Average Error: 0.4 → 0.5
Time: 19.2s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[1 \cdot \frac{\frac{1}{\sqrt{\sqrt{k}}}}{\frac{\sqrt{\sqrt{k}}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
1 \cdot \frac{\frac{1}{\sqrt{\sqrt{k}}}}{\frac{\sqrt{\sqrt{k}}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}
double f(double k, double n) {
        double r154075 = 1.0;
        double r154076 = k;
        double r154077 = sqrt(r154076);
        double r154078 = r154075 / r154077;
        double r154079 = 2.0;
        double r154080 = atan2(1.0, 0.0);
        double r154081 = r154079 * r154080;
        double r154082 = n;
        double r154083 = r154081 * r154082;
        double r154084 = r154075 - r154076;
        double r154085 = r154084 / r154079;
        double r154086 = pow(r154083, r154085);
        double r154087 = r154078 * r154086;
        return r154087;
}
↓
double f(double k, double n) {
        double r154088 = 1.0;
        double r154089 = 1.0;
        double r154090 = k;
        double r154091 = sqrt(r154090);
        double r154092 = sqrt(r154091);
        double r154093 = r154089 / r154092;
        double r154094 = 2.0;
        double r154095 = atan2(1.0, 0.0);
        double r154096 = r154094 * r154095;
        double r154097 = n;
        double r154098 = r154096 * r154097;
        double r154099 = r154088 - r154090;
        double r154100 = r154099 / r154094;
        double r154101 = pow(r154098, r154100);
        double r154102 = r154092 / r154101;
        double r154103 = r154093 / r154102;
        double r154104 = r154088 * r154103;
        return r154104;
}

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 div-inv0.4

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

  4. Applied associate-*l*0.4

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

  5. Simplified0.3

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

  7. Applied clear-num0.4

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

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

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

  10. Applied add-sqr-sqrt0.4

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

  11. Applied sqrt-prod0.5

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

  12. Applied times-frac0.5

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

  13. Applied associate-/r*0.5

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

  14. Simplified0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))