Average Error: 0.4 → 0.4
Time: 13.5s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
double f(double k, double n) {
        double r62966 = 1.0;
        double r62967 = k;
        double r62968 = sqrt(r62967);
        double r62969 = r62966 / r62968;
        double r62970 = 2.0;
        double r62971 = atan2(1.0, 0.0);
        double r62972 = r62970 * r62971;
        double r62973 = n;
        double r62974 = r62972 * r62973;
        double r62975 = r62966 - r62967;
        double r62976 = r62975 / r62970;
        double r62977 = pow(r62974, r62976);
        double r62978 = r62969 * r62977;
        return r62978;
}
↓
double f(double k, double n) {
        double r62979 = 1.0;
        double r62980 = 1.0;
        double r62981 = k;
        double r62982 = r62980 / r62981;
        double r62983 = 0.25;
        double r62984 = pow(r62982, r62983);
        double r62985 = r62979 * r62984;
        double r62986 = sqrt(r62981);
        double r62987 = sqrt(r62986);
        double r62988 = r62985 / r62987;
        double r62989 = 2.0;
        double r62990 = atan2(1.0, 0.0);
        double r62991 = r62989 * r62990;
        double r62992 = n;
        double r62993 = r62991 * r62992;
        double r62994 = r62979 - r62981;
        double r62995 = r62994 / r62989;
        double r62996 = pow(r62993, r62995);
        double r62997 = r62988 * r62996;
        return r62997;
}

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 add-sqr-sqrt0.4

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

  4. Applied sqrt-prod0.5

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

  5. Applied associate-/r*0.5

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

  7. Applied add-sqr-sqrt0.5

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

  8. Applied sqrt-prod0.5

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

  9. Applied sqrt-prod0.6

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

  10. Applied *-un-lft-identity0.6

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

  11. Applied times-frac0.6

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

  12. Taylor expanded around 0 0.4

    \[\leadsto \frac{\color{blue}{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

  13. Final simplification0.4

    \[\leadsto \frac{1 \cdot {\left(\frac{1}{k}\right)}^{\frac{1}{4}}}{\sqrt{\sqrt{k}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))