Average Error: 0.4 → 0.5
Time: 10.1s
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 r141302 = 1.0;
        double r141303 = k;
        double r141304 = sqrt(r141303);
        double r141305 = r141302 / r141304;
        double r141306 = 2.0;
        double r141307 = atan2(1.0, 0.0);
        double r141308 = r141306 * r141307;
        double r141309 = n;
        double r141310 = r141308 * r141309;
        double r141311 = r141302 - r141303;
        double r141312 = r141311 / r141306;
        double r141313 = pow(r141310, r141312);
        double r141314 = r141305 * r141313;
        return r141314;
}
↓
double f(double k, double n) {
        double r141315 = 1.0;
        double r141316 = k;
        double r141317 = sqrt(r141316);
        double r141318 = r141315 / r141317;
        double r141319 = 2.0;
        double r141320 = atan2(1.0, 0.0);
        double r141321 = r141319 * r141320;
        double r141322 = n;
        double r141323 = r141321 * r141322;
        double r141324 = r141315 - r141316;
        double r141325 = r141324 / r141319;
        double r141326 = 2.0;
        double r141327 = r141325 / r141326;
        double r141328 = r141327 / r141326;
        double r141329 = pow(r141323, r141328);
        double r141330 = r141329 * r141329;
        double r141331 = r141318 * r141330;
        double r141332 = pow(r141323, r141327);
        double r141333 = r141331 * r141332;
        return r141333;
}

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))))