Average Error: 0.4 → 0.4
Time: 21.2s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1 \cdot \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)}{\sqrt{k}}\]
\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(\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)}{\sqrt{k}}
double f(double k, double n) {
        double r148208 = 1.0;
        double r148209 = k;
        double r148210 = sqrt(r148209);
        double r148211 = r148208 / r148210;
        double r148212 = 2.0;
        double r148213 = atan2(1.0, 0.0);
        double r148214 = r148212 * r148213;
        double r148215 = n;
        double r148216 = r148214 * r148215;
        double r148217 = r148208 - r148209;
        double r148218 = r148217 / r148212;
        double r148219 = pow(r148216, r148218);
        double r148220 = r148211 * r148219;
        return r148220;
}
↓
double f(double k, double n) {
        double r148221 = 1.0;
        double r148222 = 2.0;
        double r148223 = atan2(1.0, 0.0);
        double r148224 = r148222 * r148223;
        double r148225 = n;
        double r148226 = r148224 * r148225;
        double r148227 = k;
        double r148228 = r148221 - r148227;
        double r148229 = r148228 / r148222;
        double r148230 = 2.0;
        double r148231 = r148229 / r148230;
        double r148232 = pow(r148226, r148231);
        double r148233 = r148232 * r148232;
        double r148234 = r148221 * r148233;
        double r148235 = sqrt(r148227);
        double r148236 = r148234 / r148235;
        return r148236;
}

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 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 sqr-pow0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))