Average Error: 0.9 → 0.9
Time: 10.4s
Precision: binary64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}\]
↓
\[\sqrt{0.5 + \frac{0.5}{\left|\sqrt[3]{1 + \left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}}\right| \cdot \sqrt{\sqrt[3]{1 + \left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}}}}}\]
\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}{\left|\sqrt[3]{1 + \left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}}\right| \cdot \sqrt{\sqrt[3]{1 + \left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}}}}}
(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
    (*
     (fabs
      (cbrt
       (+
        1.0
        (*
         (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))
         (pow (/ (* 2.0 l) Om) 2.0)))))
     (sqrt
      (cbrt
       (+
        1.0
        (*
         (+ (pow (sin ky) 2.0) (pow (sin kx) 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) {
	return sqrt(0.5 + (0.5 / (fabs(cbrt(1.0 + ((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(((2.0 * l) / Om), 2.0)))) * sqrt(cbrt(1.0 + ((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(((2.0 * l) / Om), 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 0.9

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

  2. Simplified0.9

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

  4. Applied add-cube-cbrt_binary640.9

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

  5. Applied sqrt-prod_binary640.9

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

  6. Simplified0.9

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

  7. Simplified0.9

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

  8. Final simplification0.9

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

Alternatives

Reproduce

herbie shell --seed 2021110 
(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))))))))))