Average Error: 0.5 → 0.4
Time: 34.3s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\left(-1\right) \cdot {\left(\sqrt{2} \cdot \sqrt{\pi}\right)}^{\left(\frac{1}{2}\right)}}{\left(-\sqrt{k}\right) \cdot {\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right)}^{\left(\frac{k}{2}\right)}} \cdot \frac{{\left(\left(\sqrt{2} \cdot \sqrt{\pi}\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}}{{\left(\left(\sqrt[3]{2} \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\left(-1\right) \cdot {\left(\sqrt{2} \cdot \sqrt{\pi}\right)}^{\left(\frac{1}{2}\right)}}{\left(-\sqrt{k}\right) \cdot {\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right)}^{\left(\frac{k}{2}\right)}} \cdot \frac{{\left(\left(\sqrt{2} \cdot \sqrt{\pi}\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}}{{\left(\left(\sqrt[3]{2} \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}
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 (((-1.0 * pow((sqrt(2.0) * sqrt(((double) M_PI))), (1.0 / 2.0))) / (-sqrt(k) * pow((cbrt(2.0) * cbrt(2.0)), (k / 2.0)))) * (pow(((sqrt(2.0) * sqrt(((double) M_PI))) * n), (1.0 / 2.0)) / pow(((cbrt(2.0) * ((double) M_PI)) * n), (k / 2.0))));
}

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. Using strategy rm

  3. Applied div-sub0.5

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

  4. Applied pow-sub0.4

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

  5. Applied frac-2neg0.4

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

  6. Applied frac-times0.4

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

  8. Applied add-cube-cbrt0.4

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

  9. Applied associate-*l*0.4

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

  10. Applied associate-*l*0.4

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

  11. Applied unpow-prod-down0.4

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

  12. Applied associate-*r*0.4

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

  13. Applied add-sqr-sqrt0.6

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

  14. Applied add-sqr-sqrt0.5

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

  15. Applied unswap-sqr0.5

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

  16. Applied associate-*l*0.5

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

  17. Applied unpow-prod-down0.4

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

  18. Applied associate-*r*0.4

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

  19. Applied times-frac0.4

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

  20. Final simplification0.4

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

Reproduce

herbie shell --seed 2020066 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))