Average Error: 0.4 → 0.5
Time: 7.8s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{k}} \cdot \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)\]
\frac{1}{\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(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)
double f(double k, double n) {
        double r174484 = 1.0;
        double r174485 = k;
        double r174486 = sqrt(r174485);
        double r174487 = r174484 / r174486;
        double r174488 = 2.0;
        double r174489 = atan2(1.0, 0.0);
        double r174490 = r174488 * r174489;
        double r174491 = n;
        double r174492 = r174490 * r174491;
        double r174493 = r174484 - r174485;
        double r174494 = r174493 / r174488;
        double r174495 = pow(r174492, r174494);
        double r174496 = r174487 * r174495;
        return r174496;
}
↓
double f(double k, double n) {
        double r174497 = 1.0;
        double r174498 = k;
        double r174499 = sqrt(r174498);
        double r174500 = r174497 / r174499;
        double r174501 = 2.0;
        double r174502 = atan2(1.0, 0.0);
        double r174503 = r174501 * r174502;
        double r174504 = n;
        double r174505 = r174503 * r174504;
        double r174506 = r174497 - r174498;
        double r174507 = r174506 / r174501;
        double r174508 = 2.0;
        double r174509 = r174507 / r174508;
        double r174510 = pow(r174505, r174509);
        double r174511 = r174510 * r174510;
        double r174512 = r174500 * r174511;
        return r174512;
}

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 sqr-pow0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))