Average Error: 0.4 → 0.5
Time: 39.9s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{\frac{1}{\sqrt{\left|\sqrt[3]{k}\right|}}}{\sqrt{\sqrt{\sqrt[3]{k}}}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\frac{\frac{1}{\sqrt{\left|\sqrt[3]{k}\right|}}}{\sqrt{\sqrt{\sqrt[3]{k}}}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r156090 = 1.0;
        double r156091 = k;
        double r156092 = sqrt(r156091);
        double r156093 = r156090 / r156092;
        double r156094 = 2.0;
        double r156095 = atan2(1.0, 0.0);
        double r156096 = r156094 * r156095;
        double r156097 = n;
        double r156098 = r156096 * r156097;
        double r156099 = r156090 - r156091;
        double r156100 = r156099 / r156094;
        double r156101 = pow(r156098, r156100);
        double r156102 = r156093 * r156101;
        return r156102;
}
↓
double f(double k, double n) {
        double r156103 = 1.0;
        double r156104 = k;
        double r156105 = cbrt(r156104);
        double r156106 = fabs(r156105);
        double r156107 = sqrt(r156106);
        double r156108 = r156103 / r156107;
        double r156109 = sqrt(r156105);
        double r156110 = sqrt(r156109);
        double r156111 = r156108 / r156110;
        double r156112 = sqrt(r156104);
        double r156113 = sqrt(r156112);
        double r156114 = r156111 / r156113;
        double r156115 = 2.0;
        double r156116 = atan2(1.0, 0.0);
        double r156117 = r156115 * r156116;
        double r156118 = n;
        double r156119 = r156117 * r156118;
        double r156120 = r156103 - r156104;
        double r156121 = r156120 / r156115;
        double r156122 = pow(r156119, r156121);
        double r156123 = r156114 * r156122;
        return r156123;
}

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-sqrt0.4

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

  4. Applied sqrt-prod0.5

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

  5. Applied associate-/r*0.5

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

  7. Applied add-cube-cbrt0.5

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

  8. Applied sqrt-prod0.5

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

  9. Applied sqrt-prod0.5

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

  10. Applied associate-/r*0.5

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

  11. Simplified0.5

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

  12. Final simplification0.5

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

Reproduce

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