Average Error: 0.4 → 0.7
Time: 20.9s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{k}} \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {n}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)\right) \cdot \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\frac{1}{\sqrt{k}} \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {n}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)\right) \cdot \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)}\right)
double f(double k, double n) {
        double r330 = 1.0;
        double r331 = k;
        double r332 = sqrt(r331);
        double r333 = r330 / r332;
        double r334 = 2.0;
        double r335 = atan2(1.0, 0.0);
        double r336 = r334 * r335;
        double r337 = n;
        double r338 = r336 * r337;
        double r339 = r330 - r331;
        double r340 = r339 / r334;
        double r341 = pow(r338, r340);
        double r342 = r333 * r341;
        return r342;
}
↓
double f(double k, double n) {
        double r343 = 1.0;
        double r344 = k;
        double r345 = sqrt(r344);
        double r346 = r343 / r345;
        double r347 = 2.0;
        double r348 = atan2(1.0, 0.0);
        double r349 = r347 * r348;
        double r350 = r343 - r344;
        double r351 = r350 / r347;
        double r352 = 2.0;
        double r353 = r351 / r352;
        double r354 = pow(r349, r353);
        double r355 = n;
        double r356 = pow(r355, r353);
        double r357 = r354 * r356;
        double r358 = r346 * r357;
        double r359 = r349 * r355;
        double r360 = r353 / r352;
        double r361 = pow(r359, r360);
        double r362 = r361 * r361;
        double r363 = r358 * r362;
        return r363;
}

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 sqr-pow0.5

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

  4. Applied associate-*r*0.5

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

  6. Applied unpow-prod-down0.6

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

  8. Applied sqr-pow0.7

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

  9. Final simplification0.7

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))