Average Error: 0.4 → 0.4
Time: 2.8m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\frac{\sqrt{k}}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\]
\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(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}
double f(double k, double n) {
        double r3217551 = 1.0;
        double r3217552 = k;
        double r3217553 = sqrt(r3217552);
        double r3217554 = r3217551 / r3217553;
        double r3217555 = 2.0;
        double r3217556 = atan2(1.0, 0.0);
        double r3217557 = r3217555 * r3217556;
        double r3217558 = n;
        double r3217559 = r3217557 * r3217558;
        double r3217560 = r3217551 - r3217552;
        double r3217561 = r3217560 / r3217555;
        double r3217562 = pow(r3217559, r3217561);
        double r3217563 = r3217554 * r3217562;
        return r3217563;
}
↓
double f(double k, double n) {
        double r3217564 = 1.0;
        double r3217565 = k;
        double r3217566 = sqrt(r3217565);
        double r3217567 = atan2(1.0, 0.0);
        double r3217568 = n;
        double r3217569 = 2.0;
        double r3217570 = r3217568 * r3217569;
        double r3217571 = r3217567 * r3217570;
        double r3217572 = 0.5;
        double r3217573 = r3217565 / r3217569;
        double r3217574 = r3217572 - r3217573;
        double r3217575 = pow(r3217571, r3217574);
        double r3217576 = r3217566 / r3217575;
        double r3217577 = r3217564 / r3217576;
        return r3217577;
}

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

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

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

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

  5. Applied associate-/l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))