Average Error: 0.4 → 0.6
Time: 16.0s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{1}{\sqrt{k}} \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\frac{1}{n}\right)}^{\left(-0.25 \cdot \left(1 - k\right)\right)}\right)\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\]
\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(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\frac{1}{n}\right)}^{\left(-0.25 \cdot \left(1 - k\right)\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 r304 = 1.0;
        double r305 = k;
        double r306 = sqrt(r305);
        double r307 = r304 / r306;
        double r308 = 2.0;
        double r309 = atan2(1.0, 0.0);
        double r310 = r308 * r309;
        double r311 = n;
        double r312 = r310 * r311;
        double r313 = r304 - r305;
        double r314 = r313 / r308;
        double r315 = pow(r312, r314);
        double r316 = r307 * r315;
        return r316;
}
↓
double f(double k, double n) {
        double r317 = 1.0;
        double r318 = k;
        double r319 = sqrt(r318);
        double r320 = r317 / r319;
        double r321 = 2.0;
        double r322 = atan2(1.0, 0.0);
        double r323 = r321 * r322;
        double r324 = r317 - r318;
        double r325 = r324 / r321;
        double r326 = 2.0;
        double r327 = r325 / r326;
        double r328 = pow(r323, r327);
        double r329 = 1.0;
        double r330 = n;
        double r331 = r329 / r330;
        double r332 = -0.25;
        double r333 = r332 * r324;
        double r334 = pow(r331, r333);
        double r335 = r328 * r334;
        double r336 = r320 * r335;
        double r337 = r323 * r330;
        double r338 = pow(r337, r327);
        double r339 = r336 * r338;
        return r339;
}

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-pow0.5

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

  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 unpow-prod-down0.6

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

  7. Taylor expanded around inf 17.2

    \[\leadsto \left(\frac{1}{\sqrt{k}} \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot \color{blue}{e^{-0.25 \cdot \left(\left(1 - k\right) \cdot \log \left(\frac{1}{n}\right)\right)}}\right)\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\]

  8. Simplified0.6

    \[\leadsto \left(\frac{1}{\sqrt{k}} \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot \color{blue}{{\left(\frac{1}{n}\right)}^{\left(-0.25 \cdot \left(1 - k\right)\right)}}\right)\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\]

  9. Final simplification0.6

    \[\leadsto \left(\frac{1}{\sqrt{k}} \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\frac{1}{n}\right)}^{\left(-0.25 \cdot \left(1 - k\right)\right)}\right)\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))