Average Error: 0.5 → 0.5
Time: 5.0s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}}{{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}}\right)}\right)}^{\left(\frac{k}{\sqrt[3]{2}}\right)} \cdot \sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}}{{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}}\right)}\right)}^{\left(\frac{k}{\sqrt[3]{2}}\right)} \cdot \sqrt{k}}
(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 PI) n) 0.5)
  (*
   (pow
    (pow (* (* 2.0 PI) n) (/ 1.0 (* (cbrt 2.0) (cbrt 2.0))))
    (/ k (cbrt 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 * ((double) M_PI)) * n), 0.5) / (pow(pow(((2.0 * ((double) M_PI)) * n), (1.0 / (cbrt(2.0) * cbrt(2.0)))), (k / cbrt(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

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. Simplified0.5

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

  4. Applied div-sub_binary640.5

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

  5. Applied pow-sub_binary640.4

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

  6. Applied associate-/l/_binary640.4

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

  7. Simplified0.4

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

  9. Applied add-cube-cbrt_binary640.5

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

  10. Applied *-un-lft-identity_binary640.5

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

  11. Applied times-frac_binary640.5

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

  12. Applied pow-unpow_binary640.5

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

  13. Final simplification0.5

    \[\leadsto \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}}{{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{2}}\right)}\right)}^{\left(\frac{k}{\sqrt[3]{2}}\right)} \cdot \sqrt{k}}\]

Reproduce

herbie shell --seed 2020224 
(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))))