Average Error: 0.4 → 0.5
Time: 27.4s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left({\left(n \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {2}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \frac{1}{\sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left({\left(n \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {2}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \frac{1}{\sqrt{k}}
double f(double k, double n) {
        double r4628070 = 1.0;
        double r4628071 = k;
        double r4628072 = sqrt(r4628071);
        double r4628073 = r4628070 / r4628072;
        double r4628074 = 2.0;
        double r4628075 = atan2(1.0, 0.0);
        double r4628076 = r4628074 * r4628075;
        double r4628077 = n;
        double r4628078 = r4628076 * r4628077;
        double r4628079 = r4628070 - r4628071;
        double r4628080 = r4628079 / r4628074;
        double r4628081 = pow(r4628078, r4628080);
        double r4628082 = r4628073 * r4628081;
        return r4628082;
}
↓
double f(double k, double n) {
        double r4628083 = n;
        double r4628084 = atan2(1.0, 0.0);
        double r4628085 = r4628083 * r4628084;
        double r4628086 = 1.0;
        double r4628087 = k;
        double r4628088 = r4628086 - r4628087;
        double r4628089 = 2.0;
        double r4628090 = r4628088 / r4628089;
        double r4628091 = pow(r4628085, r4628090);
        double r4628092 = pow(r4628089, r4628090);
        double r4628093 = r4628091 * r4628092;
        double r4628094 = sqrt(r4628087);
        double r4628095 = r4628086 / r4628094;
        double r4628096 = r4628093 * r4628095;
        return r4628096;
}

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. Taylor expanded around 0 0.4

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied associate-*r*0.5

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

  7. Applied unpow-prod-down0.4

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

  8. Simplified0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019172 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))