Average Error: 0.4 → 0.4
Time: 13.9s
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 r61047 = 1.0;
        double r61048 = k;
        double r61049 = sqrt(r61048);
        double r61050 = r61047 / r61049;
        double r61051 = 2.0;
        double r61052 = atan2(1.0, 0.0);
        double r61053 = r61051 * r61052;
        double r61054 = n;
        double r61055 = r61053 * r61054;
        double r61056 = r61047 - r61048;
        double r61057 = r61056 / r61051;
        double r61058 = pow(r61055, r61057);
        double r61059 = r61050 * r61058;
        return r61059;
}
↓
double f(double k, double n) {
        double r61060 = 1.0;
        double r61061 = 1.0;
        double r61062 = k;
        double r61063 = r61061 / r61062;
        double r61064 = 0.25;
        double r61065 = pow(r61063, r61064);
        double r61066 = r61060 * r61065;
        double r61067 = sqrt(r61062);
        double r61068 = sqrt(r61067);
        double r61069 = r61066 / r61068;
        double r61070 = 2.0;
        double r61071 = atan2(1.0, 0.0);
        double r61072 = r61070 * r61071;
        double r61073 = n;
        double r61074 = r61072 * r61073;
        double r61075 = r61060 - r61062;
        double r61076 = r61075 / r61070;
        double r61077 = pow(r61074, r61076);
        double r61078 = r61069 * r61077;
        return r61078;
}

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 add-sqr-sqrt0.6

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

  11. Applied times-frac0.6

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

  12. Taylor expanded around 0 0.4

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

  13. Simplified0.4

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

  14. 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))))