Average Error: 0.4 → 0.5
Time: 35.0s
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\sqrt{k}} \cdot \left({\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)\]
\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(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)
double f(double k, double n) {
        double r3325570 = 1.0;
        double r3325571 = k;
        double r3325572 = sqrt(r3325571);
        double r3325573 = r3325570 / r3325572;
        double r3325574 = 2.0;
        double r3325575 = atan2(1.0, 0.0);
        double r3325576 = r3325574 * r3325575;
        double r3325577 = n;
        double r3325578 = r3325576 * r3325577;
        double r3325579 = r3325570 - r3325571;
        double r3325580 = r3325579 / r3325574;
        double r3325581 = pow(r3325578, r3325580);
        double r3325582 = r3325573 * r3325581;
        return r3325582;
}
↓
double f(double k, double n) {
        double r3325583 = 1.0;
        double r3325584 = k;
        double r3325585 = sqrt(r3325584);
        double r3325586 = r3325583 / r3325585;
        double r3325587 = n;
        double r3325588 = 2.0;
        double r3325589 = atan2(1.0, 0.0);
        double r3325590 = r3325588 * r3325589;
        double r3325591 = r3325587 * r3325590;
        double r3325592 = r3325583 - r3325584;
        double r3325593 = r3325592 / r3325588;
        double r3325594 = r3325593 / r3325588;
        double r3325595 = pow(r3325591, r3325594);
        double r3325596 = r3325595 * r3325595;
        double r3325597 = r3325586 * r3325596;
        return r3325597;
}

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

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))