Average Error: 0.4 → 0.4
Time: 30.4s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{k}} \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\frac{1}{n}\right)}^{\left(-0.5 \cdot \left(1 - k\right)\right)}\right)\right)\]
\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 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot \left({\left(2 \cdot \pi\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\frac{1}{n}\right)}^{\left(-0.5 \cdot \left(1 - k\right)\right)}\right)\right)
double f(double k, double n) {
        double r79235 = 1.0;
        double r79236 = k;
        double r79237 = sqrt(r79236);
        double r79238 = r79235 / r79237;
        double r79239 = 2.0;
        double r79240 = atan2(1.0, 0.0);
        double r79241 = r79239 * r79240;
        double r79242 = n;
        double r79243 = r79241 * r79242;
        double r79244 = r79235 - r79236;
        double r79245 = r79244 / r79239;
        double r79246 = pow(r79243, r79245);
        double r79247 = r79238 * r79246;
        return r79247;
}
↓
double f(double k, double n) {
        double r79248 = 1.0;
        double r79249 = k;
        double r79250 = sqrt(r79249);
        double r79251 = r79248 / r79250;
        double r79252 = 2.0;
        double r79253 = atan2(1.0, 0.0);
        double r79254 = r79252 * r79253;
        double r79255 = r79248 - r79249;
        double r79256 = r79255 / r79252;
        double r79257 = 2.0;
        double r79258 = r79256 / r79257;
        double r79259 = pow(r79254, r79258);
        double r79260 = 1.0;
        double r79261 = n;
        double r79262 = r79260 / r79261;
        double r79263 = -0.5;
        double r79264 = r79263 * r79255;
        double r79265 = pow(r79262, r79264);
        double r79266 = r79259 * r79265;
        double r79267 = r79259 * r79266;
        double r79268 = r79251 * r79267;
        return r79268;
}

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-down0.5

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

  5. Applied sqr-pow0.5

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

  6. Applied associate-*l*0.4

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

  7. Taylor expanded around inf 17.3

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019303 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))