Average Error: 0.4 → 0.5
Time: 1.1m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\frac{\sqrt{{\left(n \cdot 2\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}} \cdot \sqrt{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}\right) \cdot \sqrt{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\frac{\sqrt{{\left(n \cdot 2\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}} \cdot \sqrt{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}\right) \cdot \sqrt{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}
double f(double k, double n) {
        double r12693057 = 1.0;
        double r12693058 = k;
        double r12693059 = sqrt(r12693058);
        double r12693060 = r12693057 / r12693059;
        double r12693061 = 2.0;
        double r12693062 = atan2(1.0, 0.0);
        double r12693063 = r12693061 * r12693062;
        double r12693064 = n;
        double r12693065 = r12693063 * r12693064;
        double r12693066 = r12693057 - r12693058;
        double r12693067 = r12693066 / r12693061;
        double r12693068 = pow(r12693065, r12693067);
        double r12693069 = r12693060 * r12693068;
        return r12693069;
}
↓
double f(double k, double n) {
        double r12693070 = n;
        double r12693071 = 2.0;
        double r12693072 = r12693070 * r12693071;
        double r12693073 = 0.5;
        double r12693074 = k;
        double r12693075 = r12693074 / r12693071;
        double r12693076 = r12693073 - r12693075;
        double r12693077 = pow(r12693072, r12693076);
        double r12693078 = sqrt(r12693077);
        double r12693079 = sqrt(r12693074);
        double r12693080 = r12693078 / r12693079;
        double r12693081 = atan2(1.0, 0.0);
        double r12693082 = r12693072 * r12693081;
        double r12693083 = pow(r12693082, r12693076);
        double r12693084 = sqrt(r12693083);
        double r12693085 = r12693080 * r12693084;
        double r12693086 = pow(r12693081, r12693076);
        double r12693087 = sqrt(r12693086);
        double r12693088 = r12693085 * r12693087;
        return r12693088;
}

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. Simplified0.3

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

  4. Applied *-un-lft-identity0.3

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied times-frac0.5

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

  8. Applied unpow-prod-down0.5

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

  9. Applied sqrt-prod0.5

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

  10. Applied associate-/l*0.5

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

  12. Applied associate-/r/0.5

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

  13. Applied associate-*r*0.5

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

  14. Simplified0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2019162 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))