Average Error: 0.5 → 0.5
Time: 2.6min
Precision: binary64
Cost: 1664
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[{2}^{\left(\frac{1 - k}{2}\right)} \cdot \left({\pi}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{{n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
{2}^{\left(\frac{1 - k}{2}\right)} \cdot \left({\pi}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{{n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)
(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))
↓
(FPCore (k n)
 :precision binary64
 (*
  (pow 2.0 (/ (- 1.0 k) 2.0))
  (* (pow PI (/ (- 1.0 k) 2.0)) (/ (pow n (/ (- 1.0 k) 2.0)) (sqrt k)))))
double code(double k, double n) {
	return (1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0));
}
↓
double code(double k, double n) {
	return pow(2.0, ((1.0 - k) / 2.0)) * (pow(((double) M_PI), ((1.0 - k) / 2.0)) * (pow(n, ((1.0 - k) / 2.0)) / sqrt(k)));
}

Error

Bits error versus k

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs
Alternative 1
Accuracy0.5
Cost1792
\[{2}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{{\pi}^{0.5} \cdot {n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k} \cdot {\pi}^{\left(\frac{k}{2}\right)}}\]
Alternative 2
Accuracy0.5
Cost1728
\[\frac{\sqrt{2} \cdot \frac{{\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}{{2}^{\left(\frac{k}{2}\right)}}\]
Alternative 3
Accuracy0.5
Cost1856
\[{2}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{\sqrt[3]{{\left({\pi}^{\left(\frac{1 - k}{2}\right)}\right)}^{3}} \cdot {n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]
Alternative 4
Accuracy0.5
Cost2304
\[{2}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{{\left({\pi}^{\left(\frac{1 + \sqrt{k}}{\sqrt{2}}\right)}\right)}^{\left(\frac{1 - \sqrt{k}}{\sqrt{2}}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]
Alternative 5
Accuracy0.5
Cost1664
\[{2}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{{\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]
Alternative 6
Accuracy0.5
Cost2048
\[{2}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{{\left({\pi}^{\left(1 + \sqrt{k}\right)}\right)}^{\left(\frac{1 - \sqrt{k}}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]
Alternative 7
Accuracy22.0
Cost4736
\[\frac{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n} \cdot \left(0.125 \cdot \left(\left(k \cdot k\right) \cdot {\log n}^{2}\right) + \left(\left(k \cdot k\right) \cdot 0.25\right) \cdot \left(\log n \cdot \log \left(2 \cdot \pi\right)\right)\right) + \sqrt{\left(2 \cdot \pi\right) \cdot n} \cdot \left(0.125 \cdot \left(\left(k \cdot k\right) \cdot {\log \left(2 \cdot \pi\right)}^{2}\right) + 1\right)\right) + \left(k \cdot -0.5\right) \cdot \left(\sqrt{\left(2 \cdot \pi\right) \cdot n} \cdot \log \left(\left(2 \cdot \pi\right) \cdot n\right)\right)}{\sqrt{k}}\]

Derivation

  1. Initial program 0.5

    \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}\]
  3. Using strategy rm

  4. Applied unpow-prod-down_binary64_4980.6

    \[\leadsto \frac{\color{blue}{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}}\]
  5. Using strategy rm

  6. Applied unpow-prod-down_binary64_4980.6

    \[\leadsto \frac{\color{blue}{\left({2}^{\left(\frac{1 - k}{2}\right)} \cdot {\pi}^{\left(\frac{1 - k}{2}\right)}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]

  7. Applied associate-*l*_binary64_3600.5

    \[\leadsto \frac{\color{blue}{{2}^{\left(\frac{1 - k}{2}\right)} \cdot \left({\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}\right)}}{\sqrt{k}}\]
  8. Using strategy rm

  9. Applied *-un-lft-identity_binary64_4190.5

    \[\leadsto \frac{{2}^{\left(\frac{1 - k}{2}\right)} \cdot \left({\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}\right)}{\sqrt{\color{blue}{1 \cdot k}}}\]

  10. Applied sqrt-prod_binary64_4350.5

    \[\leadsto \frac{{2}^{\left(\frac{1 - k}{2}\right)} \cdot \left({\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}\right)}{\color{blue}{\sqrt{1} \cdot \sqrt{k}}}\]

  11. Applied times-frac_binary64_4250.5

    \[\leadsto \color{blue}{\frac{{2}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{1}} \cdot \frac{{\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}\]

  12. Simplified0.5

    \[\leadsto \color{blue}{{2}^{\left(\frac{1 - k}{2}\right)}} \cdot \frac{{\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]
  13. Using strategy rm

  14. Applied *-un-lft-identity_binary64_4190.5

    \[\leadsto {2}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{{\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{\color{blue}{1 \cdot k}}}\]

  15. Applied sqrt-prod_binary64_4350.5

    \[\leadsto {2}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{{\pi}^{\left(\frac{1 - k}{2}\right)} \cdot {n}^{\left(\frac{1 - k}{2}\right)}}{\color{blue}{\sqrt{1} \cdot \sqrt{k}}}\]

  16. Applied times-frac_binary64_4250.5

    \[\leadsto {2}^{\left(\frac{1 - k}{2}\right)} \cdot \color{blue}{\left(\frac{{\pi}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{1}} \cdot \frac{{n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)}\]

  17. Simplified0.5

    \[\leadsto {2}^{\left(\frac{1 - k}{2}\right)} \cdot \left(\color{blue}{{\pi}^{\left(\frac{1 - k}{2}\right)}} \cdot \frac{{n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\]

  18. Final simplification0.5

    \[\leadsto {2}^{\left(\frac{1 - k}{2}\right)} \cdot \left({\pi}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{{n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\]

Reproduce

herbie shell --seed 2020322 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))