Average Error: 0.5 → 0.4
Time: 10.8s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[1 \cdot \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{k} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
1 \cdot \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)}}
double f(double k, double n) {
        double r155300 = 1.0;
        double r155301 = k;
        double r155302 = sqrt(r155301);
        double r155303 = r155300 / r155302;
        double r155304 = 2.0;
        double r155305 = atan2(1.0, 0.0);
        double r155306 = r155304 * r155305;
        double r155307 = n;
        double r155308 = r155306 * r155307;
        double r155309 = r155300 - r155301;
        double r155310 = r155309 / r155304;
        double r155311 = pow(r155308, r155310);
        double r155312 = r155303 * r155311;
        return r155312;
}
↓
double f(double k, double n) {
        double r155313 = 1.0;
        double r155314 = 2.0;
        double r155315 = atan2(1.0, 0.0);
        double r155316 = r155314 * r155315;
        double r155317 = n;
        double r155318 = r155316 * r155317;
        double r155319 = r155313 / r155314;
        double r155320 = pow(r155318, r155319);
        double r155321 = k;
        double r155322 = sqrt(r155321);
        double r155323 = r155321 / r155314;
        double r155324 = pow(r155318, r155323);
        double r155325 = r155322 * r155324;
        double r155326 = r155320 / r155325;
        double r155327 = r155313 * r155326;
        return r155327;
}

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-inv0.5

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

  4. Applied associate-*l*0.5

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

  5. Simplified0.5

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

  7. Applied div-sub0.5

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

  8. Applied pow-sub0.4

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

  9. Applied associate-/l/0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020059 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))