Average Error: 0.4 → 0.6
Time: 3.4m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{\sqrt{k}}} \cdot \frac{1}{\frac{\sqrt{\sqrt{k}}}{{\left(2 \cdot n\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}{\sqrt{\sqrt{k}}} \cdot \frac{1}{\frac{\sqrt{\sqrt{k}}}{{\left(2 \cdot n\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}
double f(double k, double n) {
        double r2870102 = 1.0;
        double r2870103 = k;
        double r2870104 = sqrt(r2870103);
        double r2870105 = r2870102 / r2870104;
        double r2870106 = 2.0;
        double r2870107 = atan2(1.0, 0.0);
        double r2870108 = r2870106 * r2870107;
        double r2870109 = n;
        double r2870110 = r2870108 * r2870109;
        double r2870111 = r2870102 - r2870103;
        double r2870112 = r2870111 / r2870106;
        double r2870113 = pow(r2870110, r2870112);
        double r2870114 = r2870105 * r2870113;
        return r2870114;
}
↓
double f(double k, double n) {
        double r2870115 = atan2(1.0, 0.0);
        double r2870116 = 0.5;
        double r2870117 = k;
        double r2870118 = 2.0;
        double r2870119 = r2870117 / r2870118;
        double r2870120 = r2870116 - r2870119;
        double r2870121 = pow(r2870115, r2870120);
        double r2870122 = sqrt(r2870117);
        double r2870123 = sqrt(r2870122);
        double r2870124 = r2870121 / r2870123;
        double r2870125 = 1.0;
        double r2870126 = n;
        double r2870127 = r2870118 * r2870126;
        double r2870128 = pow(r2870127, r2870120);
        double r2870129 = r2870123 / r2870128;
        double r2870130 = r2870125 / r2870129;
        double r2870131 = r2870124 * r2870130;
        return r2870131;
}

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

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied sqrt-prod0.5

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

  6. Applied unpow-prod-down0.6

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

  7. Applied times-frac0.6

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

  9. Applied clear-num0.6

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

  10. Final simplification0.6

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))