Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 13.3s

Alternatives: 11

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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* 2.0 (* PI n)))) (/ (pow t_0 0.5) (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 pow(t_0, 0.5) / 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.pow(t_0, 0.5) / Math.pow((k * Math.pow(t_0, k)), 0.5);
}

Python

def code(k, n):
	t_0 = 2.0 * (math.pi * n)
	return math.pow(t_0, 0.5) / 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((t_0 ^ 0.5) / (Float64(k * (t_0 ^ k)) ^ 0.5))
end

MATLAB

function tmp = code(k, n)
	t_0 = 2.0 * (pi * n);
	tmp = (t_0 ^ 0.5) / ((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[Power[t$95$0, 0.5], $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. pow-lowering-pow.f64 N/A

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

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

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

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

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

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

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 62.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{4}{k \cdot k}\right)}^{0.25} \cdot {\left(\pi \cdot n\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5e-46)
   (/ (sqrt (* 2.0 (* PI n))) (sqrt k))
   (* (pow (/ 4.0 (* k k)) 0.25) (pow (* PI n) 0.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5e-46) {
		tmp = sqrt((2.0 * (((double) M_PI) * n))) / sqrt(k);
	} else {
		tmp = pow((4.0 / (k * k)), 0.25) * pow((((double) M_PI) * n), 0.5);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 5e-46) {
		tmp = Math.sqrt((2.0 * (Math.PI * n))) / Math.sqrt(k);
	} else {
		tmp = Math.pow((4.0 / (k * k)), 0.25) * Math.pow((Math.PI * n), 0.5);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 5e-46:
		tmp = math.sqrt((2.0 * (math.pi * n))) / math.sqrt(k)
	else:
		tmp = math.pow((4.0 / (k * k)), 0.25) * math.pow((math.pi * n), 0.5)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5e-46)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi * n))) / sqrt(k));
	else
		tmp = Float64((Float64(4.0 / Float64(k * k)) ^ 0.25) * (Float64(pi * n) ^ 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 5e-46)
		tmp = sqrt((2.0 * (pi * n))) / sqrt(k);
	else
		tmp = ((4.0 / (k * k)) ^ 0.25) * ((pi * n) ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5e-46], N[(N[Sqrt[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(4.0 / N[(k * k), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[N[(Pi * n), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.99999999999999992e-46

   1. Initial program 99.3%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 76.0%

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

   5. Simplified 76.0%

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. sqrt-div N/A

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

      6. unpow1/2 N/A

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

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

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

      8. unpow1/2 N/A

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

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

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

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

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

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

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

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

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

      13. sqrt-lowering-sqrt.f64 99.5%

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

   7. Applied egg-rr 99.5%

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

   if 4.99999999999999992e-46 < k

   1. Initial program 99.7%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 13.2%

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

   5. Simplified 13.2%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   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 13.2%

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

   7. Applied egg-rr 13.2%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{k}{\pi \cdot n}}}} \]
   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. associate-/r/ N/A

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

      3. unpow-prod-down N/A

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

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

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

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

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

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

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

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

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

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

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

      9. PI-lowering-PI.f64 13.3%

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

   9. Applied egg-rr 13.3%

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

       1. sqr-pow N/A

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

       2. pow-prod-down N/A

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

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

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

       4. frac-times N/A

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

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

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

       6. metadata-eval N/A

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

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

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

       8. metadata-eval 31.6%

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

   11. Applied egg-rr 31.6%

       \[\leadsto \color{blue}{{\left(\frac{4}{k \cdot k}\right)}^{0.25}} \cdot {\left(\pi \cdot n\right)}^{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 55.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{4}{k \cdot k} \cdot \left(\pi \cdot \left(n \cdot \left(\pi \cdot n\right)\right)\right)\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.25e+154)
   (/ (sqrt (* 2.0 (* PI n))) (sqrt k))
   (pow (* (/ 4.0 (* k k)) (* PI (* n (* PI n)))) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.25e+154) {
		tmp = sqrt((2.0 * (((double) M_PI) * n))) / sqrt(k);
	} else {
		tmp = pow(((4.0 / (k * k)) * (((double) M_PI) * (n * (((double) M_PI) * n)))), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.25e+154) {
		tmp = Math.sqrt((2.0 * (Math.PI * n))) / Math.sqrt(k);
	} else {
		tmp = Math.pow(((4.0 / (k * k)) * (Math.PI * (n * (Math.PI * n)))), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.25e+154:
		tmp = math.sqrt((2.0 * (math.pi * n))) / math.sqrt(k)
	else:
		tmp = math.pow(((4.0 / (k * k)) * (math.pi * (n * (math.pi * n)))), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.25e+154)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi * n))) / sqrt(k));
	else
		tmp = Float64(Float64(4.0 / Float64(k * k)) * Float64(pi * Float64(n * Float64(pi * n)))) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.25e+154)
		tmp = sqrt((2.0 * (pi * n))) / sqrt(k);
	else
		tmp = ((4.0 / (k * k)) * (pi * (n * (pi * n)))) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.25e+154], N[(N[Sqrt[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(4.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(n * N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.25000000000000005e154

   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 49.5%

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

   5. Simplified 49.5%

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. sqrt-div N/A

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

      6. unpow1/2 N/A

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

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

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

      8. unpow1/2 N/A

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

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

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

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

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

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

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

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

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

      13. sqrt-lowering-sqrt.f64 61.9%

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

   7. Applied egg-rr 61.9%

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

   if 2.25000000000000005e154 < 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. associate-/r/ N/A

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

      3. unpow-prod-down N/A

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

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

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

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

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

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

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

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

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

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

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

      9. PI-lowering-PI.f64 2.7%

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

   9. Applied egg-rr 2.7%

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

       1. sqr-pow N/A

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

       2. pow-prod-down N/A

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

       3. sqr-pow N/A

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

       4. pow-prod-down N/A

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

       5. pow-prod-down N/A

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

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

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

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

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

       8. frac-times N/A

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

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

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

       10. metadata-eval N/A

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

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

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

       12. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

       18. metadata-eval 24.5%

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

   11. Applied egg-rr 24.5%

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

Alternative 4: 55.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{2}{\frac{k}{\pi}}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{4}{k \cdot k} \cdot \left(\pi \cdot \left(n \cdot \left(\pi \cdot n\right)\right)\right)\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.25e+154)
   (* (sqrt (/ 2.0 (/ k PI))) (sqrt n))
   (pow (* (/ 4.0 (* k k)) (* PI (* n (* PI n)))) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.25e+154) {
		tmp = sqrt((2.0 / (k / ((double) M_PI)))) * sqrt(n);
	} else {
		tmp = pow(((4.0 / (k * k)) * (((double) M_PI) * (n * (((double) M_PI) * n)))), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.25e+154) {
		tmp = Math.sqrt((2.0 / (k / Math.PI))) * Math.sqrt(n);
	} else {
		tmp = Math.pow(((4.0 / (k * k)) * (Math.PI * (n * (Math.PI * n)))), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.25e+154:
		tmp = math.sqrt((2.0 / (k / math.pi))) * math.sqrt(n)
	else:
		tmp = math.pow(((4.0 / (k * k)) * (math.pi * (n * (math.pi * n)))), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.25e+154)
		tmp = Float64(sqrt(Float64(2.0 / Float64(k / pi))) * sqrt(n));
	else
		tmp = Float64(Float64(4.0 / Float64(k * k)) * Float64(pi * Float64(n * Float64(pi * n)))) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.25e+154)
		tmp = sqrt((2.0 / (k / pi))) * sqrt(n);
	else
		tmp = ((4.0 / (k * k)) * (pi * (n * (pi * n)))) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.25e+154], N[(N[Sqrt[N[(2.0 / N[(k / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(4.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(n * N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.25000000000000005e154

   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 49.5%

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

   5. Simplified 49.5%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   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 49.6%

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

   7. Applied egg-rr 49.6%

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

      1. associate-/r/ N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

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

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

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

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

      7. associate-/r/ N/A

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

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

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

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

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

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

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

      11. pow1/2 N/A

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

      12. sqrt-lowering-sqrt.f64 61.9%

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

   9. Applied egg-rr 61.9%

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

   if 2.25000000000000005e154 < 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. associate-/r/ N/A

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

      3. unpow-prod-down N/A

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

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

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

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

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

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

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

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

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

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

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

      9. PI-lowering-PI.f64 2.7%

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

   9. Applied egg-rr 2.7%

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

       1. sqr-pow N/A

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

       2. pow-prod-down N/A

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

       3. sqr-pow N/A

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

       4. pow-prod-down N/A

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

       5. pow-prod-down N/A

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

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

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

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

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

       8. frac-times N/A

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

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

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

       10. metadata-eval N/A

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

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

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

       12. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

       18. metadata-eval 24.5%

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

   11. Applied egg-rr 24.5%

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

Alternative 5: 55.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{4}{k \cdot k} \cdot \left(\pi \cdot \left(n \cdot \left(\pi \cdot n\right)\right)\right)\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.25e+154)
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
   (pow (* (/ 4.0 (* k k)) (* PI (* n (* PI n)))) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.25e+154) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = pow(((4.0 / (k * k)) * (((double) M_PI) * (n * (((double) M_PI) * n)))), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.25e+154) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	} else {
		tmp = Math.pow(((4.0 / (k * k)) * (Math.PI * (n * (Math.PI * n)))), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.25e+154:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (math.pi / k)))
	else:
		tmp = math.pow(((4.0 / (k * k)) * (math.pi * (n * (math.pi * n)))), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.25e+154)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = Float64(Float64(4.0 / Float64(k * k)) * Float64(pi * Float64(n * Float64(pi * n)))) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.25e+154)
		tmp = sqrt(n) * sqrt((2.0 * (pi / k)));
	else
		tmp = ((4.0 / (k * k)) * (pi * (n * (pi * n)))) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.25e+154], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(4.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(n * N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.25000000000000005e154

   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 49.5%

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

   5. Simplified 49.5%

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

      1. sqrt-unprod N/A

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

      2. 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 61.9%

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

   7. Applied egg-rr 61.9%

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

   if 2.25000000000000005e154 < 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. associate-/r/ N/A

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

      3. unpow-prod-down N/A

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

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

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

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

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

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

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

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

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

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

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

      9. PI-lowering-PI.f64 2.7%

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

   9. Applied egg-rr 2.7%

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

       1. sqr-pow N/A

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

       2. pow-prod-down N/A

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

       3. sqr-pow N/A

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

       4. pow-prod-down N/A

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

       5. pow-prod-down N/A

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

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

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

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

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

       8. frac-times N/A

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

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

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

       10. metadata-eval N/A

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

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

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

       12. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

       18. metadata-eval 24.5%

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

   11. Applied egg-rr 24.5%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.25 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{4}{k \cdot k} \cdot \left(\pi \cdot \left(n \cdot \left(\pi \cdot n\right)\right)\right)\right)}^{0.25}\\ \end{array} \]
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 99.6%

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

3. Simplified 99.6%

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

Alternative 7: 44.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{\left(\frac{k}{n \cdot \left(2 \cdot \pi\right)}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{4}{k \cdot k} \cdot \left(\pi \cdot \left(n \cdot \left(\pi \cdot n\right)\right)\right)\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.35e+154)
   (pow (/ k (* n (* 2.0 PI))) -0.5)
   (pow (* (/ 4.0 (* k k)) (* PI (* n (* PI n)))) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.35e+154) {
		tmp = pow((k / (n * (2.0 * ((double) M_PI)))), -0.5);
	} else {
		tmp = pow(((4.0 / (k * k)) * (((double) M_PI) * (n * (((double) M_PI) * n)))), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.35e+154) {
		tmp = Math.pow((k / (n * (2.0 * Math.PI))), -0.5);
	} else {
		tmp = Math.pow(((4.0 / (k * k)) * (Math.PI * (n * (Math.PI * n)))), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.35e+154:
		tmp = math.pow((k / (n * (2.0 * math.pi))), -0.5)
	else:
		tmp = math.pow(((4.0 / (k * k)) * (math.pi * (n * (math.pi * n)))), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.35e+154)
		tmp = Float64(k / Float64(n * Float64(2.0 * pi))) ^ -0.5;
	else
		tmp = Float64(Float64(4.0 / Float64(k * k)) * Float64(pi * Float64(n * Float64(pi * n)))) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.35e+154)
		tmp = (k / (n * (2.0 * pi))) ^ -0.5;
	else
		tmp = ((4.0 / (k * k)) * (pi * (n * (pi * n)))) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.35e+154], N[Power[N[(k / N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[Power[N[(N[(4.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(n * N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.35000000000000003e154

   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 49.8%

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

   5. Simplified 49.8%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   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 49.9%

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

   7. Applied egg-rr 49.9%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. sqrt-pow1 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-/l/ N/A

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. metadata-eval 50.1%

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

   9. Applied egg-rr 50.1%

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

   if 1.35000000000000003e154 < 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 2.5%

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

   5. Simplified 2.5%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   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 2.5%

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

   7. Applied egg-rr 2.5%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{k}{\pi \cdot n}}}} \]
   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. associate-/r/ N/A

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

      3. unpow-prod-down N/A

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

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

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

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

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

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

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

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

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

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

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

      9. PI-lowering-PI.f64 2.7%

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

   9. Applied egg-rr 2.7%

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

       1. sqr-pow N/A

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

       2. pow-prod-down N/A

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

       3. sqr-pow N/A

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

       4. pow-prod-down N/A

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

       5. pow-prod-down N/A

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

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

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

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

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

       8. frac-times N/A

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

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

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

       10. metadata-eval N/A

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

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

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

       12. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

       18. metadata-eval 24.1%

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

   11. Applied egg-rr 24.1%

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

Alternative 8: 39.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Power[N[(k / N[(n * N[(2.0 * Pi), $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. 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 38.7%

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

5. Simplified 38.7%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
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 38.8%

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

7. Applied egg-rr 38.8%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. sqrt-pow1 N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. associate-/l/ N/A

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

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

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

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

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

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

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

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

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

   14. metadata-eval 39.0%

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

9. Applied egg-rr 39.0%

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

Alternative 9: 38.8% accurate, 2.0× speedup

Math

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

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(Float64(2.0 * pi) * Float64(n / k)))
end

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(2.0 * Pi), $MachinePrecision] * N[(n / 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 38.7%

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

5. Simplified 38.7%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
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 38.8%

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

7. Applied egg-rr 38.8%

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

   1. div-inv N/A

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

   2. clear-num N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

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

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

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

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

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

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

   9. PI-lowering-PI.f64 38.9%

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

9. Applied egg-rr 38.9%

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

10. Final simplification 38.9%

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

Alternative 10: 38.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(n * N[(2.0 / N[(k / Pi), $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 38.7%

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

5. Simplified 38.7%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
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 38.8%

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

7. Applied egg-rr 38.8%

   \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{k}{\pi \cdot n}}}} \]
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. associate-/r/ N/A

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

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

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

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

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

   7. PI-lowering-PI.f64 38.8%

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

9. Applied egg-rr 38.8%

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

10. Final simplification 38.8%

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

Alternative 11: 38.7% accurate, 2.0× speedup

Math

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

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(Float64(pi * 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[(N[(Pi * n), $MachinePrecision] * N[(2.0 / 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 38.7%

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

5. Simplified 38.7%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
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 38.8%

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

7. Applied egg-rr 38.8%

   \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{k}{\pi \cdot n}}}} \]
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. *-lowering-*.f64 N/A

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

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

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

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

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

   5. PI-lowering-PI.f64 38.8%

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

9. Applied egg-rr 38.8%

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

10. Final simplification 38.8%

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

Reproduce

herbie shell --seed 2024152 
(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))))