Average Error: 0.4 → 0.5
Time: 23.9s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{k}} \cdot \left(\left({2}^{\left(0.5 \cdot \left(1 - k\right)\right)} \cdot {\pi}^{\left(0.5 \cdot \left(1 - k\right)\right)}\right) \cdot {\left(\frac{1}{n}\right)}^{\left(-0.5 \cdot \left(1 - k\right)\right)}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1}{\sqrt{k}} \cdot \left(\left({2}^{\left(0.5 \cdot \left(1 - k\right)\right)} \cdot {\pi}^{\left(0.5 \cdot \left(1 - k\right)\right)}\right) \cdot {\left(\frac{1}{n}\right)}^{\left(-0.5 \cdot \left(1 - k\right)\right)}\right)
double f(double k, double n) {
        double r79400 = 1.0;
        double r79401 = k;
        double r79402 = sqrt(r79401);
        double r79403 = r79400 / r79402;
        double r79404 = 2.0;
        double r79405 = atan2(1.0, 0.0);
        double r79406 = r79404 * r79405;
        double r79407 = n;
        double r79408 = r79406 * r79407;
        double r79409 = r79400 - r79401;
        double r79410 = r79409 / r79404;
        double r79411 = pow(r79408, r79410);
        double r79412 = r79403 * r79411;
        return r79412;
}
↓
double f(double k, double n) {
        double r79413 = 1.0;
        double r79414 = k;
        double r79415 = sqrt(r79414);
        double r79416 = r79413 / r79415;
        double r79417 = 2.0;
        double r79418 = 0.5;
        double r79419 = r79413 - r79414;
        double r79420 = r79418 * r79419;
        double r79421 = pow(r79417, r79420);
        double r79422 = atan2(1.0, 0.0);
        double r79423 = pow(r79422, r79420);
        double r79424 = r79421 * r79423;
        double r79425 = 1.0;
        double r79426 = n;
        double r79427 = r79425 / r79426;
        double r79428 = -0.5;
        double r79429 = r79428 * r79419;
        double r79430 = pow(r79427, r79429);
        double r79431 = r79424 * r79430;
        double r79432 = r79416 * r79431;
        return r79432;
}

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 unpow-prod-down0.6

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

  5. Applied unpow-prod-down0.5

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

  6. Taylor expanded around inf 17.5

    \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\left(e^{0.5 \cdot \left(\log 2 \cdot \left(1 - k\right)\right)} \cdot \left(e^{0.5 \cdot \left(\left(1 - k\right) \cdot \log \pi\right)} \cdot e^{-0.5 \cdot \left(\left(1 - k\right) \cdot \log \left(\frac{1}{n}\right)\right)}\right)\right)}\]

  7. Simplified0.5

    \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\left(\left({2}^{\left(0.5 \cdot \left(1 - k\right)\right)} \cdot {\pi}^{\left(0.5 \cdot \left(1 - k\right)\right)}\right) \cdot {\left(\frac{1}{n}\right)}^{\left(-0.5 \cdot \left(1 - k\right)\right)}\right)}\]

  8. Final simplification0.5

    \[\leadsto \frac{1}{\sqrt{k}} \cdot \left(\left({2}^{\left(0.5 \cdot \left(1 - k\right)\right)} \cdot {\pi}^{\left(0.5 \cdot \left(1 - k\right)\right)}\right) \cdot {\left(\frac{1}{n}\right)}^{\left(-0.5 \cdot \left(1 - k\right)\right)}\right)\]

Reproduce

herbie shell --seed 2019305 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))