Average Error: 0.4 → 0.4
Time: 1.3m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\sqrt{{2}^{\left(\frac{1}{2} - \frac{k}{2}\right)}} \cdot {n}^{\left(\frac{1}{2} - \frac{k}{2}\right)}\right) \cdot \frac{\sqrt{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\frac{\sqrt{k}}{\sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\sqrt{{2}^{\left(\frac{1}{2} - \frac{k}{2}\right)}} \cdot {n}^{\left(\frac{1}{2} - \frac{k}{2}\right)}\right) \cdot \frac{\sqrt{{\pi}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\frac{\sqrt{k}}{\sqrt{{\left(2 \cdot \pi\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}}}}
double f(double k, double n) {
        double r5263111 = 1.0;
        double r5263112 = k;
        double r5263113 = sqrt(r5263112);
        double r5263114 = r5263111 / r5263113;
        double r5263115 = 2.0;
        double r5263116 = atan2(1.0, 0.0);
        double r5263117 = r5263115 * r5263116;
        double r5263118 = n;
        double r5263119 = r5263117 * r5263118;
        double r5263120 = r5263111 - r5263112;
        double r5263121 = r5263120 / r5263115;
        double r5263122 = pow(r5263119, r5263121);
        double r5263123 = r5263114 * r5263122;
        return r5263123;
}
↓
double f(double k, double n) {
        double r5263124 = 2.0;
        double r5263125 = 0.5;
        double r5263126 = k;
        double r5263127 = r5263126 / r5263124;
        double r5263128 = r5263125 - r5263127;
        double r5263129 = pow(r5263124, r5263128);
        double r5263130 = sqrt(r5263129);
        double r5263131 = n;
        double r5263132 = pow(r5263131, r5263128);
        double r5263133 = r5263130 * r5263132;
        double r5263134 = atan2(1.0, 0.0);
        double r5263135 = pow(r5263134, r5263128);
        double r5263136 = sqrt(r5263135);
        double r5263137 = sqrt(r5263126);
        double r5263138 = r5263124 * r5263134;
        double r5263139 = pow(r5263138, r5263128);
        double r5263140 = sqrt(r5263139);
        double r5263141 = r5263137 / r5263140;
        double r5263142 = r5263136 / r5263141;
        double r5263143 = r5263133 * r5263142;
        return r5263143;
}

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

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

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

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

  5. Applied unpow-prod-down0.5

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

  6. Applied times-frac0.5

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

  7. Simplified0.5

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

  9. Applied add-sqr-sqrt0.5

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

  10. Applied associate-/l*0.4

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

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

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

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

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

  14. Applied times-frac0.4

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

  15. Applied unpow-prod-down0.4

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

  16. Applied sqrt-prod0.4

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

  17. Applied times-frac0.4

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

  18. Applied associate-*r*0.4

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

  19. Simplified0.4

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

  20. Final simplification0.4

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

Reproduce

herbie shell --seed 2019149 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))