Average Error: 0.3 → 0.4
Time: 8.0s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{k}} \cdot \left(\sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}\right)\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1}{\sqrt{k}} \cdot \left(\sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}\right)
double f(double k, double n) {
        double r131110 = 1.0;
        double r131111 = k;
        double r131112 = sqrt(r131111);
        double r131113 = r131110 / r131112;
        double r131114 = 2.0;
        double r131115 = atan2(1.0, 0.0);
        double r131116 = r131114 * r131115;
        double r131117 = n;
        double r131118 = r131116 * r131117;
        double r131119 = r131110 - r131111;
        double r131120 = r131119 / r131114;
        double r131121 = pow(r131118, r131120);
        double r131122 = r131113 * r131121;
        return r131122;
}
↓
double f(double k, double n) {
        double r131123 = 1.0;
        double r131124 = k;
        double r131125 = sqrt(r131124);
        double r131126 = r131123 / r131125;
        double r131127 = 2.0;
        double r131128 = atan2(1.0, 0.0);
        double r131129 = r131127 * r131128;
        double r131130 = n;
        double r131131 = r131129 * r131130;
        double r131132 = r131123 - r131124;
        double r131133 = r131132 / r131127;
        double r131134 = pow(r131131, r131133);
        double r131135 = sqrt(r131134);
        double r131136 = r131135 * r131135;
        double r131137 = r131126 * r131136;
        return r131137;
}

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 0.3

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

  3. Applied add-sqr-sqrt0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))