Average Error: 0.6 → 0.4
Time: 11.2s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{k} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{k} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}
double f(double k, double n) {
        double r142312 = 1.0;
        double r142313 = k;
        double r142314 = sqrt(r142313);
        double r142315 = r142312 / r142314;
        double r142316 = 2.0;
        double r142317 = atan2(1.0, 0.0);
        double r142318 = r142316 * r142317;
        double r142319 = n;
        double r142320 = r142318 * r142319;
        double r142321 = r142312 - r142313;
        double r142322 = r142321 / r142316;
        double r142323 = pow(r142320, r142322);
        double r142324 = r142315 * r142323;
        return r142324;
}
↓
double f(double k, double n) {
        double r142325 = 1.0;
        double r142326 = 2.0;
        double r142327 = atan2(1.0, 0.0);
        double r142328 = r142326 * r142327;
        double r142329 = n;
        double r142330 = r142328 * r142329;
        double r142331 = r142325 / r142326;
        double r142332 = pow(r142330, r142331);
        double r142333 = r142325 * r142332;
        double r142334 = k;
        double r142335 = sqrt(r142334);
        double r142336 = r142334 / r142326;
        double r142337 = pow(r142330, r142336);
        double r142338 = r142335 * r142337;
        double r142339 = r142333 / r142338;
        return r142339;
}

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 0.6

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

  3. Applied div-sub0.6

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

  4. Applied pow-sub0.5

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

  5. Applied frac-times0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2020064 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))