Average Error: 0.4 → 0.4
Time: 26.0s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{\sqrt{\sqrt{k}}}}{\sqrt{\sqrt{k}}}} \cdot \left(\sqrt{\frac{1}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{\frac{\frac{1}{\sqrt{\sqrt{k}}}}{\sqrt{\sqrt{k}}}} \cdot \left(\sqrt{\frac{1}{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)
double f(double k, double n) {
        double r63441 = 1.0;
        double r63442 = k;
        double r63443 = sqrt(r63442);
        double r63444 = r63441 / r63443;
        double r63445 = 2.0;
        double r63446 = atan2(1.0, 0.0);
        double r63447 = r63445 * r63446;
        double r63448 = n;
        double r63449 = r63447 * r63448;
        double r63450 = r63441 - r63442;
        double r63451 = r63450 / r63445;
        double r63452 = pow(r63449, r63451);
        double r63453 = r63444 * r63452;
        return r63453;
}
↓
double f(double k, double n) {
        double r63454 = 1.0;
        double r63455 = k;
        double r63456 = sqrt(r63455);
        double r63457 = sqrt(r63456);
        double r63458 = r63454 / r63457;
        double r63459 = r63458 / r63457;
        double r63460 = sqrt(r63459);
        double r63461 = r63454 / r63456;
        double r63462 = sqrt(r63461);
        double r63463 = 2.0;
        double r63464 = atan2(1.0, 0.0);
        double r63465 = r63463 * r63464;
        double r63466 = n;
        double r63467 = r63465 * r63466;
        double r63468 = r63454 - r63455;
        double r63469 = r63468 / r63463;
        double r63470 = pow(r63467, r63469);
        double r63471 = r63462 * r63470;
        double r63472 = r63460 * r63471;
        return r63472;
}

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

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

  4. Applied associate-*l*0.5

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

  6. Applied add-sqr-sqrt0.5

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

  7. Applied sqrt-prod0.5

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

  8. Applied associate-/r*0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019304 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))