Average Error: 0.3 → 0.4
Time: 30.1s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\sqrt{\frac{\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot 1}{\sqrt{\sqrt{k}}}}{\sqrt{\sqrt{k}}}} \cdot \sqrt{\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot 1}{\sqrt{k}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\sqrt{\frac{\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot 1}{\sqrt{\sqrt{k}}}}{\sqrt{\sqrt{k}}}} \cdot \sqrt{\frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot 1}{\sqrt{k}}}
double f(double k, double n) {
        double r5274387 = 1.0;
        double r5274388 = k;
        double r5274389 = sqrt(r5274388);
        double r5274390 = r5274387 / r5274389;
        double r5274391 = 2.0;
        double r5274392 = atan2(1.0, 0.0);
        double r5274393 = r5274391 * r5274392;
        double r5274394 = n;
        double r5274395 = r5274393 * r5274394;
        double r5274396 = r5274387 - r5274388;
        double r5274397 = r5274396 / r5274391;
        double r5274398 = pow(r5274395, r5274397);
        double r5274399 = r5274390 * r5274398;
        return r5274399;
}
↓
double f(double k, double n) {
        double r5274400 = atan2(1.0, 0.0);
        double r5274401 = 2.0;
        double r5274402 = r5274400 * r5274401;
        double r5274403 = n;
        double r5274404 = r5274402 * r5274403;
        double r5274405 = 1.0;
        double r5274406 = k;
        double r5274407 = r5274405 - r5274406;
        double r5274408 = r5274407 / r5274401;
        double r5274409 = pow(r5274404, r5274408);
        double r5274410 = r5274409 * r5274405;
        double r5274411 = sqrt(r5274406);
        double r5274412 = sqrt(r5274411);
        double r5274413 = r5274410 / r5274412;
        double r5274414 = r5274413 / r5274412;
        double r5274415 = sqrt(r5274414);
        double r5274416 = r5274410 / r5274411;
        double r5274417 = sqrt(r5274416);
        double r5274418 = r5274415 * r5274417;
        return r5274418;
}

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 0.3

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

  3. Applied associate-*l/0.3

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

  5. Applied add-sqr-sqrt0.4

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

  7. Applied add-sqr-sqrt0.4

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

  8. Applied sqrt-prod0.4

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

  9. Applied associate-/r*0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019171 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))