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{2} \cdot \sqrt{{\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{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) hypot(((double) sqrt(1.0)), ((double) (((double) pow(((double) (((double) (((double) sqrt(2.0)) * ((double) cbrt(l)))) / ((double) cbrt(Om)))), 2.0)) * ((double) sqrt(((double) (((double) pow(((double) (((double) cbrt(l)) / ((double) cbrt(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 1.1

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

  3. Applied add-cube-cbrt1.1

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

  4. Applied add-cube-cbrt1.1

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

  5. Applied associate-*r*1.1

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

  6. Applied times-frac1.1

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

  7. Applied unpow-prod-down1.1

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

  8. Applied associate-*l*0.8

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

  10. Applied add-sqr-sqrt0.8

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

  11. Applied add-sqr-sqrt0.8

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

  12. Applied unswap-sqr0.8

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

  13. Applied times-frac0.8

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

  14. Applied unpow-prod-down0.8

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

  15. Applied unswap-sqr0.7

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

  16. Applied add-sqr-sqrt0.7

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

  17. Applied hypot-def0.7

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

  18. Final simplification0.7

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

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  :precision binary64
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))