Average Error: 0.4 → 0.4
Time: 1.0m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\frac{\sqrt{k}}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\]
\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(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}
double f(double k, double n) {
        double r2730553 = 1.0;
        double r2730554 = k;
        double r2730555 = sqrt(r2730554);
        double r2730556 = r2730553 / r2730555;
        double r2730557 = 2.0;
        double r2730558 = atan2(1.0, 0.0);
        double r2730559 = r2730557 * r2730558;
        double r2730560 = n;
        double r2730561 = r2730559 * r2730560;
        double r2730562 = r2730553 - r2730554;
        double r2730563 = r2730562 / r2730557;
        double r2730564 = pow(r2730561, r2730563);
        double r2730565 = r2730556 * r2730564;
        return r2730565;
}
↓
double f(double k, double n) {
        double r2730566 = 1.0;
        double r2730567 = k;
        double r2730568 = sqrt(r2730567);
        double r2730569 = n;
        double r2730570 = 2.0;
        double r2730571 = atan2(1.0, 0.0);
        double r2730572 = r2730570 * r2730571;
        double r2730573 = r2730569 * r2730572;
        double r2730574 = 0.5;
        double r2730575 = r2730567 / r2730570;
        double r2730576 = r2730574 - r2730575;
        double r2730577 = pow(r2730573, r2730576);
        double r2730578 = r2730568 / r2730577;
        double r2730579 = r2730566 / r2730578;
        return r2730579;
}

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. Simplified0.3

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

  4. Applied *-un-lft-identity0.3

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

  5. Applied associate-/l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))