Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 11.8s

Alternatives: 11

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 6.597597552727012e+64
  2. n = -4.774950306878296e+218

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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. div-sub N/A

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

   4. metadata-eval N/A

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

   5. pow-sub N/A

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

   6. associate-/l/ N/A

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

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

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

   8. unpow1/2 N/A

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

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

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

   10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

4. Applied egg-rr 99.7%

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

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

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

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

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

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

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

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

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

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

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

   8. pow-unpow N/A

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

   9. unpow1/2 N/A

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

   10. sqrt-unprod N/A

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

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

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

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

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

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

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

   14. *-commutative N/A

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

   15. associate-*l* N/A

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

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

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

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

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

   18. *-lowering-*.f64 99.7

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

6. Applied egg-rr 99.7%

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

Alternative 2: 67.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{{\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}^{-0.25}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot \left(n \cdot 2\right)} \cdot \sqrt{\frac{1}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* (/ 1.0 (sqrt k)) (pow (* n (* PI 2.0)) (/ (- 1.0 k) 2.0))) 0.0)
   (sqrt (pow (* k (* k (* k k))) -0.25))
   (* (sqrt (* PI (* n 2.0))) (sqrt (/ 1.0 k)))))

C

double code(double k, double n) {
	double tmp;
	if (((1.0 / sqrt(k)) * pow((n * (((double) M_PI) * 2.0)), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = sqrt(pow((k * (k * (k * k))), -0.25));
	} else {
		tmp = sqrt((((double) M_PI) * (n * 2.0))) * sqrt((1.0 / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (((1.0 / Math.sqrt(k)) * Math.pow((n * (Math.PI * 2.0)), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = Math.sqrt(Math.pow((k * (k * (k * k))), -0.25));
	} else {
		tmp = Math.sqrt((Math.PI * (n * 2.0))) * Math.sqrt((1.0 / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if ((1.0 / math.sqrt(k)) * math.pow((n * (math.pi * 2.0)), ((1.0 - k) / 2.0))) <= 0.0:
		tmp = math.sqrt(math.pow((k * (k * (k * k))), -0.25))
	else:
		tmp = math.sqrt((math.pi * (n * 2.0))) * math.sqrt((1.0 / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(k)) * (Float64(n * Float64(pi * 2.0)) ^ Float64(Float64(1.0 - k) / 2.0))) <= 0.0)
		tmp = sqrt((Float64(k * Float64(k * Float64(k * k))) ^ -0.25));
	else
		tmp = Float64(sqrt(Float64(pi * Float64(n * 2.0))) * sqrt(Float64(1.0 / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (((1.0 / sqrt(k)) * ((n * (pi * 2.0)) ^ ((1.0 - k) / 2.0))) <= 0.0)
		tmp = sqrt(((k * (k * (k * k))) ^ -0.25));
	else
		tmp = sqrt((pi * (n * 2.0))) * sqrt((1.0 / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[Sqrt[N[Power[N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.25], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 0.0

   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 inf

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\color{blue}{\left(\frac{-1}{2} \cdot k\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 100.0

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   5. Simplified 100.0%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]

      2. /-lowering-/.f64 3.8

         \[\leadsto \sqrt{\color{blue}{\frac{1}{k}}} \]

   8. Simplified 3.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k}} \cdot \sqrt{\frac{1}{k}}}} \]

      2. sqrt-div N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{1}}{\sqrt{k}}} \cdot \sqrt{\frac{1}{k}}} \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{1}}{\sqrt{k}} \cdot \sqrt{\frac{1}{k}}} \]

      4. sqrt-div N/A

         \[\leadsto \sqrt{\frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{k}}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{\sqrt{k}} \cdot \frac{\color{blue}{1}}{\sqrt{k}}} \]

      6. frac-times N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot 1}{\sqrt{k} \cdot \sqrt{k}}}} \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{1}}{\sqrt{k} \cdot \sqrt{k}}} \]

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

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt{k} \cdot \sqrt{k}}}} \]

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

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\sqrt{k} \cdot \sqrt{k}}}} \]

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

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\sqrt{k}} \cdot \sqrt{k}}} \]

      11. sqrt-lowering-sqrt.f64 3.8

          \[\leadsto \sqrt{\frac{1}{\sqrt{k} \cdot \color{blue}{\sqrt{k}}}} \]

   10. Applied egg-rr 3.8%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt{k} \cdot \sqrt{k}}}} \]
   11. Step-by-step derivation

       1. inv-pow N/A

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

       2. sqrt-unprod N/A

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

       3. sqrt-pow2 N/A

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

       4. rem-square-sqrt N/A

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

       5. sqrt-unprod N/A

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

       6. metadata-eval N/A

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

       7. sqrt-pow2 N/A

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

       8. metadata-eval N/A

          \[\leadsto \sqrt{{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}^{\color{blue}{\frac{-1}{4}}}} \]

       9. metadata-eval N/A

          \[\leadsto \sqrt{{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}}} \]

       10. metadata-eval N/A

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

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

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

       12. associate-*l* N/A

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

       13. cube-mult N/A

           \[\leadsto \sqrt{{\left(k \cdot \color{blue}{{k}^{3}}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{2}\right)\right)}} \]

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

           \[\leadsto \sqrt{{\color{blue}{\left(k \cdot {k}^{3}\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{2}\right)\right)}} \]

       15. cube-mult N/A

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

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

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

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

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

       18. metadata-eval N/A

           \[\leadsto \sqrt{{\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right)}} \]

       19. metadata-eval 79.4

           \[\leadsto \sqrt{{\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}^{\color{blue}{-0.25}}} \]

   12. Applied egg-rr 79.4%

       \[\leadsto \sqrt{\color{blue}{{\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}^{-0.25}}} \]

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   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

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 50.0

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

   5. Simplified 50.0%

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. sqrt-undiv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. sqrt-div N/A

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

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

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

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

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

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

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

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

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

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

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

      18. *-lowering-*.f64 65.4

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

   7. Applied egg-rr 65.4%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{{\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}^{-0.25}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot \left(n \cdot 2\right)} \cdot \sqrt{\frac{1}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot \left(n \cdot 2\right)} \cdot \sqrt{\frac{1}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* (/ 1.0 (sqrt k)) (pow (* n (* PI 2.0)) (/ (- 1.0 k) 2.0))) 0.0)
   (sqrt (sqrt (/ 1.0 (* k k))))
   (* (sqrt (* PI (* n 2.0))) (sqrt (/ 1.0 k)))))

C

double code(double k, double n) {
	double tmp;
	if (((1.0 / sqrt(k)) * pow((n * (((double) M_PI) * 2.0)), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = sqrt(sqrt((1.0 / (k * k))));
	} else {
		tmp = sqrt((((double) M_PI) * (n * 2.0))) * sqrt((1.0 / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (((1.0 / Math.sqrt(k)) * Math.pow((n * (Math.PI * 2.0)), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = Math.sqrt(Math.sqrt((1.0 / (k * k))));
	} else {
		tmp = Math.sqrt((Math.PI * (n * 2.0))) * Math.sqrt((1.0 / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if ((1.0 / math.sqrt(k)) * math.pow((n * (math.pi * 2.0)), ((1.0 - k) / 2.0))) <= 0.0:
		tmp = math.sqrt(math.sqrt((1.0 / (k * k))))
	else:
		tmp = math.sqrt((math.pi * (n * 2.0))) * math.sqrt((1.0 / k))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (((1.0 / sqrt(k)) * ((n * (pi * 2.0)) ^ ((1.0 - k) / 2.0))) <= 0.0)
		tmp = sqrt(sqrt((1.0 / (k * k))));
	else
		tmp = sqrt((pi * (n * 2.0))) * sqrt((1.0 / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[Sqrt[N[Sqrt[N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 0.0

   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 inf

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\color{blue}{\left(\frac{-1}{2} \cdot k\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 100.0

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   5. Simplified 100.0%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]

      2. /-lowering-/.f64 3.8

         \[\leadsto \sqrt{\color{blue}{\frac{1}{k}}} \]

   8. Simplified 3.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k}} \cdot \sqrt{\frac{1}{k}}}} \]

      2. sqrt-div N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{1}}{\sqrt{k}}} \cdot \sqrt{\frac{1}{k}}} \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{1}}{\sqrt{k}} \cdot \sqrt{\frac{1}{k}}} \]

      4. sqrt-div N/A

         \[\leadsto \sqrt{\frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{k}}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{\sqrt{k}} \cdot \frac{\color{blue}{1}}{\sqrt{k}}} \]

      6. frac-times N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot 1}{\sqrt{k} \cdot \sqrt{k}}}} \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{1}}{\sqrt{k} \cdot \sqrt{k}}} \]

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

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt{k} \cdot \sqrt{k}}}} \]

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

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\sqrt{k} \cdot \sqrt{k}}}} \]

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

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\sqrt{k}} \cdot \sqrt{k}}} \]

      11. sqrt-lowering-sqrt.f64 3.8

          \[\leadsto \sqrt{\frac{1}{\sqrt{k} \cdot \color{blue}{\sqrt{k}}}} \]

   10. Applied egg-rr 3.8%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt{k} \cdot \sqrt{k}}}} \]
   11. Step-by-step derivation

       1. pow2 N/A

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

       2. pow-flip N/A

          \[\leadsto \sqrt{\color{blue}{{\left(\sqrt{k}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}} \]

       3. metadata-eval N/A

          \[\leadsto \sqrt{{\left(\sqrt{k}\right)}^{\color{blue}{-2}}} \]

       4. rem-square-sqrt N/A

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

       5. sqrt-unprod N/A

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

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

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

       7. sqrt-pow2 N/A

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

       8. metadata-eval N/A

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

       9. inv-pow N/A

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

       10. sqrt-pow2 N/A

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

       11. metadata-eval N/A

           \[\leadsto \sqrt{\sqrt{\frac{1}{k} \cdot {k}^{\color{blue}{-1}}}} \]

       12. inv-pow N/A

           \[\leadsto \sqrt{\sqrt{\frac{1}{k} \cdot \color{blue}{\frac{1}{k}}}} \]

       13. frac-times N/A

           \[\leadsto \sqrt{\sqrt{\color{blue}{\frac{1 \cdot 1}{k \cdot k}}}} \]

       14. metadata-eval N/A

           \[\leadsto \sqrt{\sqrt{\frac{\color{blue}{1}}{k \cdot k}}} \]

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

           \[\leadsto \sqrt{\sqrt{\color{blue}{\frac{1}{k \cdot k}}}} \]

       16. *-lowering-*.f64 50.9

           \[\leadsto \sqrt{\sqrt{\frac{1}{\color{blue}{k \cdot k}}}} \]

   12. Applied egg-rr 50.9%

       \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k \cdot k}}}} \]

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   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

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 50.0

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

   5. Simplified 50.0%

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. sqrt-undiv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. sqrt-div N/A

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

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

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

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

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

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

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

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

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

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

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

      18. *-lowering-*.f64 65.4

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

   7. Applied egg-rr 65.4%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot \left(n \cdot 2\right)} \cdot \sqrt{\frac{1}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot n}}{\sqrt{k \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* (/ 1.0 (sqrt k)) (pow (* n (* PI 2.0)) (/ (- 1.0 k) 2.0))) 0.0)
   (sqrt (sqrt (/ 1.0 (* k k))))
   (/ (sqrt (* PI n)) (sqrt (* k 0.5)))))

C

double code(double k, double n) {
	double tmp;
	if (((1.0 / sqrt(k)) * pow((n * (((double) M_PI) * 2.0)), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = sqrt(sqrt((1.0 / (k * k))));
	} else {
		tmp = sqrt((((double) M_PI) * n)) / sqrt((k * 0.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (((1.0 / Math.sqrt(k)) * Math.pow((n * (Math.PI * 2.0)), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = Math.sqrt(Math.sqrt((1.0 / (k * k))));
	} else {
		tmp = Math.sqrt((Math.PI * n)) / Math.sqrt((k * 0.5));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if ((1.0 / math.sqrt(k)) * math.pow((n * (math.pi * 2.0)), ((1.0 - k) / 2.0))) <= 0.0:
		tmp = math.sqrt(math.sqrt((1.0 / (k * k))))
	else:
		tmp = math.sqrt((math.pi * n)) / math.sqrt((k * 0.5))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[Sqrt[N[Sqrt[N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(Pi * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 0.0

   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 inf

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\color{blue}{\left(\frac{-1}{2} \cdot k\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 100.0

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   5. Simplified 100.0%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]

      2. /-lowering-/.f64 3.8

         \[\leadsto \sqrt{\color{blue}{\frac{1}{k}}} \]

   8. Simplified 3.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k}} \cdot \sqrt{\frac{1}{k}}}} \]

      2. sqrt-div N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{1}}{\sqrt{k}}} \cdot \sqrt{\frac{1}{k}}} \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{1}}{\sqrt{k}} \cdot \sqrt{\frac{1}{k}}} \]

      4. sqrt-div N/A

         \[\leadsto \sqrt{\frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{k}}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{\sqrt{k}} \cdot \frac{\color{blue}{1}}{\sqrt{k}}} \]

      6. frac-times N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot 1}{\sqrt{k} \cdot \sqrt{k}}}} \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{1}}{\sqrt{k} \cdot \sqrt{k}}} \]

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

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt{k} \cdot \sqrt{k}}}} \]

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

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\sqrt{k} \cdot \sqrt{k}}}} \]

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

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\sqrt{k}} \cdot \sqrt{k}}} \]

      11. sqrt-lowering-sqrt.f64 3.8

          \[\leadsto \sqrt{\frac{1}{\sqrt{k} \cdot \color{blue}{\sqrt{k}}}} \]

   10. Applied egg-rr 3.8%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt{k} \cdot \sqrt{k}}}} \]
   11. Step-by-step derivation

       1. pow2 N/A

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

       2. pow-flip N/A

          \[\leadsto \sqrt{\color{blue}{{\left(\sqrt{k}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}} \]

       3. metadata-eval N/A

          \[\leadsto \sqrt{{\left(\sqrt{k}\right)}^{\color{blue}{-2}}} \]

       4. rem-square-sqrt N/A

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

       5. sqrt-unprod N/A

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

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

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

       7. sqrt-pow2 N/A

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

       8. metadata-eval N/A

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

       9. inv-pow N/A

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

       10. sqrt-pow2 N/A

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

       11. metadata-eval N/A

           \[\leadsto \sqrt{\sqrt{\frac{1}{k} \cdot {k}^{\color{blue}{-1}}}} \]

       12. inv-pow N/A

           \[\leadsto \sqrt{\sqrt{\frac{1}{k} \cdot \color{blue}{\frac{1}{k}}}} \]

       13. frac-times N/A

           \[\leadsto \sqrt{\sqrt{\color{blue}{\frac{1 \cdot 1}{k \cdot k}}}} \]

       14. metadata-eval N/A

           \[\leadsto \sqrt{\sqrt{\frac{\color{blue}{1}}{k \cdot k}}} \]

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

           \[\leadsto \sqrt{\sqrt{\color{blue}{\frac{1}{k \cdot k}}}} \]

       16. *-lowering-*.f64 50.9

           \[\leadsto \sqrt{\sqrt{\frac{1}{\color{blue}{k \cdot k}}}} \]

   12. Applied egg-rr 50.9%

       \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k \cdot k}}}} \]

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   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

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 50.0

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

   5. Simplified 50.0%

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. sqrt-undiv N/A

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

      6. div-inv N/A

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

      7. sqrt-prod N/A

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

      8. pow1/2 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. div-inv N/A

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

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

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

      13. pow1/2 N/A

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

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

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

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

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

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

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

      17. sqrt-undiv N/A

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

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

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

      19. /-lowering-/.f64 65.4

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

   7. Applied egg-rr 65.4%

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

      1. sqrt-unprod N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. sqrt-div N/A

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

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

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

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

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

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

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

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

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

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

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lowering-*.f64 65.4

          \[\leadsto \frac{\sqrt{\pi \cdot n}}{\sqrt{\color{blue}{k \cdot 0.5}}} \]

   9. Applied egg-rr 65.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\pi \cdot n}}{\sqrt{k \cdot 0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot n}}{\sqrt{k \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* (/ 1.0 (sqrt k)) (pow (* n (* PI 2.0)) (/ (- 1.0 k) 2.0))) 0.0)
   (sqrt (sqrt (/ 1.0 (* k k))))
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))))

C

double code(double k, double n) {
	double tmp;
	if (((1.0 / sqrt(k)) * pow((n * (((double) M_PI) * 2.0)), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = sqrt(sqrt((1.0 / (k * k))));
	} else {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (((1.0 / Math.sqrt(k)) * Math.pow((n * (Math.PI * 2.0)), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = Math.sqrt(Math.sqrt((1.0 / (k * k))));
	} else {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if ((1.0 / math.sqrt(k)) * math.pow((n * (math.pi * 2.0)), ((1.0 - k) / 2.0))) <= 0.0:
		tmp = math.sqrt(math.sqrt((1.0 / (k * k))))
	else:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (math.pi / k)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (((1.0 / sqrt(k)) * ((n * (pi * 2.0)) ^ ((1.0 - k) / 2.0))) <= 0.0)
		tmp = sqrt(sqrt((1.0 / (k * k))));
	else
		tmp = sqrt(n) * sqrt((2.0 * (pi / k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[Sqrt[N[Sqrt[N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 0.0

   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 inf

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\color{blue}{\left(\frac{-1}{2} \cdot k\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 100.0

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   5. Simplified 100.0%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]

      2. /-lowering-/.f64 3.8

         \[\leadsto \sqrt{\color{blue}{\frac{1}{k}}} \]

   8. Simplified 3.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k}} \cdot \sqrt{\frac{1}{k}}}} \]

      2. sqrt-div N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{1}}{\sqrt{k}}} \cdot \sqrt{\frac{1}{k}}} \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{1}}{\sqrt{k}} \cdot \sqrt{\frac{1}{k}}} \]

      4. sqrt-div N/A

         \[\leadsto \sqrt{\frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{k}}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{\sqrt{k}} \cdot \frac{\color{blue}{1}}{\sqrt{k}}} \]

      6. frac-times N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot 1}{\sqrt{k} \cdot \sqrt{k}}}} \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{1}}{\sqrt{k} \cdot \sqrt{k}}} \]

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

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt{k} \cdot \sqrt{k}}}} \]

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

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\sqrt{k} \cdot \sqrt{k}}}} \]

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

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\sqrt{k}} \cdot \sqrt{k}}} \]

      11. sqrt-lowering-sqrt.f64 3.8

          \[\leadsto \sqrt{\frac{1}{\sqrt{k} \cdot \color{blue}{\sqrt{k}}}} \]

   10. Applied egg-rr 3.8%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt{k} \cdot \sqrt{k}}}} \]
   11. Step-by-step derivation

       1. pow2 N/A

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

       2. pow-flip N/A

          \[\leadsto \sqrt{\color{blue}{{\left(\sqrt{k}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}} \]

       3. metadata-eval N/A

          \[\leadsto \sqrt{{\left(\sqrt{k}\right)}^{\color{blue}{-2}}} \]

       4. rem-square-sqrt N/A

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

       5. sqrt-unprod N/A

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

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

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

       7. sqrt-pow2 N/A

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

       8. metadata-eval N/A

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

       9. inv-pow N/A

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

       10. sqrt-pow2 N/A

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

       11. metadata-eval N/A

           \[\leadsto \sqrt{\sqrt{\frac{1}{k} \cdot {k}^{\color{blue}{-1}}}} \]

       12. inv-pow N/A

           \[\leadsto \sqrt{\sqrt{\frac{1}{k} \cdot \color{blue}{\frac{1}{k}}}} \]

       13. frac-times N/A

           \[\leadsto \sqrt{\sqrt{\color{blue}{\frac{1 \cdot 1}{k \cdot k}}}} \]

       14. metadata-eval N/A

           \[\leadsto \sqrt{\sqrt{\frac{\color{blue}{1}}{k \cdot k}}} \]

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

           \[\leadsto \sqrt{\sqrt{\color{blue}{\frac{1}{k \cdot k}}}} \]

       16. *-lowering-*.f64 50.9

           \[\leadsto \sqrt{\sqrt{\frac{1}{\color{blue}{k \cdot k}}}} \]

   12. Applied egg-rr 50.9%

       \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k \cdot k}}}} \]

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   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

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 50.0

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

   5. Simplified 50.0%

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

      1. sqrt-unprod N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

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

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

      7. pow1/2 N/A

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

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

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

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

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

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

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

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

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

      12. PI-lowering-PI.f64 65.4

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

   7. Applied egg-rr 65.4%

      \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{\frac{\pi}{k} \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;k \leq 1:\\ \;\;\;\;\sqrt{t\_0} \cdot \sqrt{\frac{1}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_0}^{\left(k \cdot -0.5\right)}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* n 2.0))))
   (if (<= k 1.0)
     (* (sqrt t_0) (sqrt (/ 1.0 k)))
     (/ (pow t_0 (* k -0.5)) (sqrt k)))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (n * 2.0);
	double tmp;
	if (k <= 1.0) {
		tmp = sqrt(t_0) * sqrt((1.0 / k));
	} else {
		tmp = pow(t_0, (k * -0.5)) / sqrt(k);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (n * 2.0);
	double tmp;
	if (k <= 1.0) {
		tmp = Math.sqrt(t_0) * Math.sqrt((1.0 / k));
	} else {
		tmp = Math.pow(t_0, (k * -0.5)) / Math.sqrt(k);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.pi * (n * 2.0)
	tmp = 0
	if k <= 1.0:
		tmp = math.sqrt(t_0) * math.sqrt((1.0 / k))
	else:
		tmp = math.pow(t_0, (k * -0.5)) / math.sqrt(k)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(n * 2.0))
	tmp = 0.0
	if (k <= 1.0)
		tmp = Float64(sqrt(t_0) * sqrt(Float64(1.0 / k)));
	else
		tmp = Float64((t_0 ^ Float64(k * -0.5)) / sqrt(k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = pi * (n * 2.0);
	tmp = 0.0;
	if (k <= 1.0)
		tmp = sqrt(t_0) * sqrt((1.0 / k));
	else
		tmp = (t_0 ^ (k * -0.5)) / sqrt(k);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.0], N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[t$95$0, N[(k * -0.5), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 99.0%

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

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 73.4

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

   5. Simplified 73.4%

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. sqrt-undiv N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

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

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

      9. metadata-eval N/A

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

      10. sqrt-div N/A

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

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

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

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

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

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

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

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

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

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

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

      18. *-lowering-*.f64 96.3

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

   7. Applied egg-rr 96.3%

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

   if 1 < 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 inf

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\color{blue}{\left(\frac{-1}{2} \cdot k\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 100.0

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   5. Simplified 100.0%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. div-inv N/A

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

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

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

      4. associate-*r* N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. sqrt-lowering-sqrt.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 98.1%

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

Alternative 7: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (sqrt (/ 1.0 k)) (pow (* n (* PI 2.0)) (fma -0.5 k 0.5))))

C

double code(double k, double n) {
	return sqrt((1.0 / k)) * pow((n * (((double) M_PI) * 2.0)), fma(-0.5, k, 0.5));
}

Julia

function code(k, n)
	return Float64(sqrt(Float64(1.0 / k)) * (Float64(n * Float64(pi * 2.0)) ^ fma(-0.5, k, 0.5)))
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\sqrt{\frac{1}{k}} \cdot e^{\frac{1}{2} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(1 - k\right)\right)}} \]
4. Step-by-step derivation

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

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

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

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

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

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. exp-prod N/A

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

   7. *-commutative N/A

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

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

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

   9. rem-exp-log N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. rem-exp-log N/A

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

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

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

   15. rem-exp-log N/A

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

   16. *-commutative N/A

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

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

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

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

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

   19. sub-neg N/A

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

   20. mul-1-neg N/A

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}} \]
6. Add Preprocessing

Alternative 8: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. div-sub N/A

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

   4. metadata-eval N/A

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

   5. pow-sub N/A

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

   6. associate-/l/ N/A

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

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

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

   8. unpow1/2 N/A

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

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

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

   10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

4. Applied egg-rr 99.7%

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

6. Applied egg-rr 99.5%

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

Alternative 9: 49.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.5:\\ \;\;\;\;\sqrt{\frac{\pi \cdot \left(n \cdot 2\right)}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 0.5) (sqrt (/ (* PI (* n 2.0)) k)) (sqrt (sqrt (/ 1.0 (* k k))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 0.5) {
		tmp = sqrt(((((double) M_PI) * (n * 2.0)) / k));
	} else {
		tmp = sqrt(sqrt((1.0 / (k * k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 0.5) {
		tmp = Math.sqrt(((Math.PI * (n * 2.0)) / k));
	} else {
		tmp = Math.sqrt(Math.sqrt((1.0 / (k * k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 0.5:
		tmp = math.sqrt(((math.pi * (n * 2.0)) / k))
	else:
		tmp = math.sqrt(math.sqrt((1.0 / (k * k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 0.5)
		tmp = sqrt(Float64(Float64(pi * Float64(n * 2.0)) / k));
	else
		tmp = sqrt(sqrt(Float64(1.0 / Float64(k * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 0.5)
		tmp = sqrt(((pi * (n * 2.0)) / k));
	else
		tmp = sqrt(sqrt((1.0 / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 0.5], N[Sqrt[N[(N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Sqrt[N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.5

   1. Initial program 99.0%

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

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 73.4

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

   5. Simplified 73.4%

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

      1. sqrt-unprod N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. *-lowering-*.f64 73.0

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

   7. Applied egg-rr 73.0%

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

   if 0.5 < 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 inf

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\color{blue}{\left(\frac{-1}{2} \cdot k\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 100.0

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   5. Simplified 100.0%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]

      2. /-lowering-/.f64 3.2

         \[\leadsto \sqrt{\color{blue}{\frac{1}{k}}} \]

   8. Simplified 3.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k}} \cdot \sqrt{\frac{1}{k}}}} \]

      2. sqrt-div N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{1}}{\sqrt{k}}} \cdot \sqrt{\frac{1}{k}}} \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{1}}{\sqrt{k}} \cdot \sqrt{\frac{1}{k}}} \]

      4. sqrt-div N/A

         \[\leadsto \sqrt{\frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{k}}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{\sqrt{k}} \cdot \frac{\color{blue}{1}}{\sqrt{k}}} \]

      6. frac-times N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot 1}{\sqrt{k} \cdot \sqrt{k}}}} \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{1}}{\sqrt{k} \cdot \sqrt{k}}} \]

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

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt{k} \cdot \sqrt{k}}}} \]

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

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\sqrt{k} \cdot \sqrt{k}}}} \]

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

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\sqrt{k}} \cdot \sqrt{k}}} \]

      11. sqrt-lowering-sqrt.f64 3.2

          \[\leadsto \sqrt{\frac{1}{\sqrt{k} \cdot \color{blue}{\sqrt{k}}}} \]

   10. Applied egg-rr 3.2%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt{k} \cdot \sqrt{k}}}} \]
   11. Step-by-step derivation

       1. pow2 N/A

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

       2. pow-flip N/A

          \[\leadsto \sqrt{\color{blue}{{\left(\sqrt{k}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}} \]

       3. metadata-eval N/A

          \[\leadsto \sqrt{{\left(\sqrt{k}\right)}^{\color{blue}{-2}}} \]

       4. rem-square-sqrt N/A

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

       5. sqrt-unprod N/A

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

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

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

       7. sqrt-pow2 N/A

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

       8. metadata-eval N/A

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

       9. inv-pow N/A

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

       10. sqrt-pow2 N/A

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

       11. metadata-eval N/A

           \[\leadsto \sqrt{\sqrt{\frac{1}{k} \cdot {k}^{\color{blue}{-1}}}} \]

       12. inv-pow N/A

           \[\leadsto \sqrt{\sqrt{\frac{1}{k} \cdot \color{blue}{\frac{1}{k}}}} \]

       13. frac-times N/A

           \[\leadsto \sqrt{\sqrt{\color{blue}{\frac{1 \cdot 1}{k \cdot k}}}} \]

       14. metadata-eval N/A

           \[\leadsto \sqrt{\sqrt{\frac{\color{blue}{1}}{k \cdot k}}} \]

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

           \[\leadsto \sqrt{\sqrt{\color{blue}{\frac{1}{k \cdot k}}}} \]

       16. *-lowering-*.f64 25.9

           \[\leadsto \sqrt{\sqrt{\frac{1}{\color{blue}{k \cdot k}}}} \]

   12. Applied egg-rr 25.9%

       \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k \cdot k}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 37.1% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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

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

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

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

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

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

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

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

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

   6. sqrt-lowering-sqrt.f64 38.8

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

5. Simplified 38.8%

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

   1. sqrt-unprod N/A

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

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

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

   3. *-commutative N/A

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

   4. associate-*l/ N/A

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

   5. *-commutative N/A

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

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

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

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

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

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

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

   11. *-lowering-*.f64 38.6

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

7. Applied egg-rr 38.6%

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

Alternative 11: 5.1% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{k}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (/ 1.0 k)))

C

double code(double k, double n) {
	return sqrt((1.0 / k));
}

Fortran

real(8) function code(k, n)
    real(8), intent (in) :: k
    real(8), intent (in) :: n
    code = sqrt((1.0d0 / k))
end function

Java

public static double code(double k, double n) {
	return Math.sqrt((1.0 / k));
}

Python

def code(k, n):
	return math.sqrt((1.0 / k))

Julia

function code(k, n)
	return sqrt(Float64(1.0 / k))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt((1.0 / k));
end

Wolfram

code[k_, n_] := N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around inf

   \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\color{blue}{\left(\frac{-1}{2} \cdot k\right)}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 52.5

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

5. Simplified 52.5%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
7. Step-by-step derivation

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

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]

   2. /-lowering-/.f64 5.2

      \[\leadsto \sqrt{\color{blue}{\frac{1}{k}}} \]

8. Simplified 5.2%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(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))))