Average Error: 0.4 → 0.4
Time: 30.1s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\frac{\sqrt{k}}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1}{\frac{\sqrt{k}}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)}}}
double f(double k, double n) {
        double r3125175 = 1.0;
        double r3125176 = k;
        double r3125177 = sqrt(r3125176);
        double r3125178 = r3125175 / r3125177;
        double r3125179 = 2.0;
        double r3125180 = atan2(1.0, 0.0);
        double r3125181 = r3125179 * r3125180;
        double r3125182 = n;
        double r3125183 = r3125181 * r3125182;
        double r3125184 = r3125175 - r3125176;
        double r3125185 = r3125184 / r3125179;
        double r3125186 = pow(r3125183, r3125185);
        double r3125187 = r3125178 * r3125186;
        return r3125187;
}
↓
double f(double k, double n) {
        double r3125188 = 1.0;
        double r3125189 = k;
        double r3125190 = sqrt(r3125189);
        double r3125191 = n;
        double r3125192 = 2.0;
        double r3125193 = atan2(1.0, 0.0);
        double r3125194 = r3125192 * r3125193;
        double r3125195 = r3125191 * r3125194;
        double r3125196 = r3125188 - r3125189;
        double r3125197 = r3125196 / r3125192;
        double r3125198 = pow(r3125195, r3125197);
        double r3125199 = r3125190 / r3125198;
        double r3125200 = r3125188 / r3125199;
        return r3125200;
}

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. Simplified0.4

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

  4. Applied *-un-lft-identity0.4

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

  5. Applied associate-/l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))