Average Error: 0.4 → 1.0
Time: 2.3m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\frac{\sqrt{k}}{{\left({\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 + \sqrt{k}\right)}\right)}^{\left(\frac{1 - \sqrt{k}}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1}{\frac{\sqrt{k}}{{\left({\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 + \sqrt{k}\right)}\right)}^{\left(\frac{1 - \sqrt{k}}{2}\right)}}}
double f(double k, double n) {
        double r25674634 = 1.0;
        double r25674635 = k;
        double r25674636 = sqrt(r25674635);
        double r25674637 = r25674634 / r25674636;
        double r25674638 = 2.0;
        double r25674639 = atan2(1.0, 0.0);
        double r25674640 = r25674638 * r25674639;
        double r25674641 = n;
        double r25674642 = r25674640 * r25674641;
        double r25674643 = r25674634 - r25674635;
        double r25674644 = r25674643 / r25674638;
        double r25674645 = pow(r25674642, r25674644);
        double r25674646 = r25674637 * r25674645;
        return r25674646;
}
↓
double f(double k, double n) {
        double r25674647 = 1.0;
        double r25674648 = k;
        double r25674649 = sqrt(r25674648);
        double r25674650 = n;
        double r25674651 = 2.0;
        double r25674652 = atan2(1.0, 0.0);
        double r25674653 = r25674651 * r25674652;
        double r25674654 = r25674650 * r25674653;
        double r25674655 = r25674647 + r25674649;
        double r25674656 = pow(r25674654, r25674655);
        double r25674657 = r25674647 - r25674649;
        double r25674658 = r25674657 / r25674651;
        double r25674659 = pow(r25674656, r25674658);
        double r25674660 = r25674649 / r25674659;
        double r25674661 = r25674647 / r25674660;
        return r25674661;
}

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. Simplified0.4

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

  4. Applied *-un-lft-identity0.4

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

  5. Applied add-sqr-sqrt0.4

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

  6. Applied *-un-lft-identity0.4

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

  7. Applied difference-of-squares0.5

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

  8. Applied times-frac0.5

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

  9. Applied pow-unpow0.9

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

  11. Applied *-un-lft-identity0.9

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

  12. Applied associate-/l*1.0

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

  13. Final simplification1.0

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

Reproduce

herbie shell --seed 2019107 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))