Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 11.8s

Alternatives: 12

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 6.955666019189133e+106
  2. n = -6.928567391650891e+75

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-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

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

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

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

   5. *-commutative N/A

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

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

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

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

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

   8. pow-unpow N/A

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

   9. unpow1/2 N/A

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

   10. sqrt-unprod N/A

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

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

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

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

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

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

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

   14. *-commutative N/A

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

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

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

   16. *-commutative N/A

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

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

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

   18. PI-lowering-PI.f64 99.7

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

6. Applied egg-rr 99.7%

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

   1. sqrt-undiv N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. sqrt-prod N/A

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

   5. pow1/2 N/A

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

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

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

   7. pow1/2 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

8. Applied egg-rr 99.7%

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

Alternative 2: 68.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}^{-0.125}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi \cdot 2}{k}} \cdot \sqrt{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* (/ 1.0 (sqrt k)) (pow (* n (* PI 2.0)) (/ (- 1.0 k) 2.0))) 0.0)
   (pow (* (* k k) (* k k)) -0.125)
   (* (sqrt (/ (* PI 2.0) k)) (sqrt n))))

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 = pow(((k * k) * (k * k)), -0.125);
	} else {
		tmp = sqrt(((((double) M_PI) * 2.0) / k)) * sqrt(n);
	}
	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.pow(((k * k) * (k * k)), -0.125);
	} else {
		tmp = Math.sqrt(((Math.PI * 2.0) / k)) * Math.sqrt(n);
	}
	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.pow(((k * k) * (k * k)), -0.125)
	else:
		tmp = math.sqrt(((math.pi * 2.0) / k)) * math.sqrt(n)
	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 = Float64(Float64(k * k) * Float64(k * k)) ^ -0.125;
	else
		tmp = Float64(sqrt(Float64(Float64(pi * 2.0) / k)) * sqrt(n));
	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 = ((k * k) * (k * k)) ^ -0.125;
	else
		tmp = sqrt(((pi * 2.0) / k)) * sqrt(n);
	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[Power[N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision], -0.125], $MachinePrecision], N[(N[Sqrt[N[(N[(Pi * 2.0), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $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. sqrt-div N/A

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

      2. metadata-eval N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 3.8

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

   10. Applied egg-rr 3.8%

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

       1. pow1/2 N/A

          \[\leadsto \frac{1}{\color{blue}{{k}^{\frac{1}{2}}}} \]

       2. pow-flip N/A

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

       3. metadata-eval N/A

          \[\leadsto {k}^{\color{blue}{\frac{-1}{2}}} \]

       4. sqr-pow N/A

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

       5. unpow-prod-down N/A

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

       6. sqr-pow N/A

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

       7. pow-prod-down N/A

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

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

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

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

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

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

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

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

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

       12. metadata-eval N/A

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

       13. metadata-eval 75.8

           \[\leadsto {\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}^{\color{blue}{-0.125}} \]

   12. Applied egg-rr 75.8%

       \[\leadsto \color{blue}{{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}^{-0.125}} \]

   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 99.4%

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

   3. Taylor expanded in k around 0

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

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

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

      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 47.8

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

   5. Simplified 47.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. PI-lowering-PI.f64 47.8

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

   7. Applied egg-rr 47.8%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. associate-*l* N/A

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

      4. unpow-prod-down N/A

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

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

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

      10. pow-prod-down N/A

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

      11. pow1/2 N/A

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

      12. pow1/2 N/A

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

      13. sqrt-unprod N/A

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

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

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

      15. *-lowering-*.f64 64.8

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

   9. Applied egg-rr 64.8%

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

       1. *-commutative N/A

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

       2. pow1/2 N/A

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

       3. unpow-prod-down N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. pow1/2 N/A

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

       8. sqrt-unprod N/A

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

       9. *-commutative N/A

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

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

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

       11. associate-*l/ N/A

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

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

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

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

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

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

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

       15. pow1/2 N/A

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

       16. sqrt-lowering-sqrt.f64 64.8

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

   11. Applied egg-rr 64.8%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}^{-0.125}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi \cdot 2}{k}} \cdot \sqrt{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := If[LessEqual[k, 1.0], N[(N[Sqrt[N[(N[(Pi * 2.0), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(Pi * n), $MachinePrecision] * 2.0), $MachinePrecision], 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 99.1%

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

   3. Taylor expanded in k around 0

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

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

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

      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 71.1

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

   5. Simplified 71.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. PI-lowering-PI.f64 71.1

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

   7. Applied egg-rr 71.1%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. associate-*l* N/A

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

      4. unpow-prod-down N/A

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

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

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

      10. pow-prod-down N/A

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

      11. pow1/2 N/A

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

      12. pow1/2 N/A

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

      13. sqrt-unprod N/A

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

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

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

      15. *-lowering-*.f64 96.5

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

   9. Applied egg-rr 96.5%

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

       1. *-commutative N/A

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

       2. pow1/2 N/A

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

       3. unpow-prod-down N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. pow1/2 N/A

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

       8. sqrt-unprod N/A

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

       9. *-commutative N/A

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

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

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

       11. associate-*l/ N/A

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

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

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

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

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

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

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

       15. pow1/2 N/A

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

       16. sqrt-lowering-sqrt.f64 96.6

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

   11. Applied egg-rr 96.6%

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

   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. un-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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. sqrt-lowering-sqrt.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 98.4%

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

Alternative 4: 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 99.6%

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

3. 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 5: 99.5% accurate, 1.1× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-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. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. pow1/2 N/A

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

   4. metadata-eval N/A

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

   5. pow-flip N/A

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

   6. associate-/r* N/A

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

   7. unpow-prod-up N/A

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

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

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 6: 62.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 0.5

   1. Initial program 99.1%

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

   3. Taylor expanded in k around 0

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

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

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

      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 71.1

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

   5. Simplified 71.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. PI-lowering-PI.f64 71.1

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

   7. Applied egg-rr 71.1%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. associate-*l* N/A

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

      4. unpow-prod-down N/A

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

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

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

      10. pow-prod-down N/A

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

      11. pow1/2 N/A

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

      12. pow1/2 N/A

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

      13. sqrt-unprod N/A

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

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

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

      15. *-lowering-*.f64 96.5

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

   9. Applied egg-rr 96.5%

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

       1. *-commutative N/A

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

       2. pow1/2 N/A

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

       3. unpow-prod-down N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. pow1/2 N/A

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

       8. sqrt-unprod N/A

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

       9. *-commutative N/A

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

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

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

       11. associate-*l/ N/A

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

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

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

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

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

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

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

       15. pow1/2 N/A

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

       16. sqrt-lowering-sqrt.f64 96.6

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

   11. Applied egg-rr 96.6%

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

   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 3.3

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

   8. Simplified 3.3%

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

      1. inv-pow N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

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

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

      6. *-lowering-*.f64 27.4

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

   10. Applied egg-rr 27.4%

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

Alternative 7: 62.1% accurate, 3.2× speedup

Math

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

FPCore

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

C

double code(double k, double n) {
	double tmp;
	if (k <= 0.5) {
		tmp = sqrt(((((double) M_PI) * 2.0) / k)) * sqrt(n);
	} 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 * 2.0) / k)) * Math.sqrt(n);
	} 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 * 2.0) / k)) * math.sqrt(n)
	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 = Float64(sqrt(Float64(Float64(pi * 2.0) / k)) * sqrt(n));
	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 * 2.0) / k)) * sqrt(n);
	else
		tmp = sqrt(sqrt((1.0 / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 0.5], N[(N[Sqrt[N[(N[(Pi * 2.0), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $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 99.1%

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

   3. Taylor expanded in k around 0

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

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

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

      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 71.1

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

   5. Simplified 71.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. PI-lowering-PI.f64 71.1

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

   7. Applied egg-rr 71.1%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. associate-*l* N/A

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

      4. unpow-prod-down N/A

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

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

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

      10. pow-prod-down N/A

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

      11. pow1/2 N/A

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

      12. pow1/2 N/A

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

      13. sqrt-unprod N/A

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

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

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

      15. *-lowering-*.f64 96.5

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

   9. Applied egg-rr 96.5%

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

       1. *-commutative N/A

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

       2. pow1/2 N/A

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

       3. unpow-prod-down N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. pow1/2 N/A

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

       8. sqrt-unprod N/A

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

       9. *-commutative N/A

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

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

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

       11. associate-*l/ N/A

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

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

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

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

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

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

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

       15. pow1/2 N/A

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

       16. sqrt-lowering-sqrt.f64 96.6

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

   11. Applied egg-rr 96.6%

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

   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 3.3

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

   8. Simplified 3.3%

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

      1. inv-pow N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

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

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

      6. *-lowering-*.f64 27.4

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

   10. Applied egg-rr 27.4%

       \[\leadsto \sqrt{\color{blue}{{\left(k \cdot k\right)}^{-0.5}}} \]
   11. Step-by-step derivation

       1. pow2 N/A

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

       2. pow-pow N/A

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

       3. metadata-eval N/A

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

       4. inv-pow N/A

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

       5. rem-square-sqrt N/A

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

       6. sqrt-unprod N/A

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

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

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

       8. frac-times N/A

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

       9. metadata-eval N/A

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

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

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

       11. *-lowering-*.f64 27.4

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

   12. Applied egg-rr 27.4%

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

Alternative 8: 62.1% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 0.5)
   (* (sqrt (* PI n)) (sqrt (/ 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)) * sqrt((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)) * Math.sqrt((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)) * math.sqrt((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 = Float64(sqrt(Float64(pi * n)) * sqrt(Float64(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)) * sqrt((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[(N[Sqrt[N[(Pi * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 / k), $MachinePrecision]], $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 99.1%

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

   3. Taylor expanded in k around 0

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

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

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

      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 71.1

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

   5. Simplified 71.1%

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

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      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. sqrt-prod N/A

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

      9. *-commutative N/A

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

      10. pow1/2 N/A

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

      11. associate-*r* N/A

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

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

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

   7. Applied egg-rr 96.6%

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

   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 3.3

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

   8. Simplified 3.3%

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

      1. inv-pow N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

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

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

      6. *-lowering-*.f64 27.4

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

   10. Applied egg-rr 27.4%

       \[\leadsto \sqrt{\color{blue}{{\left(k \cdot k\right)}^{-0.5}}} \]
   11. Step-by-step derivation

       1. pow2 N/A

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

       2. pow-pow N/A

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

       3. metadata-eval N/A

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

       4. inv-pow N/A

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

       5. rem-square-sqrt N/A

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

       6. sqrt-unprod N/A

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

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

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

       8. frac-times N/A

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

       9. metadata-eval N/A

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

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

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

       11. *-lowering-*.f64 27.4

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

   12. Applied egg-rr 27.4%

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

4. Final simplification 60.4%

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

Alternative 9: 62.1% accurate, 3.2× speedup

Math

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

FPCore

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

C

double code(double k, double n) {
	double tmp;
	if (k <= 0.5) {
		tmp = sqrt(n) * sqrt((((double) M_PI) * (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(n) * Math.sqrt((Math.PI * (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(n) * math.sqrt((math.pi * (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 = Float64(sqrt(n) * sqrt(Float64(pi * Float64(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(n) * sqrt((pi * (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[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]], $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 99.1%

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

   3. Taylor expanded in k around 0

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

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

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

      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 71.1

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

   5. Simplified 71.1%

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

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      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. associate-*r* N/A

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

      8. unpow1/2 N/A

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

      9. *-commutative N/A

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

      10. unpow-prod-down N/A

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

      11. associate-*r* N/A

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

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

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

   7. Applied egg-rr 96.6%

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

   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 3.3

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

   8. Simplified 3.3%

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

      1. inv-pow N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

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

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

      6. *-lowering-*.f64 27.4

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

   10. Applied egg-rr 27.4%

       \[\leadsto \sqrt{\color{blue}{{\left(k \cdot k\right)}^{-0.5}}} \]
   11. Step-by-step derivation

       1. pow2 N/A

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

       2. pow-pow N/A

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

       3. metadata-eval N/A

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

       4. inv-pow N/A

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

       5. rem-square-sqrt N/A

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

       6. sqrt-unprod N/A

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

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

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

       8. frac-times N/A

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

       9. metadata-eval N/A

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

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

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

       11. *-lowering-*.f64 27.4

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

   12. Applied egg-rr 27.4%

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

4. Final simplification 60.4%

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

Alternative 10: 49.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

double code(double k, double n) {
	double tmp;
	if (k <= 0.5) {
		tmp = sqrt((2.0 * ((((double) M_PI) * n) / 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((2.0 * ((Math.PI * n) / 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((2.0 * ((math.pi * n) / 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(2.0 * Float64(Float64(pi * n) / 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((2.0 * ((pi * n) / 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[(2.0 * N[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision]), $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 99.1%

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

   3. Taylor expanded in k around 0

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

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

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

      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 71.1

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

   5. Simplified 71.1%

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

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. PI-lowering-PI.f64 71.4

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

   7. Applied egg-rr 71.4%

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

   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 3.3

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

   8. Simplified 3.3%

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

      1. inv-pow N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

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

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

      6. *-lowering-*.f64 27.4

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

   10. Applied egg-rr 27.4%

       \[\leadsto \sqrt{\color{blue}{{\left(k \cdot k\right)}^{-0.5}}} \]
   11. Step-by-step derivation

       1. pow2 N/A

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

       2. pow-pow N/A

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

       3. metadata-eval N/A

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

       4. inv-pow N/A

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

       5. rem-square-sqrt N/A

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

       6. sqrt-unprod N/A

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

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

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

       8. frac-times N/A

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

       9. metadata-eval N/A

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

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

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

       11. *-lowering-*.f64 27.4

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

   12. Applied egg-rr 27.4%

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

Alternative 11: 37.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. 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 35.3

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

5. Simplified 35.3%

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

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

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

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

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

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

   6. *-commutative N/A

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

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

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

   8. PI-lowering-PI.f64 35.4

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

7. Applied egg-rr 35.4%

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

Alternative 12: 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 99.6%

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

3. 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 55.8

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

5. Simplified 55.8%

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

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 2024205 
(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))))