Average Error: 0.4 → 0.5
Time: 8.3s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{\sqrt{1}}{\sqrt{\sqrt{k}}} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt{k}}}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\frac{\sqrt{1}}{\sqrt{\sqrt{k}}} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt{k}}}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r137412 = 1.0;
        double r137413 = k;
        double r137414 = sqrt(r137413);
        double r137415 = r137412 / r137414;
        double r137416 = 2.0;
        double r137417 = atan2(1.0, 0.0);
        double r137418 = r137416 * r137417;
        double r137419 = n;
        double r137420 = r137418 * r137419;
        double r137421 = r137412 - r137413;
        double r137422 = r137421 / r137416;
        double r137423 = pow(r137420, r137422);
        double r137424 = r137415 * r137423;
        return r137424;
}
↓
double f(double k, double n) {
        double r137425 = 1.0;
        double r137426 = sqrt(r137425);
        double r137427 = k;
        double r137428 = sqrt(r137427);
        double r137429 = sqrt(r137428);
        double r137430 = r137426 / r137429;
        double r137431 = r137430 * r137430;
        double r137432 = 2.0;
        double r137433 = atan2(1.0, 0.0);
        double r137434 = r137432 * r137433;
        double r137435 = n;
        double r137436 = r137434 * r137435;
        double r137437 = r137425 - r137427;
        double r137438 = r137437 / r137432;
        double r137439 = pow(r137436, r137438);
        double r137440 = r137431 * r137439;
        return r137440;
}

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-sqrt0.4

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

  4. Applied sqrt-prod0.5

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied times-frac0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2020049 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))