Average Error: 0.4 → 0.4
Time: 18.0s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(2 \cdot \frac{\frac{1 - k}{2}}{2}\right)} \cdot \frac{1}{\sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(2 \cdot \frac{\frac{1 - k}{2}}{2}\right)} \cdot \frac{1}{\sqrt{k}}
double f(double k, double n) {
        double r86310 = 1.0;
        double r86311 = k;
        double r86312 = sqrt(r86311);
        double r86313 = r86310 / r86312;
        double r86314 = 2.0;
        double r86315 = atan2(1.0, 0.0);
        double r86316 = r86314 * r86315;
        double r86317 = n;
        double r86318 = r86316 * r86317;
        double r86319 = r86310 - r86311;
        double r86320 = r86319 / r86314;
        double r86321 = pow(r86318, r86320);
        double r86322 = r86313 * r86321;
        return r86322;
}
↓
double f(double k, double n) {
        double r86323 = 2.0;
        double r86324 = atan2(1.0, 0.0);
        double r86325 = r86323 * r86324;
        double r86326 = n;
        double r86327 = r86325 * r86326;
        double r86328 = 2.0;
        double r86329 = 1.0;
        double r86330 = k;
        double r86331 = r86329 - r86330;
        double r86332 = r86331 / r86323;
        double r86333 = r86332 / r86328;
        double r86334 = r86328 * r86333;
        double r86335 = pow(r86327, r86334);
        double r86336 = sqrt(r86330);
        double r86337 = r86329 / r86336;
        double r86338 = r86335 * r86337;
        return r86338;
}

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 sqr-pow0.5

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019308 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))