Average Error: 0.4 → 0.4
Time: 38.8s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left({\left(n \cdot 2\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)} \cdot \sqrt{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}\right) \cdot \frac{\sqrt{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left({\left(n \cdot 2\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)} \cdot \sqrt{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}\right) \cdot \frac{\sqrt{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}}
double f(double k, double n) {
        double r3459222 = 1.0;
        double r3459223 = k;
        double r3459224 = sqrt(r3459223);
        double r3459225 = r3459222 / r3459224;
        double r3459226 = 2.0;
        double r3459227 = atan2(1.0, 0.0);
        double r3459228 = r3459226 * r3459227;
        double r3459229 = n;
        double r3459230 = r3459228 * r3459229;
        double r3459231 = r3459222 - r3459223;
        double r3459232 = r3459231 / r3459226;
        double r3459233 = pow(r3459230, r3459232);
        double r3459234 = r3459225 * r3459233;
        return r3459234;
}
↓
double f(double k, double n) {
        double r3459235 = n;
        double r3459236 = 2.0;
        double r3459237 = r3459235 * r3459236;
        double r3459238 = 0.5;
        double r3459239 = k;
        double r3459240 = r3459239 / r3459236;
        double r3459241 = r3459238 - r3459240;
        double r3459242 = pow(r3459237, r3459241);
        double r3459243 = atan2(1.0, 0.0);
        double r3459244 = pow(r3459243, r3459241);
        double r3459245 = sqrt(r3459244);
        double r3459246 = r3459242 * r3459245;
        double r3459247 = sqrt(r3459239);
        double r3459248 = r3459245 / r3459247;
        double r3459249 = r3459246 * r3459248;
        return r3459249;
}

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 0.4

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

  2. Simplified0.4

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

  4. Applied *-un-lft-identity0.4

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

  5. Applied sqrt-prod0.4

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

  6. Applied unpow-prod-down0.5

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

  7. Applied times-frac0.5

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

  8. Simplified0.5

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

  10. Applied *-un-lft-identity0.5

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

  11. Applied sqrt-prod0.5

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

  12. Applied add-sqr-sqrt0.4

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

  13. Applied times-frac0.5

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

  14. Applied associate-*r*0.4

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2019155 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))