Average Error: 0.4 → 0.5
Time: 23.9s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\sqrt{\frac{\sqrt{1}}{\left|\sqrt[3]{k}\right|} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt[3]{k}}}} \cdot \sqrt{\frac{1}{\sqrt{k}}}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\sqrt{\frac{\sqrt{1}}{\left|\sqrt[3]{k}\right|} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt[3]{k}}}} \cdot \sqrt{\frac{1}{\sqrt{k}}}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r67304 = 1.0;
        double r67305 = k;
        double r67306 = sqrt(r67305);
        double r67307 = r67304 / r67306;
        double r67308 = 2.0;
        double r67309 = atan2(1.0, 0.0);
        double r67310 = r67308 * r67309;
        double r67311 = n;
        double r67312 = r67310 * r67311;
        double r67313 = r67304 - r67305;
        double r67314 = r67313 / r67308;
        double r67315 = pow(r67312, r67314);
        double r67316 = r67307 * r67315;
        return r67316;
}
↓
double f(double k, double n) {
        double r67317 = 1.0;
        double r67318 = sqrt(r67317);
        double r67319 = k;
        double r67320 = cbrt(r67319);
        double r67321 = fabs(r67320);
        double r67322 = r67318 / r67321;
        double r67323 = sqrt(r67320);
        double r67324 = r67318 / r67323;
        double r67325 = r67322 * r67324;
        double r67326 = sqrt(r67325);
        double r67327 = sqrt(r67319);
        double r67328 = r67317 / r67327;
        double r67329 = sqrt(r67328);
        double r67330 = r67326 * r67329;
        double r67331 = 2.0;
        double r67332 = atan2(1.0, 0.0);
        double r67333 = r67331 * r67332;
        double r67334 = n;
        double r67335 = r67333 * r67334;
        double r67336 = r67317 - r67319;
        double r67337 = r67336 / r67331;
        double r67338 = pow(r67335, r67337);
        double r67339 = r67330 * r67338;
        return r67339;
}

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 add-sqr-sqrt0.5

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

  5. Applied add-cube-cbrt0.5

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

  6. Applied sqrt-prod0.5

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

  7. Applied add-sqr-sqrt0.5

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

  8. Applied times-frac0.5

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

  9. Simplified0.5

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

  10. Final simplification0.5

    \[\leadsto \left(\sqrt{\frac{\sqrt{1}}{\left|\sqrt[3]{k}\right|} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt[3]{k}}}} \cdot \sqrt{\frac{1}{\sqrt{k}}}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))