Average Error: 0.4 → 0.5
Time: 10.0s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{k}} \cdot \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)}\right)\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\frac{1}{\sqrt{k}} \cdot \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{\frac{1 - k}{2}}{2}}{2}\right)}\right)\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}
double f(double k, double n) {
        double r141575 = 1.0;
        double r141576 = k;
        double r141577 = sqrt(r141576);
        double r141578 = r141575 / r141577;
        double r141579 = 2.0;
        double r141580 = atan2(1.0, 0.0);
        double r141581 = r141579 * r141580;
        double r141582 = n;
        double r141583 = r141581 * r141582;
        double r141584 = r141575 - r141576;
        double r141585 = r141584 / r141579;
        double r141586 = pow(r141583, r141585);
        double r141587 = r141578 * r141586;
        return r141587;
}
↓
double f(double k, double n) {
        double r141588 = 1.0;
        double r141589 = k;
        double r141590 = sqrt(r141589);
        double r141591 = r141588 / r141590;
        double r141592 = 2.0;
        double r141593 = atan2(1.0, 0.0);
        double r141594 = r141592 * r141593;
        double r141595 = n;
        double r141596 = r141594 * r141595;
        double r141597 = r141588 - r141589;
        double r141598 = r141597 / r141592;
        double r141599 = 2.0;
        double r141600 = r141598 / r141599;
        double r141601 = r141600 / r141599;
        double r141602 = pow(r141596, r141601);
        double r141603 = r141602 * r141602;
        double r141604 = r141591 * r141603;
        double r141605 = pow(r141596, r141600);
        double r141606 = r141604 * r141605;
        return r141606;
}

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 sqr-pow0.4

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

  4. Applied associate-*r*0.5

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

  6. Applied sqr-pow0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))