Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 13.9s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))

C

double code(double k, double n) {
	return (1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0));
}

Java

public static double code(double k, double n) {
	return (1.0 / Math.sqrt(k)) * Math.pow(((2.0 * Math.PI) * n), ((1.0 - k) / 2.0));
}

Python

def code(k, n):
	return (1.0 / math.sqrt(k)) * math.pow(((2.0 * math.pi) * n), ((1.0 - k) / 2.0))

Julia

function code(k, n)
	return Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0)))
end

MATLAB

function tmp = code(k, n)
	tmp = (1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0));
end

Wolfram

code[k_, n_] := N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))

C

double code(double k, double n) {
	return (1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0));
}

Java

public static double code(double k, double n) {
	return (1.0 / Math.sqrt(k)) * Math.pow(((2.0 * Math.PI) * n), ((1.0 - k) / 2.0));
}

Python

def code(k, n):
	return (1.0 / math.sqrt(k)) * math.pow(((2.0 * math.pi) * n), ((1.0 - k) / 2.0))

Julia

function code(k, n)
	return Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0)))
end

MATLAB

function tmp = code(k, n)
	tmp = (1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0));
end

Wolfram

code[k_, n_] := N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot n\right)\\ \frac{\sqrt{t\_0}}{{\left(k \cdot {t\_0}^{k}\right)}^{0.5}} \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* 2.0 (* PI n)))) (/ (sqrt t_0) (pow (* k (pow t_0 k)) 0.5))))

C

double code(double k, double n) {
	double t_0 = 2.0 * (((double) M_PI) * n);
	return sqrt(t_0) / pow((k * pow(t_0, k)), 0.5);
}

Java

public static double code(double k, double n) {
	double t_0 = 2.0 * (Math.PI * n);
	return Math.sqrt(t_0) / Math.pow((k * Math.pow(t_0, k)), 0.5);
}

Python

def code(k, n):
	t_0 = 2.0 * (math.pi * n)
	return math.sqrt(t_0) / math.pow((k * math.pow(t_0, k)), 0.5)

Julia

function code(k, n)
	t_0 = Float64(2.0 * Float64(pi * n))
	return Float64(sqrt(t_0) / (Float64(k * (t_0 ^ k)) ^ 0.5))
end

MATLAB

function tmp = code(k, n)
	t_0 = 2.0 * (pi * n);
	tmp = sqrt(t_0) / ((k * (t_0 ^ k)) ^ 0.5);
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]}, N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Power[N[(k * N[Power[t$95$0, k], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*r* N/A

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

   2. div-sub N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. pow-sub N/A

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

   6. frac-times N/A

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

   7. *-rgt-identity N/A

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

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)}\right) \]

   9. unpow1/2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}\right), \left(\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \cdot \sqrt{k}\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \left(\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \cdot \sqrt{k}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \left({\color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right), \left({\left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot n\right)}\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

   13. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \left({\left(2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot n\right)\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{k} \cdot k\right)}^{0.5}}} \]

5. Final simplification 99.7%

   \[\leadsto \frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(k \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{k}\right)}^{0.5}} \]
6. Add Preprocessing

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-58}:\\ \;\;\;\;{\left(\frac{\pi}{k}\right)}^{0.5} \cdot {\left(2 \cdot n\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k + -1\right)}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6e-58)
   (* (pow (/ PI k) 0.5) (pow (* 2.0 n) 0.5))
   (pow (* k (pow (* 2.0 (* PI n)) (+ k -1.0))) -0.5)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6e-58) {
		tmp = pow((((double) M_PI) / k), 0.5) * pow((2.0 * n), 0.5);
	} else {
		tmp = pow((k * pow((2.0 * (((double) M_PI) * n)), (k + -1.0))), -0.5);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 6e-58) {
		tmp = Math.pow((Math.PI / k), 0.5) * Math.pow((2.0 * n), 0.5);
	} else {
		tmp = Math.pow((k * Math.pow((2.0 * (Math.PI * n)), (k + -1.0))), -0.5);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 6e-58:
		tmp = math.pow((math.pi / k), 0.5) * math.pow((2.0 * n), 0.5)
	else:
		tmp = math.pow((k * math.pow((2.0 * (math.pi * n)), (k + -1.0))), -0.5)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 6e-58)
		tmp = Float64((Float64(pi / k) ^ 0.5) * (Float64(2.0 * n) ^ 0.5));
	else
		tmp = Float64(k * (Float64(2.0 * Float64(pi * n)) ^ Float64(k + -1.0))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 6e-58)
		tmp = ((pi / k) ^ 0.5) * ((2.0 * n) ^ 0.5);
	else
		tmp = (k * ((2.0 * (pi * n)) ^ (k + -1.0))) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6e-58], N[(N[Power[N[(Pi / k), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(2.0 * n), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[Power[N[(k * N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(k + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.00000000000000015e-58

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 72.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 72.5%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}} \cdot 2\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1 \cdot 2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{n \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{\mathsf{PI}\left(\right) \cdot n}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right)\right) \]

      10. PI-lowering-PI.f64 72.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right) \]

   7. Applied egg-rr 72.7%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{k}{\pi \cdot n}}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \sqrt{2 \cdot \frac{1}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}} \]

      2. sqrt-prod N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\sqrt{\frac{1}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}}} \]

      3. clear-num N/A

         \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k}} \]

      4. associate-*l/ N/A

         \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\mathsf{PI}\left(\right)}{k} \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{\mathsf{PI}\left(\right)}{k} \cdot n} \cdot \color{blue}{\sqrt{2}} \]

      6. pow1/2 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{\mathsf{PI}\left(\right)}{k}\right)}^{\frac{1}{2}}\right), \color{blue}{\left({n}^{\frac{1}{2}} \cdot \sqrt{2}\right)}\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{k}\right), \frac{1}{2}\right), \left(\color{blue}{{n}^{\frac{1}{2}}} \cdot \sqrt{2}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), k\right), \frac{1}{2}\right), \left({\color{blue}{n}}^{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), k\right), \frac{1}{2}\right), \left({n}^{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \]

      13. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), k\right), \frac{1}{2}\right), \left({n}^{\frac{1}{2}} \cdot {2}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      14. pow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), k\right), \frac{1}{2}\right), \left({\left(n \cdot 2\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      15. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), k\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\left(n \cdot 2\right), \color{blue}{\frac{1}{2}}\right)\right) \]

      16. *-lowering-*.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), k\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, 2\right), \frac{1}{2}\right)\right) \]

   9. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{{\left(\frac{\pi}{k}\right)}^{0.5} \cdot {\left(n \cdot 2\right)}^{0.5}} \]

   if 6.00000000000000015e-58 < k

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r* N/A

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

      2. div-sub N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. pow-sub N/A

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

      6. frac-times N/A

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

      7. *-rgt-identity N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}\right), \left(\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \cdot \sqrt{k}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \left(\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \cdot \sqrt{k}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \left({\color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right), \left({\left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot n\right)}\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \left({\left(2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot n\right)\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{k} \cdot k\right)}^{0.5}}} \]
   5. Step-by-step derivation

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. unpow1/2 N/A

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

      4. sqrt-undiv N/A

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

      5. sqrt-pow2 N/A

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

      6. metadata-eval N/A

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

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{k} \cdot k}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}\right), \color{blue}{\frac{-1}{2}}\right) \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\left(\frac{k \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{k}}{2 \cdot \left(\pi \cdot n\right)}\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(k \cdot \frac{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{k}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}\right), \frac{-1}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{k}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)} \cdot k\right), \frac{-1}{2}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{k}}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}\right), k\right), \frac{-1}{2}\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{k} \cdot \frac{1}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}\right), k\right), \frac{-1}{2}\right) \]

      5. inv-pow N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{k} \cdot {\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{-1}\right), k\right), \frac{-1}{2}\right) \]

      6. pow-prod-up N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(k + -1\right)}\right), k\right), \frac{-1}{2}\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), \left(k + -1\right)\right), k\right), \frac{-1}{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot n\right)\right), \left(k + -1\right)\right), k\right), \frac{-1}{2}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right), \left(k + -1\right)\right), k\right), \frac{-1}{2}\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(k + -1\right)\right), k\right), \frac{-1}{2}\right) \]

      11. +-lowering-+.f64 99.7%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{+.f64}\left(k, -1\right)\right), k\right), \frac{-1}{2}\right) \]

   8. Applied egg-rr 99.7%

      \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k + -1\right)} \cdot k\right)}}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-58}:\\ \;\;\;\;{\left(\frac{\pi}{k}\right)}^{0.5} \cdot {\left(2 \cdot n\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k + -1\right)}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\frac{k}{\pi \cdot n}}\\ \mathbf{if}\;k \leq 1.22 \cdot 10^{+184}:\\ \;\;\;\;{\left(\frac{\pi}{k}\right)}^{0.5} \cdot {\left(2 \cdot n\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(t\_0 \cdot t\_0\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (/ 2.0 (/ k (* PI n)))))
   (if (<= k 1.22e+184)
     (* (pow (/ PI k) 0.5) (pow (* 2.0 n) 0.5))
     (pow (* t_0 t_0) 0.25))))

C

double code(double k, double n) {
	double t_0 = 2.0 / (k / (((double) M_PI) * n));
	double tmp;
	if (k <= 1.22e+184) {
		tmp = pow((((double) M_PI) / k), 0.5) * pow((2.0 * n), 0.5);
	} else {
		tmp = pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = 2.0 / (k / (Math.PI * n));
	double tmp;
	if (k <= 1.22e+184) {
		tmp = Math.pow((Math.PI / k), 0.5) * Math.pow((2.0 * n), 0.5);
	} else {
		tmp = Math.pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = 2.0 / (k / (math.pi * n))
	tmp = 0
	if k <= 1.22e+184:
		tmp = math.pow((math.pi / k), 0.5) * math.pow((2.0 * n), 0.5)
	else:
		tmp = math.pow((t_0 * t_0), 0.25)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(2.0 / Float64(k / Float64(pi * n)))
	tmp = 0.0
	if (k <= 1.22e+184)
		tmp = Float64((Float64(pi / k) ^ 0.5) * (Float64(2.0 * n) ^ 0.5));
	else
		tmp = Float64(t_0 * t_0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = 2.0 / (k / (pi * n));
	tmp = 0.0;
	if (k <= 1.22e+184)
		tmp = ((pi / k) ^ 0.5) * ((2.0 * n) ^ 0.5);
	else
		tmp = (t_0 * t_0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 / N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.22e+184], N[(N[Power[N[(Pi / k), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(2.0 * n), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 0.25], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.22000000000000006e184

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 46.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 46.2%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}} \cdot 2\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1 \cdot 2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{n \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{\mathsf{PI}\left(\right) \cdot n}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right)\right) \]

      10. PI-lowering-PI.f64 46.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right) \]

   7. Applied egg-rr 46.3%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{k}{\pi \cdot n}}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \sqrt{2 \cdot \frac{1}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}} \]

      2. sqrt-prod N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\sqrt{\frac{1}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}}} \]

      3. clear-num N/A

         \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k}} \]

      4. associate-*l/ N/A

         \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\mathsf{PI}\left(\right)}{k} \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{\mathsf{PI}\left(\right)}{k} \cdot n} \cdot \color{blue}{\sqrt{2}} \]

      6. pow1/2 N/A

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

      7. unpow-prod-down N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{\mathsf{PI}\left(\right)}{k}\right)}^{\frac{1}{2}}\right), \color{blue}{\left({n}^{\frac{1}{2}} \cdot \sqrt{2}\right)}\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{k}\right), \frac{1}{2}\right), \left(\color{blue}{{n}^{\frac{1}{2}}} \cdot \sqrt{2}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), k\right), \frac{1}{2}\right), \left({\color{blue}{n}}^{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), k\right), \frac{1}{2}\right), \left({n}^{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \]

      13. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), k\right), \frac{1}{2}\right), \left({n}^{\frac{1}{2}} \cdot {2}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      14. pow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), k\right), \frac{1}{2}\right), \left({\left(n \cdot 2\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      15. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), k\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\left(n \cdot 2\right), \color{blue}{\frac{1}{2}}\right)\right) \]

      16. *-lowering-*.f64 59.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), k\right), \frac{1}{2}\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, 2\right), \frac{1}{2}\right)\right) \]

   9. Applied egg-rr 59.3%

      \[\leadsto \color{blue}{{\left(\frac{\pi}{k}\right)}^{0.5} \cdot {\left(n \cdot 2\right)}^{0.5}} \]

   if 1.22000000000000006e184 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 2.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 2.6%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}} \cdot 2\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1 \cdot 2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{n \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{\mathsf{PI}\left(\right) \cdot n}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right)\right) \]

      10. PI-lowering-PI.f64 2.6%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right) \]

   7. Applied egg-rr 2.6%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{k}{\pi \cdot n}}}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto {\left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)}^{\color{blue}{\frac{1}{2}}} \]

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow-prod-down N/A

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

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}} \cdot \frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1}{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right), \left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{-1}{2}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{\mathsf{PI}\left(\right) \cdot n}\right)\right), \left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right), \left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \mathsf{/.f64}\left(2, \left(\frac{k}{\mathsf{PI}\left(\right) \cdot n}\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      15. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      16. metadata-eval 18.9%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right), \frac{1}{4}\right) \]

   9. Applied egg-rr 18.9%

      \[\leadsto \color{blue}{{\left(\frac{2}{\frac{k}{\pi \cdot n}} \cdot \frac{2}{\frac{k}{\pi \cdot n}}\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.22 \cdot 10^{+184}:\\ \;\;\;\;{\left(\frac{\pi}{k}\right)}^{0.5} \cdot {\left(2 \cdot n\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{2}{\frac{k}{\pi \cdot n}} \cdot \frac{2}{\frac{k}{\pi \cdot n}}\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 + \frac{k}{-2}\right)} \cdot {k}^{-0.5} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (pow (* 2.0 (* PI n)) (+ 0.5 (/ k -2.0))) (pow k -0.5)))

C

double code(double k, double n) {
	return pow((2.0 * (((double) M_PI) * n)), (0.5 + (k / -2.0))) * pow(k, -0.5);
}

Java

public static double code(double k, double n) {
	return Math.pow((2.0 * (Math.PI * n)), (0.5 + (k / -2.0))) * Math.pow(k, -0.5);
}

Python

def code(k, n):
	return math.pow((2.0 * (math.pi * n)), (0.5 + (k / -2.0))) * math.pow(k, -0.5)

Julia

function code(k, n)
	return Float64((Float64(2.0 * Float64(pi * n)) ^ Float64(0.5 + Float64(k / -2.0))) * (k ^ -0.5))
end

MATLAB

function tmp = code(k, n)
	tmp = ((2.0 * (pi * n)) ^ (0.5 + (k / -2.0))) * (k ^ -0.5);
end

Wolfram

code[k_, n_] := N[(N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(k / -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[k, -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. div-sub N/A

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

   4. metadata-eval N/A

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

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}\right), \color{blue}{\left(\frac{1}{\sqrt{k}}\right)}\right) \]

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 + \frac{k}{-2}\right)} \cdot {k}^{-0.5}} \]
5. Add Preprocessing

Alternative 5: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\frac{k}{\pi \cdot n}}\\ \mathbf{if}\;k \leq 4.2 \cdot 10^{+185}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(t\_0 \cdot t\_0\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (/ 2.0 (/ k (* PI n)))))
   (if (<= k 4.2e+185)
     (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
     (pow (* t_0 t_0) 0.25))))

C

double code(double k, double n) {
	double t_0 = 2.0 / (k / (((double) M_PI) * n));
	double tmp;
	if (k <= 4.2e+185) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = 2.0 / (k / (Math.PI * n));
	double tmp;
	if (k <= 4.2e+185) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	} else {
		tmp = Math.pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = 2.0 / (k / (math.pi * n))
	tmp = 0
	if k <= 4.2e+185:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (math.pi / k)))
	else:
		tmp = math.pow((t_0 * t_0), 0.25)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(2.0 / Float64(k / Float64(pi * n)))
	tmp = 0.0
	if (k <= 4.2e+185)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = Float64(t_0 * t_0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = 2.0 / (k / (pi * n));
	tmp = 0.0;
	if (k <= 4.2e+185)
		tmp = sqrt(n) * sqrt((2.0 * (pi / k)));
	else
		tmp = (t_0 * t_0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 / N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 4.2e+185], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 0.25], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.2e185

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 46.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 46.2%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(n \cdot \frac{\mathsf{PI}\left(\right)}{k}\right) \cdot 2} \]

      3. associate-*l* N/A

         \[\leadsto \sqrt{n \cdot \left(\frac{\mathsf{PI}\left(\right)}{k} \cdot 2\right)} \]

      4. sqrt-prod N/A

         \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{\frac{\mathsf{PI}\left(\right)}{k} \cdot 2}} \]

      5. pow1/2 N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({n}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{\frac{\mathsf{PI}\left(\right)}{k} \cdot 2}\right)}\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{n}\right), \left(\sqrt{\color{blue}{\frac{\mathsf{PI}\left(\right)}{k} \cdot 2}}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \left(\sqrt{\color{blue}{\frac{\mathsf{PI}\left(\right)}{k} \cdot 2}}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{k} \cdot 2\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{k}\right), 2\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), k\right), 2\right)\right)\right) \]

      12. PI-lowering-PI.f64 59.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), k\right), 2\right)\right)\right) \]

   7. Applied egg-rr 59.3%

      \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{\frac{\pi}{k} \cdot 2}} \]

   if 4.2e185 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 2.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 2.6%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}} \cdot 2\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1 \cdot 2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{n \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{\mathsf{PI}\left(\right) \cdot n}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right)\right) \]

      10. PI-lowering-PI.f64 2.6%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right) \]

   7. Applied egg-rr 2.6%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{k}{\pi \cdot n}}}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto {\left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)}^{\color{blue}{\frac{1}{2}}} \]

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow-prod-down N/A

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

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}} \cdot \frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1}{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right), \left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{-1}{2}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{\mathsf{PI}\left(\right) \cdot n}\right)\right), \left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right), \left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \mathsf{/.f64}\left(2, \left(\frac{k}{\mathsf{PI}\left(\right) \cdot n}\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      15. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      16. metadata-eval 18.9%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right), \frac{1}{4}\right) \]

   9. Applied egg-rr 18.9%

      \[\leadsto \color{blue}{{\left(\frac{2}{\frac{k}{\pi \cdot n}} \cdot \frac{2}{\frac{k}{\pi \cdot n}}\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{+185}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{2}{\frac{k}{\pi \cdot n}} \cdot \frac{2}{\frac{k}{\pi \cdot n}}\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (/ (pow (* 2.0 (* PI n)) (- 0.5 (/ k 2.0))) (sqrt k)))

C

double code(double k, double n) {
	return pow((2.0 * (((double) M_PI) * n)), (0.5 - (k / 2.0))) / sqrt(k);
}

Java

public static double code(double k, double n) {
	return Math.pow((2.0 * (Math.PI * n)), (0.5 - (k / 2.0))) / Math.sqrt(k);
}

Python

def code(k, n):
	return math.pow((2.0 * (math.pi * n)), (0.5 - (k / 2.0))) / math.sqrt(k)

Julia

function code(k, n)
	return Float64((Float64(2.0 * Float64(pi * n)) ^ Float64(0.5 - Float64(k / 2.0))) / sqrt(k))
end

MATLAB

function tmp = code(k, n)
	tmp = ((2.0 * (pi * n)) ^ (0.5 - (k / 2.0))) / sqrt(k);
end

Wolfram

code[k_, n_] := N[(N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(0.5 - N[(k / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. *-lft-identity N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right), \color{blue}{\left(\sqrt{k}\right)}\right) \]

   4. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{\color{blue}{k}}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   8. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   9. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\frac{1}{2} - \frac{k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{\_.f64}\left(\left(\frac{1}{2}\right), \left(\frac{k}{2}\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{k}{2}\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, 2\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   13. sqrt-lowering-sqrt.f64 99.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, 2\right)\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 7: 41.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\frac{k}{\pi \cdot n}}\\ \mathbf{if}\;k \leq 5.5 \cdot 10^{+182}:\\ \;\;\;\;{\left(\frac{k}{2 \cdot \left(\pi \cdot n\right)}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(t\_0 \cdot t\_0\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (/ 2.0 (/ k (* PI n)))))
   (if (<= k 5.5e+182)
     (pow (/ k (* 2.0 (* PI n))) -0.5)
     (pow (* t_0 t_0) 0.25))))

C

double code(double k, double n) {
	double t_0 = 2.0 / (k / (((double) M_PI) * n));
	double tmp;
	if (k <= 5.5e+182) {
		tmp = pow((k / (2.0 * (((double) M_PI) * n))), -0.5);
	} else {
		tmp = pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = 2.0 / (k / (Math.PI * n));
	double tmp;
	if (k <= 5.5e+182) {
		tmp = Math.pow((k / (2.0 * (Math.PI * n))), -0.5);
	} else {
		tmp = Math.pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = 2.0 / (k / (math.pi * n))
	tmp = 0
	if k <= 5.5e+182:
		tmp = math.pow((k / (2.0 * (math.pi * n))), -0.5)
	else:
		tmp = math.pow((t_0 * t_0), 0.25)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(2.0 / Float64(k / Float64(pi * n)))
	tmp = 0.0
	if (k <= 5.5e+182)
		tmp = Float64(k / Float64(2.0 * Float64(pi * n))) ^ -0.5;
	else
		tmp = Float64(t_0 * t_0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = 2.0 / (k / (pi * n));
	tmp = 0.0;
	if (k <= 5.5e+182)
		tmp = (k / (2.0 * (pi * n))) ^ -0.5;
	else
		tmp = (t_0 * t_0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 / N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 5.5e+182], N[Power[N[(k / N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 0.25], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.49999999999999977e182

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r* N/A

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

      2. div-sub N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. pow-sub N/A

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

      6. frac-times N/A

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

      7. *-rgt-identity N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}\right), \left(\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \cdot \sqrt{k}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \left(\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \cdot \sqrt{k}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \left({\color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right), \left({\left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot n\right)}\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \left({\left(2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot n\right)\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{k} \cdot k\right)}^{0.5}}} \]
   5. Step-by-step derivation

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. unpow1/2 N/A

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

      4. sqrt-undiv N/A

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

      5. sqrt-pow2 N/A

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

      6. metadata-eval N/A

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

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{k} \cdot k}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}\right), \color{blue}{\frac{-1}{2}}\right) \]

   6. Applied egg-rr 88.0%

      \[\leadsto \color{blue}{{\left(\frac{k \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{k}}{2 \cdot \left(\pi \cdot n\right)}\right)}^{-0.5}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\color{blue}{k}, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \frac{-1}{2}\right) \]
   8. Step-by-step derivation

   9. Simplified 48.1%

      \[\leadsto {\left(\frac{\color{blue}{k}}{2 \cdot \left(\pi \cdot n\right)}\right)}^{-0.5} \]

   if 5.49999999999999977e182 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 2.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 2.6%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}} \cdot 2\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1 \cdot 2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{n \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{\mathsf{PI}\left(\right) \cdot n}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right)\right) \]

      10. PI-lowering-PI.f64 2.6%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right) \]

   7. Applied egg-rr 2.6%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{k}{\pi \cdot n}}}} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto {\left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)}^{\color{blue}{\frac{1}{2}}} \]

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow-prod-down N/A

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

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}} \cdot \frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1}{2}\right)}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right), \left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{-1}{2}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{\mathsf{PI}\left(\right) \cdot n}\right)\right), \left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right), \left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right) \cdot n}}\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \mathsf{/.f64}\left(2, \left(\frac{k}{\mathsf{PI}\left(\right) \cdot n}\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      15. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)\right) \]

      16. metadata-eval 18.2%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right), \frac{1}{4}\right) \]

   9. Applied egg-rr 18.2%

      \[\leadsto \color{blue}{{\left(\frac{2}{\frac{k}{\pi \cdot n}} \cdot \frac{2}{\frac{k}{\pi \cdot n}}\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 38.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{k}{2 \cdot \left(\pi \cdot n\right)}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (pow (/ k (* 2.0 (* PI n))) -0.5))

C

double code(double k, double n) {
	return pow((k / (2.0 * (((double) M_PI) * n))), -0.5);
}

Java

public static double code(double k, double n) {
	return Math.pow((k / (2.0 * (Math.PI * n))), -0.5);
}

Python

def code(k, n):
	return math.pow((k / (2.0 * (math.pi * n))), -0.5)

Julia

function code(k, n)
	return Float64(k / Float64(2.0 * Float64(pi * n))) ^ -0.5
end

MATLAB

function tmp = code(k, n)
	tmp = (k / (2.0 * (pi * n))) ^ -0.5;
end

Wolfram

code[k_, n_] := N[Power[N[(k / N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*r* N/A

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

   2. div-sub N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. pow-sub N/A

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

   6. frac-times N/A

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

   7. *-rgt-identity N/A

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

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)}\right) \]

   9. unpow1/2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}\right), \left(\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \cdot \sqrt{k}\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \left(\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \cdot \sqrt{k}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \left({\color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right), \left({\left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot n\right)}\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

   13. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \left({\left(2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot n\right)\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{k} \cdot k\right)}^{0.5}}} \]
5. Step-by-step derivation

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. unpow1/2 N/A

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

   4. sqrt-undiv N/A

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

   5. sqrt-pow2 N/A

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

   6. metadata-eval N/A

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

   7. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{k} \cdot k}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}\right), \color{blue}{\frac{-1}{2}}\right) \]

6. Applied egg-rr 90.3%

   \[\leadsto \color{blue}{{\left(\frac{k \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{k}}{2 \cdot \left(\pi \cdot n\right)}\right)}^{-0.5}} \]

7. Taylor expanded in k around 0

   \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\color{blue}{k}, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \frac{-1}{2}\right) \]
8. Step-by-step derivation

9. Simplified 39.4%

   \[\leadsto {\left(\frac{\color{blue}{k}}{2 \cdot \left(\pi \cdot n\right)}\right)}^{-0.5} \]
10. Add Preprocessing

Alternative 9: 37.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \frac{\pi \cdot n}{k}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (* 2.0 (/ (* PI n) k))))

C

double code(double k, double n) {
	return sqrt((2.0 * ((((double) M_PI) * n) / k)));
}

Java

public static double code(double k, double n) {
	return Math.sqrt((2.0 * ((Math.PI * n) / k)));
}

Python

def code(k, n):
	return math.sqrt((2.0 * ((math.pi * n) / k)))

Julia

function code(k, n)
	return sqrt(Float64(2.0 * Float64(Float64(pi * n) / k)))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt((2.0 * ((pi * n) / k)));
end

Wolfram

code[k_, n_] := N[Sqrt[N[(2.0 * N[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 38.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 38.2%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
6. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}} \cdot 2\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1 \cdot 2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{n \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{\mathsf{PI}\left(\right) \cdot n}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right)\right) \]

   10. PI-lowering-PI.f64 38.3%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right) \]

7. Applied egg-rr 38.3%

   \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{k}{\pi \cdot n}}}} \]
8. Step-by-step derivation

   1. associate-/r/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{k} \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{k} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{2}{k} \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k} \cdot n\right), \mathsf{PI}\left(\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), n\right), \mathsf{PI}\left(\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), n\right), \mathsf{PI}\left(\right)\right)\right) \]

   7. PI-lowering-PI.f64 38.3%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), n\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

9. Applied egg-rr 38.3%

   \[\leadsto \sqrt{\color{blue}{\left(\frac{2}{k} \cdot n\right) \cdot \pi}} \]
10. Step-by-step derivation

    1. associate-*l* N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{k} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{k} \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot \frac{2}{k}\right)\right) \]

    4. clear-num N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot \frac{1}{\frac{k}{2}}\right)\right) \]

    5. associate-/r/ N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot \left(\frac{1}{k} \cdot 2\right)\right)\right) \]

    6. associate-*r* N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot \frac{1}{k}\right) \cdot 2\right)\right) \]

    7. div-inv N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2\right)\right) \]

    8. *-lft-identity N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k} \cdot 2\right)\right) \]

    9. *-rgt-identity N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k \cdot 1} \cdot 2\right)\right) \]

    10. frac-times N/A

        \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{k} \cdot \frac{\mathsf{PI}\left(\right) \cdot n}{1}\right) \cdot 2\right)\right) \]

    11. /-rgt-identity N/A

        \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{k} \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right) \cdot 2\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{k} \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), 2\right)\right) \]

    13. /-rgt-identity N/A

        \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{k} \cdot \frac{\mathsf{PI}\left(\right) \cdot n}{1}\right), 2\right)\right) \]

    14. frac-times N/A

        \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{1 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k \cdot 1}\right), 2\right)\right) \]

    15. *-lft-identity N/A

        \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot n}{k \cdot 1}\right), 2\right)\right) \]

    16. *-rgt-identity N/A

        \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot n}{k}\right), 2\right)\right) \]

    17. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot n\right), k\right), 2\right)\right) \]

    18. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right), k\right), 2\right)\right) \]

    19. PI-lowering-PI.f64 38.3%

        \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right), k\right), 2\right)\right) \]

11. Applied egg-rr 38.3%

    \[\leadsto \sqrt{\color{blue}{\frac{\pi \cdot n}{k} \cdot 2}} \]

12. Final simplification 38.3%

    \[\leadsto \sqrt{2 \cdot \frac{\pi \cdot n}{k}} \]
13. Add Preprocessing

Alternative 10: 37.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\pi \cdot \frac{2 \cdot n}{k}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (* PI (/ (* 2.0 n) k))))

C

double code(double k, double n) {
	return sqrt((((double) M_PI) * ((2.0 * n) / k)));
}

Java

public static double code(double k, double n) {
	return Math.sqrt((Math.PI * ((2.0 * n) / k)));
}

Python

def code(k, n):
	return math.sqrt((math.pi * ((2.0 * n) / k)))

Julia

function code(k, n)
	return sqrt(Float64(pi * Float64(Float64(2.0 * n) / k)))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt((pi * ((2.0 * n) / k)));
end

Wolfram

code[k_, n_] := N[Sqrt[N[(Pi * N[(N[(2.0 * n), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 38.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 38.2%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
6. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}} \cdot 2\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1 \cdot 2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{n \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{\mathsf{PI}\left(\right) \cdot n}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right)\right) \]

   10. PI-lowering-PI.f64 38.3%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right) \]

7. Applied egg-rr 38.3%

   \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{k}{\pi \cdot n}}}} \]
8. Step-by-step derivation

   1. associate-/r/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{k} \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{k} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{2}{k} \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k} \cdot n\right), \mathsf{PI}\left(\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), n\right), \mathsf{PI}\left(\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), n\right), \mathsf{PI}\left(\right)\right)\right) \]

   7. PI-lowering-PI.f64 38.3%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), n\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

9. Applied egg-rr 38.3%

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

10. Taylor expanded in k around 0

    \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \frac{n}{k}\right)}, \mathsf{PI.f64}\left(\right)\right)\right) \]
11. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 \cdot n}{k}\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot n\right), k\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(n \cdot 2\right), k\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

    4. *-lowering-*.f64 38.3%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, 2\right), k\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

12. Simplified 38.3%

    \[\leadsto \sqrt{\color{blue}{\frac{n \cdot 2}{k}} \cdot \pi} \]

13. Final simplification 38.3%

    \[\leadsto \sqrt{\pi \cdot \frac{2 \cdot n}{k}} \]
14. Add Preprocessing

Alternative 11: 37.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\pi \cdot \frac{2}{\frac{k}{n}}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (* PI (/ 2.0 (/ k n)))))

C

double code(double k, double n) {
	return sqrt((((double) M_PI) * (2.0 / (k / n))));
}

Java

public static double code(double k, double n) {
	return Math.sqrt((Math.PI * (2.0 / (k / n))));
}

Python

def code(k, n):
	return math.sqrt((math.pi * (2.0 / (k / n))))

Julia

function code(k, n)
	return sqrt(Float64(pi * Float64(2.0 / Float64(k / n))))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt((pi * (2.0 / (k / n))));
end

Wolfram

code[k_, n_] := N[Sqrt[N[(Pi * N[(2.0 / N[(k / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 38.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 38.2%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
6. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}} \cdot 2\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1 \cdot 2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{n \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{\mathsf{PI}\left(\right) \cdot n}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right)\right) \]

   10. PI-lowering-PI.f64 38.3%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right) \]

7. Applied egg-rr 38.3%

   \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{k}{\pi \cdot n}}}} \]
8. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{\frac{k}{n}}{\mathsf{PI}\left(\right)}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{k}{n}} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{\frac{k}{n}}\right), \mathsf{PI}\left(\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{n}\right)\right), \mathsf{PI}\left(\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, n\right)\right), \mathsf{PI}\left(\right)\right)\right) \]

   6. PI-lowering-PI.f64 38.3%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, n\right)\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

9. Applied egg-rr 38.3%

   \[\leadsto \sqrt{\color{blue}{\frac{2}{\frac{k}{n}} \cdot \pi}} \]

10. Final simplification 38.3%

    \[\leadsto \sqrt{\pi \cdot \frac{2}{\frac{k}{n}}} \]
11. Add Preprocessing

Alternative 12: 37.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{n \cdot \left(\pi \cdot \frac{2}{k}\right)} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (* n (* PI (/ 2.0 k)))))

C

double code(double k, double n) {
	return sqrt((n * (((double) M_PI) * (2.0 / k))));
}

Java

public static double code(double k, double n) {
	return Math.sqrt((n * (Math.PI * (2.0 / k))));
}

Python

def code(k, n):
	return math.sqrt((n * (math.pi * (2.0 / k))))

Julia

function code(k, n)
	return sqrt(Float64(n * Float64(pi * Float64(2.0 / k))))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt((n * (pi * (2.0 / k))));
end

Wolfram

code[k_, n_] := N[Sqrt[N[(n * N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 38.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 38.2%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
6. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}} \cdot 2\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1 \cdot 2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{n \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{\mathsf{PI}\left(\right) \cdot n}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right)\right) \]

   10. PI-lowering-PI.f64 38.3%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right) \]

7. Applied egg-rr 38.3%

   \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{k}{\pi \cdot n}}}} \]
8. Step-by-step derivation

   1. associate-/r/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{k} \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{2}{k} \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k} \cdot \mathsf{PI}\left(\right)\right), n\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), \mathsf{PI}\left(\right)\right), n\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{PI}\left(\right)\right), n\right)\right) \]

   6. PI-lowering-PI.f64 38.3%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{PI.f64}\left(\right)\right), n\right)\right) \]

9. Applied egg-rr 38.3%

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

10. Final simplification 38.3%

    \[\leadsto \sqrt{n \cdot \left(\pi \cdot \frac{2}{k}\right)} \]
11. Add Preprocessing

Alternative 13: 37.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\pi \cdot \left(n \cdot \frac{2}{k}\right)} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (* PI (* n (/ 2.0 k)))))

C

double code(double k, double n) {
	return sqrt((((double) M_PI) * (n * (2.0 / k))));
}

Java

public static double code(double k, double n) {
	return Math.sqrt((Math.PI * (n * (2.0 / k))));
}

Python

def code(k, n):
	return math.sqrt((math.pi * (n * (2.0 / k))))

Julia

function code(k, n)
	return sqrt(Float64(pi * Float64(n * Float64(2.0 / k))))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt((pi * (n * (2.0 / k))));
end

Wolfram

code[k_, n_] := N[Sqrt[N[(Pi * N[(n * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 38.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 38.2%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
6. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2} \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}} \cdot 2\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1 \cdot 2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{k}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{n \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{\mathsf{PI}\left(\right) \cdot n}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right)\right) \]

   10. PI-lowering-PI.f64 38.3%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right) \]

7. Applied egg-rr 38.3%

   \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{k}{\pi \cdot n}}}} \]
8. Step-by-step derivation

   1. associate-/r/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{k} \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{k} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{2}{k} \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k} \cdot n\right), \mathsf{PI}\left(\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), n\right), \mathsf{PI}\left(\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), n\right), \mathsf{PI}\left(\right)\right)\right) \]

   7. PI-lowering-PI.f64 38.3%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), n\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

9. Applied egg-rr 38.3%

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

10. Final simplification 38.3%

    \[\leadsto \sqrt{\pi \cdot \left(n \cdot \frac{2}{k}\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))