Average Error: 0.3 → 0.4
Time: 31.0s
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 r5198414 = 1.0;
        double r5198415 = k;
        double r5198416 = sqrt(r5198415);
        double r5198417 = r5198414 / r5198416;
        double r5198418 = 2.0;
        double r5198419 = atan2(1.0, 0.0);
        double r5198420 = r5198418 * r5198419;
        double r5198421 = n;
        double r5198422 = r5198420 * r5198421;
        double r5198423 = r5198414 - r5198415;
        double r5198424 = r5198423 / r5198418;
        double r5198425 = pow(r5198422, r5198424);
        double r5198426 = r5198417 * r5198425;
        return r5198426;
}
↓
double f(double k, double n) {
        double r5198427 = atan2(1.0, 0.0);
        double r5198428 = 2.0;
        double r5198429 = r5198427 * r5198428;
        double r5198430 = n;
        double r5198431 = r5198429 * r5198430;
        double r5198432 = 1.0;
        double r5198433 = k;
        double r5198434 = r5198432 - r5198433;
        double r5198435 = r5198434 / r5198428;
        double r5198436 = pow(r5198431, r5198435);
        double r5198437 = r5198436 * r5198432;
        double r5198438 = sqrt(r5198433);
        double r5198439 = sqrt(r5198438);
        double r5198440 = r5198437 / r5198439;
        double r5198441 = r5198440 / r5198439;
        double r5198442 = sqrt(r5198441);
        double r5198443 = r5198437 / r5198438;
        double r5198444 = sqrt(r5198443);
        double r5198445 = r5198442 * r5198444;
        return r5198445;
}

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 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))