Average Error: 0.3 → 0.4
Time: 32.0s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot 1}{\sqrt{k}}} \cdot \sqrt{\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot 1}{\sqrt{k}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot 1}{\sqrt{k}}} \cdot \sqrt{\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot 1}{\sqrt{k}}}
double f(double k, double n) {
        double r3618322 = 1.0;
        double r3618323 = k;
        double r3618324 = sqrt(r3618323);
        double r3618325 = r3618322 / r3618324;
        double r3618326 = 2.0;
        double r3618327 = atan2(1.0, 0.0);
        double r3618328 = r3618326 * r3618327;
        double r3618329 = n;
        double r3618330 = r3618328 * r3618329;
        double r3618331 = r3618322 - r3618323;
        double r3618332 = r3618331 / r3618326;
        double r3618333 = pow(r3618330, r3618332);
        double r3618334 = r3618325 * r3618333;
        return r3618334;
}
↓
double f(double k, double n) {
        double r3618335 = atan2(1.0, 0.0);
        double r3618336 = 2.0;
        double r3618337 = r3618335 * r3618336;
        double r3618338 = n;
        double r3618339 = r3618337 * r3618338;
        double r3618340 = 1.0;
        double r3618341 = k;
        double r3618342 = r3618340 - r3618341;
        double r3618343 = r3618342 / r3618336;
        double r3618344 = pow(r3618339, r3618343);
        double r3618345 = r3618344 * r3618340;
        double r3618346 = sqrt(r3618341);
        double r3618347 = r3618345 / r3618346;
        double r3618348 = sqrt(r3618347);
        double r3618349 = r3618348 * r3618348;
        return r3618349;
}

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 0.3

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

  3. Applied associate-*l/0.3

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

  5. Applied add-sqr-sqrt0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))