Average Error: 0.4 → 0.6
Time: 8.7s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{\sqrt{1}}{\sqrt{\sqrt{k}}} \cdot \sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}\right) \cdot \left(\frac{\sqrt{1}}{\sqrt{\sqrt{k}}} \cdot \sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\frac{\sqrt{1}}{\sqrt{\sqrt{k}}} \cdot \sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}\right) \cdot \left(\frac{\sqrt{1}}{\sqrt{\sqrt{k}}} \cdot \sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}\right)
double f(double k, double n) {
        double r116341 = 1.0;
        double r116342 = k;
        double r116343 = sqrt(r116342);
        double r116344 = r116341 / r116343;
        double r116345 = 2.0;
        double r116346 = atan2(1.0, 0.0);
        double r116347 = r116345 * r116346;
        double r116348 = n;
        double r116349 = r116347 * r116348;
        double r116350 = r116341 - r116342;
        double r116351 = r116350 / r116345;
        double r116352 = pow(r116349, r116351);
        double r116353 = r116344 * r116352;
        return r116353;
}
↓
double f(double k, double n) {
        double r116354 = 1.0;
        double r116355 = sqrt(r116354);
        double r116356 = k;
        double r116357 = sqrt(r116356);
        double r116358 = sqrt(r116357);
        double r116359 = r116355 / r116358;
        double r116360 = 2.0;
        double r116361 = atan2(1.0, 0.0);
        double r116362 = r116360 * r116361;
        double r116363 = n;
        double r116364 = r116362 * r116363;
        double r116365 = r116354 - r116356;
        double r116366 = r116365 / r116360;
        double r116367 = pow(r116364, r116366);
        double r116368 = sqrt(r116367);
        double r116369 = r116359 * r116368;
        double r116370 = r116369 * r116369;
        return r116370;
}

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 add-sqr-sqrt0.5

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied sqrt-prod0.6

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

  6. Applied add-sqr-sqrt0.6

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

  7. Applied times-frac0.6

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

  8. Applied unswap-sqr0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019356 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))