Migdal et al, Equation (51)

Percentage Accurate: 98.6% → 98.2%

Time: 19.7s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. k = 4.94066e-324
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.6% 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: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;k \leq 5.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{t\_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{t\_0}^{\left(1 - k\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* n (* 2.0 PI))))
   (if (<= k 5.5e-76)
     (/ (sqrt t_0) (sqrt k))
     (/ 1.0 (sqrt (/ k (pow t_0 (- 1.0 k))))))))

C

double code(double k, double n) {
	double t_0 = n * (2.0 * ((double) M_PI));
	double tmp;
	if (k <= 5.5e-76) {
		tmp = sqrt(t_0) / sqrt(k);
	} else {
		tmp = 1.0 / sqrt((k / pow(t_0, (1.0 - k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = n * (2.0 * Math.PI);
	double tmp;
	if (k <= 5.5e-76) {
		tmp = Math.sqrt(t_0) / Math.sqrt(k);
	} else {
		tmp = 1.0 / Math.sqrt((k / Math.pow(t_0, (1.0 - k))));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = n * (2.0 * math.pi)
	tmp = 0
	if k <= 5.5e-76:
		tmp = math.sqrt(t_0) / math.sqrt(k)
	else:
		tmp = 1.0 / math.sqrt((k / math.pow(t_0, (1.0 - k))))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(n * Float64(2.0 * pi))
	tmp = 0.0
	if (k <= 5.5e-76)
		tmp = Float64(sqrt(t_0) / sqrt(k));
	else
		tmp = Float64(1.0 / sqrt(Float64(k / (t_0 ^ Float64(1.0 - k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = n * (2.0 * pi);
	tmp = 0.0;
	if (k <= 5.5e-76)
		tmp = sqrt(t_0) / sqrt(k);
	else
		tmp = 1.0 / sqrt((k / (t_0 ^ (1.0 - k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 5.5e-76], N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(k / N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.50000000000000014e-76

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 99.0%

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

      1. associate-*l/ 99.2%

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

      2. *-un-lft-identity 99.2%

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

      3. sqrt-unprod 99.5%

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

      4. *-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*r* 99.5%

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

   5. Applied egg-rr 99.5%

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

   if 5.50000000000000014e-76 < k

   1. Initial program 98.7%

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

      1. associate-/r/ 98.7%

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

      2. associate-*l* 98.7%

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

      3. div-inv 98.7%

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

      4. metadata-eval 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. add-sqr-sqrt 98.6%

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

      2. sqrt-unprod 98.7%

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

      3. frac-times 98.7%

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

      4. add-sqr-sqrt 98.7%

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

      5. pow-unpow 98.7%

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

      6. *-commutative 98.7%

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

      7. associate-*r* 98.7%

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

      8. pow-unpow 98.7%

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

      9. *-commutative 98.7%

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

      10. associate-*r* 98.7%

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

   6. Applied egg-rr 98.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.15e-75)
   (/ (sqrt (* n (* 2.0 PI))) (sqrt k))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.15e-75) {
		tmp = sqrt((n * (2.0 * ((double) M_PI)))) / sqrt(k);
	} else {
		tmp = sqrt((pow((((double) M_PI) * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.15e-75) {
		tmp = Math.sqrt((n * (2.0 * Math.PI))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.15e-75:
		tmp = math.sqrt((n * (2.0 * math.pi))) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow((math.pi * (2.0 * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.15e-75)
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * pi))) / sqrt(k));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(2.0 * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.15e-75)
		tmp = sqrt((n * (2.0 * pi))) / sqrt(k);
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.15e-75], N[(N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.15e-75

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 99.0%

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

      1. associate-*l/ 99.2%

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

      2. *-un-lft-identity 99.2%

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

      3. sqrt-unprod 99.5%

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

      4. *-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*r* 99.5%

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

   5. Applied egg-rr 99.5%

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

   if 2.15e-75 < k

   1. Initial program 98.7%

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

      1. add-sqr-sqrt 98.6%

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

      2. sqrt-unprod 98.7%

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

      3. *-commutative 98.7%

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

      4. div-sub 98.7%

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

      5. metadata-eval 98.7%

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

      6. div-inv 98.6%

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

      7. *-commutative 98.6%

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

      8. div-sub 98.6%

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

      9. metadata-eval 98.6%

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

      10. div-inv 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. sqr-pow 98.7%

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

      2. pow-sqr 98.7%

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

      3. associate-*r* 98.7%

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

      4. associate-/l* 98.7%

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

      5. metadata-eval 98.7%

         \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(2 \cdot \left(\left(2 \cdot \left(1 - k\right)\right) \cdot \color{blue}{0.25}\right)\right)}}{k}} \]

      6. associate-*r* 98.7%

         \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(2 \cdot \color{blue}{\left(2 \cdot \left(\left(1 - k\right) \cdot 0.25\right)\right)}\right)}}{k}} \]

      7. *-commutative 98.7%

         \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(2 \cdot \left(2 \cdot \color{blue}{\left(0.25 \cdot \left(1 - k\right)\right)}\right)\right)}}{k}} \]

      8. associate-*r* 98.7%

         \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(2 \cdot \color{blue}{\left(\left(2 \cdot 0.25\right) \cdot \left(1 - k\right)\right)}\right)}}{k}} \]

      9. metadata-eval 98.7%

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

      10. associate-*r* 98.7%

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

      11. metadata-eval 98.7%

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

      12. *-lft-identity 98.7%

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

      13. *-commutative 98.7%

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

      14. associate-*l* 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2e+61)
   (/ (sqrt (* n (* 2.0 PI))) (sqrt k))
   (sqrt (* 2.0 (+ -1.0 (fma n (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2e+61) {
		tmp = sqrt((n * (2.0 * ((double) M_PI)))) / sqrt(k);
	} else {
		tmp = sqrt((2.0 * (-1.0 + fma(n, (((double) M_PI) / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2e+61)
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * pi))) / sqrt(k));
	else
		tmp = sqrt(Float64(2.0 * Float64(-1.0 + fma(n, Float64(pi / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2e+61], N[(N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(-1.0 + N[(n * N[(Pi / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9999999999999999e61

   1. Initial program 97.8%

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

   3. Taylor expanded in k around 0 72.1%

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

      1. associate-*l/ 72.1%

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

      2. *-un-lft-identity 72.1%

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

      3. sqrt-unprod 72.3%

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

      4. *-commutative 72.3%

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

      5. *-commutative 72.3%

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

      6. associate-*r* 72.3%

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

   5. Applied egg-rr 72.3%

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

   if 1.9999999999999999e61 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 1.7%

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

      1. pow1 1.7%

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

      2. associate-*l/ 1.7%

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

      3. *-un-lft-identity 1.7%

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

      4. *-commutative 1.7%

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

      5. sqrt-prod 1.7%

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

      6. associate-*r* 1.7%

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

      7. sqrt-undiv 1.7%

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

      8. associate-*r* 1.7%

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

      9. *-commutative 1.7%

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

      10. associate-*r* 1.7%

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

   5. Applied egg-rr 1.7%

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

      1. unpow1 1.7%

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

      2. associate-*l* 1.7%

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

      3. *-commutative 1.7%

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

      4. associate-*r/ 1.7%

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

   7. Simplified 1.7%

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

      1. expm1-log1p-u 1.7%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n \cdot \pi}{k}\right)\right)}} \]

      2. expm1-undefine 31.3%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{n \cdot \pi}{k}\right)} - 1\right)}} \]

      3. associate-/l* 31.3%

         \[\leadsto \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{n \cdot \frac{\pi}{k}}\right)} - 1\right)} \]

   9. Applied egg-rr 31.3%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} - 1\right)}} \]
   10. Step-by-step derivation

       1. sub-neg 31.3%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} + \left(-1\right)\right)}} \]

       2. metadata-eval 31.3%

          \[\leadsto \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} + \color{blue}{-1}\right)} \]

       3. +-commutative 31.3%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)}\right)}} \]

       4. log1p-undefine 31.3%

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

       5. rem-exp-log 31.3%

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

       6. +-commutative 31.3%

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

       7. fma-define 31.3%

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

   11. Simplified 31.3%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+61}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{k}}{\sqrt{2 \cdot \left(\pi \cdot n\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.9e+62)
   (/ 1.0 (/ (sqrt k) (sqrt (* 2.0 (* PI n)))))
   (sqrt (* 2.0 (+ -1.0 (fma n (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.9e+62) {
		tmp = 1.0 / (sqrt(k) / sqrt((2.0 * (((double) M_PI) * n))));
	} else {
		tmp = sqrt((2.0 * (-1.0 + fma(n, (((double) M_PI) / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.9e+62)
		tmp = Float64(1.0 / Float64(sqrt(k) / sqrt(Float64(2.0 * Float64(pi * n)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(-1.0 + fma(n, Float64(pi / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.9e+62], N[(1.0 / N[(N[Sqrt[k], $MachinePrecision] / N[Sqrt[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(-1.0 + N[(n * N[(Pi / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.89999999999999992e62

   1. Initial program 97.8%

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

   3. Taylor expanded in k around 0 72.1%

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

      1. pow1 72.1%

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

      2. associate-*l/ 72.1%

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

      3. *-un-lft-identity 72.1%

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

      4. *-commutative 72.1%

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

      5. sqrt-prod 72.3%

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

      6. associate-*r* 72.3%

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

      7. sqrt-undiv 57.2%

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

      8. associate-*r* 57.2%

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

      9. *-commutative 57.2%

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

      10. associate-*r* 57.2%

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

   5. Applied egg-rr 57.2%

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

      1. unpow1 57.2%

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

      2. associate-*l* 57.2%

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

      3. *-commutative 57.2%

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

      4. associate-*r/ 57.2%

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

   7. Simplified 57.2%

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

      1. associate-*r/ 57.2%

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

      2. *-commutative 57.2%

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

      3. associate-*r* 57.2%

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

      4. sqrt-div 72.3%

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

      5. clear-num 72.4%

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

      6. associate-*r* 72.4%

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

   9. Applied egg-rr 72.4%

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

   if 1.89999999999999992e62 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 1.7%

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

      1. pow1 1.7%

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

      2. associate-*l/ 1.7%

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

      3. *-un-lft-identity 1.7%

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

      4. *-commutative 1.7%

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

      5. sqrt-prod 1.7%

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

      6. associate-*r* 1.7%

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

      7. sqrt-undiv 1.7%

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

      8. associate-*r* 1.7%

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

      9. *-commutative 1.7%

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

      10. associate-*r* 1.7%

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

   5. Applied egg-rr 1.7%

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

      1. unpow1 1.7%

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

      2. associate-*l* 1.7%

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

      3. *-commutative 1.7%

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

      4. associate-*r/ 1.7%

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

   7. Simplified 1.7%

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

      1. expm1-log1p-u 1.7%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n \cdot \pi}{k}\right)\right)}} \]

      2. expm1-undefine 31.3%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{n \cdot \pi}{k}\right)} - 1\right)}} \]

      3. associate-/l* 31.3%

         \[\leadsto \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{n \cdot \frac{\pi}{k}}\right)} - 1\right)} \]

   9. Applied egg-rr 31.3%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} - 1\right)}} \]
   10. Step-by-step derivation

       1. sub-neg 31.3%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} + \left(-1\right)\right)}} \]

       2. metadata-eval 31.3%

          \[\leadsto \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} + \color{blue}{-1}\right)} \]

       3. +-commutative 31.3%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)}\right)}} \]

       4. log1p-undefine 31.3%

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

       5. rem-exp-log 31.3%

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

       6. +-commutative 31.3%

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

       7. fma-define 31.3%

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

   11. Simplified 31.3%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{k}}{\sqrt{2 \cdot \left(\pi \cdot n\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) * 0.5)));
}

Python

def code(k, n):
	return 1.0 / (math.sqrt(k) / math.pow((2.0 * (math.pi * n)), ((1.0 - k) * 0.5)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

   1. associate-/r/ 98.9%

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

   2. associate-*l* 98.9%

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

   3. div-inv 98.9%

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

   4. metadata-eval 98.9%

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

4. Applied egg-rr 98.9%

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

5. Final simplification 98.9%

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

Alternative 6: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

   1. associate-*l/ 98.9%

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

   2. *-lft-identity 98.9%

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

   3. div-sub 98.9%

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

   4. metadata-eval 98.9%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 7: 30.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in k around 0 38.5%

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

   1. pow1 38.5%

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

   2. associate-*l/ 38.6%

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

   3. *-un-lft-identity 38.6%

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

   4. *-commutative 38.6%

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

   5. sqrt-prod 38.7%

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

   6. associate-*r* 38.7%

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

   7. sqrt-undiv 30.7%

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

   8. associate-*r* 30.7%

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

   9. *-commutative 30.7%

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

   10. associate-*r* 30.7%

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

5. Applied egg-rr 30.7%

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

   1. unpow1 30.7%

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

   2. associate-*l* 30.7%

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

   3. *-commutative 30.7%

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

   4. associate-*r/ 30.7%

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

7. Simplified 30.7%

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

   1. add-sqr-sqrt 30.7%

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

   2. sqrt-unprod 22.4%

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

   3. pow2 22.4%

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

   4. associate-/l* 22.4%

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

9. Applied egg-rr 22.4%

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

    1. unpow2 22.4%

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

    2. rem-sqrt-square 31.4%

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

11. Simplified 31.4%

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

12. Final simplification 31.4%

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

Alternative 8: 30.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

   1. add-sqr-sqrt 98.7%

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

   2. sqrt-unprod 90.9%

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

   3. *-commutative 90.9%

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

   4. div-sub 90.9%

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

   5. metadata-eval 90.9%

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

   6. div-inv 90.9%

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

   7. *-commutative 90.9%

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

   8. div-sub 90.9%

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

   9. metadata-eval 90.9%

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

   10. div-inv 90.9%

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

4. Applied egg-rr 91.0%

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

   1. sqr-pow 90.9%

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

   2. pow-sqr 91.0%

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

   3. associate-*r* 91.0%

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

   4. associate-/l* 91.0%

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

   5. metadata-eval 91.0%

      \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(2 \cdot \left(\left(2 \cdot \left(1 - k\right)\right) \cdot \color{blue}{0.25}\right)\right)}}{k}} \]

   6. associate-*r* 91.0%

      \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(2 \cdot \color{blue}{\left(2 \cdot \left(\left(1 - k\right) \cdot 0.25\right)\right)}\right)}}{k}} \]

   7. *-commutative 91.0%

      \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(2 \cdot \left(2 \cdot \color{blue}{\left(0.25 \cdot \left(1 - k\right)\right)}\right)\right)}}{k}} \]

   8. associate-*r* 91.0%

      \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(2 \cdot \color{blue}{\left(\left(2 \cdot 0.25\right) \cdot \left(1 - k\right)\right)}\right)}}{k}} \]

   9. metadata-eval 91.0%

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

   10. associate-*r* 91.0%

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

   11. metadata-eval 91.0%

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

   12. *-lft-identity 91.0%

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

   13. *-commutative 91.0%

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

   14. associate-*l* 91.0%

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

6. Simplified 91.0%

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

   1. add-sqr-sqrt 69.8%

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

   2. pow1/2 69.8%

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

   3. pow1/2 69.8%

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

   4. pow-prod-down 80.7%

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

   5. pow2 80.7%

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

   6. associate-*r* 80.7%

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

   7. *-commutative 80.7%

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

   8. associate-*r* 80.7%

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

8. Applied egg-rr 80.7%

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

   1. unpow1/2 80.7%

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

   2. unpow2 80.7%

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

   3. rem-sqrt-square 91.7%

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

   4. *-commutative 91.7%

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

10. Simplified 91.7%

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

11. Taylor expanded in k around 0 31.4%

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

12. Final simplification 31.4%

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

Alternative 9: 39.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in k around 0 38.5%

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

   1. pow1 38.5%

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

   2. associate-*l/ 38.6%

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

   3. *-un-lft-identity 38.6%

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

   4. *-commutative 38.6%

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

   5. sqrt-prod 38.7%

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

   6. associate-*r* 38.7%

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

   7. sqrt-undiv 30.7%

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

   8. associate-*r* 30.7%

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

   9. *-commutative 30.7%

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

   10. associate-*r* 30.7%

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

5. Applied egg-rr 30.7%

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

   1. unpow1 30.7%

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

   2. associate-*l* 30.7%

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

   3. *-commutative 30.7%

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

   4. associate-*r/ 30.7%

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

7. Simplified 30.7%

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

   1. *-commutative 30.7%

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

   2. associate-/l* 30.7%

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

9. Applied egg-rr 30.7%

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

    1. associate-*r* 30.7%

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

    2. associate-*r/ 30.7%

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

    3. *-commutative 30.7%

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

    4. associate-*r/ 30.7%

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

    5. sqrt-prod 38.7%

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

    6. *-commutative 38.7%

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

    7. associate-/l* 38.6%

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

11. Applied egg-rr 38.6%

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

    1. associate-*r/ 38.7%

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

    2. associate-*l/ 38.7%

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

    3. *-commutative 38.7%

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

13. Simplified 38.7%

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

14. Final simplification 38.7%

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

Alternative 10: 38.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in k around 0 38.5%

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

   1. associate-*l/ 38.6%

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

   2. *-un-lft-identity 38.6%

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

   3. sqrt-unprod 38.7%

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

   4. *-commutative 38.7%

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

   5. *-commutative 38.7%

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

   6. associate-*r* 38.7%

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

5. Applied egg-rr 38.7%

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

6. Final simplification 38.7%

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

Alternative 11: 30.0% accurate, 2.0× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in k around 0 38.5%

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

   1. pow1 38.5%

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

   2. associate-*l/ 38.6%

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

   3. *-un-lft-identity 38.6%

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

   4. *-commutative 38.6%

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

   5. sqrt-prod 38.7%

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

   6. associate-*r* 38.7%

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

   7. sqrt-undiv 30.7%

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

   8. associate-*r* 30.7%

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

   9. *-commutative 30.7%

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

   10. associate-*r* 30.7%

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

5. Applied egg-rr 30.7%

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

   1. unpow1 30.7%

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

   2. associate-*l* 30.7%

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

   3. *-commutative 30.7%

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

   4. associate-*r/ 30.7%

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

7. Simplified 30.7%

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

   1. *-commutative 30.7%

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

   2. associate-/l* 30.7%

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

9. Applied egg-rr 30.7%

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

10. Final simplification 30.7%

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

Alternative 12: 30.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in k around 0 38.5%

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

   1. pow1 38.5%

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

   2. associate-*l/ 38.6%

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

   3. *-un-lft-identity 38.6%

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

   4. *-commutative 38.6%

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

   5. sqrt-prod 38.7%

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

   6. associate-*r* 38.7%

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

   7. sqrt-undiv 30.7%

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

   8. associate-*r* 30.7%

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

   9. *-commutative 30.7%

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

   10. associate-*r* 30.7%

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

5. Applied egg-rr 30.7%

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

   1. unpow1 30.7%

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

   2. associate-*l* 30.7%

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

   3. *-commutative 30.7%

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

   4. associate-*r/ 30.7%

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

7. Simplified 30.7%

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

   1. *-commutative 30.7%

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

   2. associate-/l* 30.7%

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

9. Applied egg-rr 30.7%

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

    1. *-commutative 30.7%

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

    2. associate-*l/ 30.7%

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

    3. associate-*r/ 30.7%

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

    4. clear-num 30.7%

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

    5. un-div-inv 30.7%

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

11. Applied egg-rr 30.7%

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

12. Final simplification 30.7%

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

Reproduce

herbie shell --seed 2024062 
(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))))