Average Error: 0.4 → 0.5
Time: 19.8s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{1}{\sqrt{\sqrt{k}}}}{\frac{\sqrt{\sqrt{k}}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\frac{1}{\sqrt{\sqrt{k}}}}{\frac{\sqrt{\sqrt{k}}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}
double f(double k, double n) {
        double r154324 = 1.0;
        double r154325 = k;
        double r154326 = sqrt(r154325);
        double r154327 = r154324 / r154326;
        double r154328 = 2.0;
        double r154329 = atan2(1.0, 0.0);
        double r154330 = r154328 * r154329;
        double r154331 = n;
        double r154332 = r154330 * r154331;
        double r154333 = r154324 - r154325;
        double r154334 = r154333 / r154328;
        double r154335 = pow(r154332, r154334);
        double r154336 = r154327 * r154335;
        return r154336;
}
↓
double f(double k, double n) {
        double r154337 = 1.0;
        double r154338 = k;
        double r154339 = sqrt(r154338);
        double r154340 = sqrt(r154339);
        double r154341 = r154337 / r154340;
        double r154342 = 2.0;
        double r154343 = atan2(1.0, 0.0);
        double r154344 = r154342 * r154343;
        double r154345 = n;
        double r154346 = r154344 * r154345;
        double r154347 = r154337 - r154338;
        double r154348 = r154347 / r154342;
        double r154349 = pow(r154346, r154348);
        double r154350 = r154340 / r154349;
        double r154351 = r154341 / r154350;
        return r154351;
}

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. Using strategy rm

  3. Applied associate-*l/0.3

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

  5. Applied clear-num0.4

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

  7. Applied add-sqr-sqrt0.4

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

  8. Applied sqrt-prod0.5

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

  9. Applied times-frac0.5

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

  10. Applied associate-/r*0.5

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

  11. Simplified0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))