Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 11.7s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. k = 1.0698509585460927e+288
  2. n = -6.6147412124811755e-31

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. div-sub N/A

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

   4. metadata-eval N/A

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

   5. pow-sub N/A

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

   6. associate-/l/ N/A

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

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

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

   8. unpow1/2 N/A

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

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

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

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

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

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

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

   15. *-commutative N/A

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

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

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

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

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

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

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

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

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

4. Applied egg-rr 99.7%

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

Alternative 2: 54.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.7 \cdot 10^{+165}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\pi}{k \cdot \frac{k}{\pi}} \cdot \left(4 \cdot \left(n \cdot n\right)\right)\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6.7e+165)
   (/ (sqrt (* PI (* 2.0 n))) (sqrt k))
   (pow (* (/ PI (* k (/ k PI))) (* 4.0 (* n n))) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6.7e+165) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) / sqrt(k);
	} else {
		tmp = pow(((((double) M_PI) / (k * (k / ((double) M_PI)))) * (4.0 * (n * n))), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 6.7e+165) {
		tmp = Math.sqrt((Math.PI * (2.0 * n))) / Math.sqrt(k);
	} else {
		tmp = Math.pow(((Math.PI / (k * (k / Math.PI))) * (4.0 * (n * n))), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 6.7e+165:
		tmp = math.sqrt((math.pi * (2.0 * n))) / math.sqrt(k)
	else:
		tmp = math.pow(((math.pi / (k * (k / math.pi))) * (4.0 * (n * n))), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 6.7e+165)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) / sqrt(k));
	else
		tmp = Float64(Float64(pi / Float64(k * Float64(k / pi))) * Float64(4.0 * Float64(n * n))) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 6.7e+165)
		tmp = sqrt((pi * (2.0 * n))) / sqrt(k);
	else
		tmp = ((pi / (k * (k / pi))) * (4.0 * (n * n))) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6.7e+165], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(Pi / N[(k * N[(k / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.70000000000000037e165

   1. Initial program 99.1%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 52.7%

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

   5. Simplified 52.7%

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

      1. sqrt-unprod N/A

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

      2. associate-*l/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. sqrt-undiv N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. pow-flip N/A

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

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

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

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

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

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

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

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

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

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

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

      15. pow-flip N/A

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

      16. metadata-eval N/A

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

      17. pow1/2 N/A

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

      18. sqrt-lowering-sqrt.f64 67.5%

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

   7. Applied egg-rr 67.5%

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

   if 6.70000000000000037e165 < 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 2.9%

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

   5. Simplified 2.9%

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

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

      8. *-commutative 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 2.9%

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

   7. Applied egg-rr 2.9%

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

      1. associate-/r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. PI-lowering-PI.f64 2.9%

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

   9. Applied egg-rr 2.9%

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

       1. pow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*l/ N/A

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

       4. associate-*r/ N/A

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

       5. associate-*l/ N/A

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

       6. pow-prod-down N/A

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

       7. sqr-pow N/A

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

       8. sqr-pow N/A

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

       9. metadata-eval N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. pow-prod-down N/A

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

       14. metadata-eval N/A

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

       15. metadata-eval N/A

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

       16. metadata-eval N/A

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

       17. metadata-eval N/A

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

   11. Applied egg-rr 37.1%

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

Alternative 3: 54.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.7 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{\frac{2}{\frac{k}{\pi}}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\pi}{k \cdot \frac{k}{\pi}} \cdot \left(4 \cdot \left(n \cdot n\right)\right)\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6.7e+165)
   (* (sqrt (/ 2.0 (/ k PI))) (sqrt n))
   (pow (* (/ PI (* k (/ k PI))) (* 4.0 (* n n))) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6.7e+165) {
		tmp = sqrt((2.0 / (k / ((double) M_PI)))) * sqrt(n);
	} else {
		tmp = pow(((((double) M_PI) / (k * (k / ((double) M_PI)))) * (4.0 * (n * n))), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 6.7e+165) {
		tmp = Math.sqrt((2.0 / (k / Math.PI))) * Math.sqrt(n);
	} else {
		tmp = Math.pow(((Math.PI / (k * (k / Math.PI))) * (4.0 * (n * n))), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 6.7e+165:
		tmp = math.sqrt((2.0 / (k / math.pi))) * math.sqrt(n)
	else:
		tmp = math.pow(((math.pi / (k * (k / math.pi))) * (4.0 * (n * n))), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 6.7e+165)
		tmp = Float64(sqrt(Float64(2.0 / Float64(k / pi))) * sqrt(n));
	else
		tmp = Float64(Float64(pi / Float64(k * Float64(k / pi))) * Float64(4.0 * Float64(n * n))) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 6.7e+165)
		tmp = sqrt((2.0 / (k / pi))) * sqrt(n);
	else
		tmp = ((pi / (k * (k / pi))) * (4.0 * (n * n))) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6.7e+165], N[(N[Sqrt[N[(2.0 / N[(k / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(Pi / N[(k * N[(k / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.70000000000000037e165

   1. Initial program 99.1%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 52.7%

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

   5. Simplified 52.7%

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

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

      8. *-commutative 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 52.8%

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

   7. Applied egg-rr 52.8%

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. PI-lowering-PI.f64 52.7%

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

   9. Applied egg-rr 52.7%

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. associate-*r/ N/A

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

       4. associate-*l/ N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. associate-/r/ N/A

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

       8. *-commutative N/A

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

       9. associate-/r/ N/A

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

       10. associate-*r* N/A

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

       11. sqrt-prod N/A

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

       12. pow1/2 N/A

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

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

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

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

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

       15. clear-num N/A

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

       16. un-div-inv N/A

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

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

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

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

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

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

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

       20. pow1/2 N/A

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

       21. sqrt-lowering-sqrt.f64 67.3%

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

   11. Applied egg-rr 67.3%

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

   if 6.70000000000000037e165 < 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 2.9%

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

   5. Simplified 2.9%

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

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

      8. *-commutative 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 2.9%

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

   7. Applied egg-rr 2.9%

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

      1. associate-/r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. PI-lowering-PI.f64 2.9%

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

   9. Applied egg-rr 2.9%

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

       1. pow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*l/ N/A

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

       4. associate-*r/ N/A

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

       5. associate-*l/ N/A

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

       6. pow-prod-down N/A

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

       7. sqr-pow N/A

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

       8. sqr-pow N/A

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

       9. metadata-eval N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. pow-prod-down N/A

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

       14. metadata-eval N/A

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

       15. metadata-eval N/A

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

       16. metadata-eval N/A

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

       17. metadata-eval N/A

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

   11. Applied egg-rr 37.1%

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

Alternative 4: 54.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.7 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\pi}{k \cdot \frac{k}{\pi}} \cdot \left(4 \cdot \left(n \cdot n\right)\right)\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6.7e+165)
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
   (pow (* (/ PI (* k (/ k PI))) (* 4.0 (* n n))) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6.7e+165) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = pow(((((double) M_PI) / (k * (k / ((double) M_PI)))) * (4.0 * (n * n))), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 6.7e+165) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	} else {
		tmp = Math.pow(((Math.PI / (k * (k / Math.PI))) * (4.0 * (n * n))), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 6.7e+165:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (math.pi / k)))
	else:
		tmp = math.pow(((math.pi / (k * (k / math.pi))) * (4.0 * (n * n))), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 6.7e+165)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = Float64(Float64(pi / Float64(k * Float64(k / pi))) * Float64(4.0 * Float64(n * n))) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 6.7e+165)
		tmp = sqrt(n) * sqrt((2.0 * (pi / k)));
	else
		tmp = ((pi / (k * (k / pi))) * (4.0 * (n * n))) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6.7e+165], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(Pi / N[(k * N[(k / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.70000000000000037e165

   1. Initial program 99.1%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 52.7%

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

   5. Simplified 52.7%

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

      1. sqrt-unprod N/A

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

      2. 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 67.3%

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

   7. Applied egg-rr 67.3%

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

   if 6.70000000000000037e165 < 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 2.9%

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

   5. Simplified 2.9%

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

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

      8. *-commutative 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 2.9%

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

   7. Applied egg-rr 2.9%

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

      1. associate-/r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. PI-lowering-PI.f64 2.9%

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

   9. Applied egg-rr 2.9%

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

       1. pow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*l/ N/A

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

       4. associate-*r/ N/A

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

       5. associate-*l/ N/A

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

       6. pow-prod-down N/A

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

       7. sqr-pow N/A

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

       8. sqr-pow N/A

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

       9. metadata-eval N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. pow-prod-down N/A

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

       14. metadata-eval N/A

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

       15. metadata-eval N/A

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

       16. metadata-eval N/A

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

       17. metadata-eval N/A

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

   11. Applied egg-rr 37.1%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.7 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\pi}{k \cdot \frac{k}{\pi}} \cdot \left(4 \cdot \left(n \cdot n\right)\right)\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. div-sub N/A

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

   4. metadata-eval N/A

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

   5. pow-sub N/A

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

   6. associate-/l/ N/A

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

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

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

   8. unpow1/2 N/A

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

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

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

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

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

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

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

   15. *-commutative N/A

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

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

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

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

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

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

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

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

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

4. Applied egg-rr 99.7%

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

   1. pow1/2 N/A

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

   2. metadata-eval N/A

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

   3. pow-flip N/A

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

   4. *-commutative N/A

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

   5. associate-/r* N/A

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

   6. pow1/2 N/A

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

   7. pow-sub N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. div-sub N/A

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

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

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

6. Applied egg-rr 99.4%

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

Alternative 6: 42.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.7 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi \cdot n}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\pi}{k \cdot \frac{k}{\pi}} \cdot \left(4 \cdot \left(n \cdot n\right)\right)\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6.7e+165)
   (sqrt (* 2.0 (/ (* PI n) k)))
   (pow (* (/ PI (* k (/ k PI))) (* 4.0 (* n n))) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6.7e+165) {
		tmp = sqrt((2.0 * ((((double) M_PI) * n) / k)));
	} else {
		tmp = pow(((((double) M_PI) / (k * (k / ((double) M_PI)))) * (4.0 * (n * n))), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 6.7e+165) {
		tmp = Math.sqrt((2.0 * ((Math.PI * n) / k)));
	} else {
		tmp = Math.pow(((Math.PI / (k * (k / Math.PI))) * (4.0 * (n * n))), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 6.7e+165:
		tmp = math.sqrt((2.0 * ((math.pi * n) / k)))
	else:
		tmp = math.pow(((math.pi / (k * (k / math.pi))) * (4.0 * (n * n))), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 6.7e+165)
		tmp = sqrt(Float64(2.0 * Float64(Float64(pi * n) / k)));
	else
		tmp = Float64(Float64(pi / Float64(k * Float64(k / pi))) * Float64(4.0 * Float64(n * n))) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 6.7e+165)
		tmp = sqrt((2.0 * ((pi * n) / k)));
	else
		tmp = ((pi / (k * (k / pi))) * (4.0 * (n * n))) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6.7e+165], N[Sqrt[N[(2.0 * N[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(Pi / N[(k * N[(k / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.70000000000000037e165

   1. Initial program 99.1%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 52.7%

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

   5. Simplified 52.7%

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

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

      8. *-commutative 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 52.8%

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

   7. Applied egg-rr 52.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

      4. associate-*l/ N/A

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

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

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

      6. associate-*l/ N/A

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

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

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

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

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

      9. PI-lowering-PI.f64 52.8%

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

   9. Applied egg-rr 52.8%

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

   if 6.70000000000000037e165 < 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 2.9%

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

   5. Simplified 2.9%

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

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

      8. *-commutative 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 2.9%

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

   7. Applied egg-rr 2.9%

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

      1. associate-/r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. PI-lowering-PI.f64 2.9%

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

   9. Applied egg-rr 2.9%

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

       1. pow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*l/ N/A

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

       4. associate-*r/ N/A

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

       5. associate-*l/ N/A

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

       6. pow-prod-down N/A

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

       7. sqr-pow N/A

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

       8. sqr-pow N/A

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

       9. metadata-eval N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. pow-prod-down N/A

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

       14. metadata-eval N/A

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

       15. metadata-eval N/A

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

       16. metadata-eval N/A

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

       17. metadata-eval N/A

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

   11. Applied egg-rr 37.1%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.7 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi \cdot n}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\pi}{k \cdot \frac{k}{\pi}} \cdot \left(4 \cdot \left(n \cdot n\right)\right)\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 37.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 42.6%

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

5. Simplified 42.6%

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

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

   8. *-commutative 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 42.7%

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

7. Applied egg-rr 42.7%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. clear-num N/A

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

   4. associate-*l/ N/A

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

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

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

   6. associate-*l/ N/A

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

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

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

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

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

   9. PI-lowering-PI.f64 42.7%

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

9. Applied egg-rr 42.7%

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

10. Final simplification 42.7%

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

Alternative 8: 37.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 42.6%

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

5. Simplified 42.6%

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

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

   8. *-commutative 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 42.7%

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

7. Applied egg-rr 42.7%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. clear-num N/A

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

   4. associate-*l/ N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

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

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

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

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

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

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

   10. *-lowering-*.f64 42.7%

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

9. Applied egg-rr 42.7%

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

10. Final simplification 42.7%

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

Alternative 9: 37.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 42.6%

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

5. Simplified 42.6%

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

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

   8. *-commutative 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 42.7%

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

7. Applied egg-rr 42.7%

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

   1. associate-/r/ N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

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

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

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

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

   7. PI-lowering-PI.f64 42.6%

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

9. Applied egg-rr 42.6%

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

    1. *-commutative N/A

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

    2. clear-num N/A

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

    3. associate-*r/ N/A

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

    4. div-inv N/A

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

    5. metadata-eval N/A

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

    6. times-frac N/A

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

    7. metadata-eval N/A

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

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

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

    9. /-lowering-/.f64 42.7%

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

11. Applied egg-rr 42.7%

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

12. Final simplification 42.7%

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

Alternative 10: 37.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 42.6%

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

5. Simplified 42.6%

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

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

   8. *-commutative 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 42.7%

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

7. Applied egg-rr 42.7%

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

   1. associate-/r/ N/A

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

   6. PI-lowering-PI.f64 42.6%

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

9. Applied egg-rr 42.6%

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

10. Final simplification 42.6%

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

Alternative 11: 37.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 42.6%

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

5. Simplified 42.6%

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

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

   8. *-commutative 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 42.7%

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

7. Applied egg-rr 42.7%

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

   1. associate-/r/ N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

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

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

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

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

   7. PI-lowering-PI.f64 42.6%

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

9. Applied egg-rr 42.6%

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

10. Final simplification 42.6%

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

Reproduce

herbie shell --seed 2024194 
(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))))