Average Error: 1.0 → 0.7
Time: 38.0s
Precision: binary64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}\]
↓
\[\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{2}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}} \cdot \left(\left(2 \cdot \frac{\ell \cdot \left({\sin ky}^{2} + {\sin kx}^{2}\right)}{Om}\right) \cdot \frac{\ell}{\sqrt[3]{Om}}\right)}}}\]
\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 + \frac{2}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}} \cdot \left(\left(2 \cdot \frac{\ell \cdot \left({\sin ky}^{2} + {\sin kx}^{2}\right)}{Om}\right) \cdot \frac{\ell}{\sqrt[3]{Om}}\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
      (*
       (/ 2.0 (* (cbrt Om) (cbrt Om)))
       (*
        (* 2.0 (/ (* l (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))) Om))
        (/ l (cbrt 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 + ((2.0 / (cbrt(Om) * cbrt(Om))) * ((2.0 * ((l * (pow(sin(ky), 2.0) + pow(sin(kx), 2.0))) / Om)) * (l / cbrt(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

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 unpow2_binary64_1431.0

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

  5. Applied associate-*l*_binary64_190.7

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

  6. Simplified0.7

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

  8. Applied add-cube-cbrt_binary64_1130.7

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

  9. Applied times-frac_binary64_840.7

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

  10. Applied associate-*l*_binary64_190.7

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

  11. Simplified0.7

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

  12. Final simplification0.7

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

Reproduce

herbie shell --seed 2021075 
(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))))))))))