Migdal et al, Equation (51)

Percentage Accurate: 98.4% → 99.7%

Time: 18.7s

Alternatives: 8

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: 98.4% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot n\right)\\ \mathbf{if}\;\left(2 \cdot \pi\right) \cdot n \leq 2 \cdot 10^{-213}:\\ \;\;\;\;\frac{{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \left(\pi \cdot n\right)\right)\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{t\_0}}{{t\_0}^{\left(0.5 \cdot k\right)}}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* 2.0 n))))
   (if (<= (* (* 2.0 PI) n) 2e-213)
     (/ (pow (log1p (expm1 (* 2.0 (* PI n)))) (- 0.5 (/ k 2.0))) (sqrt k))
     (/ (/ (sqrt t_0) (pow t_0 (* 0.5 k))) (sqrt k)))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (2.0 * n);
	double tmp;
	if (((2.0 * ((double) M_PI)) * n) <= 2e-213) {
		tmp = pow(log1p(expm1((2.0 * (((double) M_PI) * n)))), (0.5 - (k / 2.0))) / sqrt(k);
	} else {
		tmp = (sqrt(t_0) / pow(t_0, (0.5 * k))) / sqrt(k);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (2.0 * n);
	double tmp;
	if (((2.0 * Math.PI) * n) <= 2e-213) {
		tmp = Math.pow(Math.log1p(Math.expm1((2.0 * (Math.PI * n)))), (0.5 - (k / 2.0))) / Math.sqrt(k);
	} else {
		tmp = (Math.sqrt(t_0) / Math.pow(t_0, (0.5 * k))) / Math.sqrt(k);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.pi * (2.0 * n)
	tmp = 0
	if ((2.0 * math.pi) * n) <= 2e-213:
		tmp = math.pow(math.log1p(math.expm1((2.0 * (math.pi * n)))), (0.5 - (k / 2.0))) / math.sqrt(k)
	else:
		tmp = (math.sqrt(t_0) / math.pow(t_0, (0.5 * k))) / math.sqrt(k)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(2.0 * n))
	tmp = 0.0
	if (Float64(Float64(2.0 * pi) * n) <= 2e-213)
		tmp = Float64((log1p(expm1(Float64(2.0 * Float64(pi * n)))) ^ Float64(0.5 - Float64(k / 2.0))) / sqrt(k));
	else
		tmp = Float64(Float64(sqrt(t_0) / (t_0 ^ Float64(0.5 * k))) / sqrt(k));
	end
	return tmp
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], 2e-213], N[(N[Power[N[Log[1 + N[(Exp[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], N[(0.5 - N[(k / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Power[t$95$0, N[(0.5 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 2 (PI.f64)) n) < 1.9999999999999999e-213

   1. Initial program 95.2%

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

      1. associate-*l/ 95.2%

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

      2. *-lft-identity 95.2%

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

      3. div-sub 95.2%

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

      4. metadata-eval 95.2%

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

   3. Simplified 95.2%

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

      1. log1p-expm1-u 99.8%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(2 \cdot \pi\right) \cdot n\right)\right)\right)}}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

      2. associate-*l* 99.8%

         \[\leadsto \frac{{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{2 \cdot \left(\pi \cdot n\right)}\right)\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{{\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \left(\pi \cdot n\right)\right)\right)\right)}}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

   if 1.9999999999999999e-213 < (*.f64 (*.f64 2 (PI.f64)) n)

   1. Initial program 99.3%

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

      1. associate-*l/ 99.4%

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

      2. *-un-lft-identity 99.4%

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

      3. unpow-prod-down 80.9%

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

      4. unpow-prod-down 99.4%

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

      5. div-sub 99.4%

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

      6. metadata-eval 99.4%

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

      7. pow-div 99.7%

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

      8. pow1/2 99.7%

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

      9. associate-/l/ 99.6%

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

      10. associate-*l* 99.6%

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

      11. associate-*l* 99.6%

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

      12. div-inv 99.6%

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

      13. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. associate-/l/ 99.7%

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

      2. remove-double-neg 99.7%

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

      3. distribute-frac-neg2 99.7%

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

      4. distribute-frac-neg 99.7%

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

      5. distribute-frac-neg2 99.7%

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

      6. distribute-frac-neg2 99.7%

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

      7. distribute-frac-neg 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot n \leq 2 \cdot 10^{-213}:\\ \;\;\;\;\frac{{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \left(\pi \cdot n\right)\right)\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 \cdot k\right)}}}{\sqrt{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ \mathbf{if}\;t\_0 \leq 10^{-278}:\\ \;\;\;\;\sqrt{\frac{{\left(\left|2 \cdot \left(\pi \cdot n\right)\right|\right)}^{\left(1 - k\right)}}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_0}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* 2.0 PI) n)))
   (if (<= t_0 1e-278)
     (sqrt (/ (pow (fabs (* 2.0 (* PI n))) (- 1.0 k)) k))
     (/ (pow t_0 (- 0.5 (/ k 2.0))) (sqrt k)))))

C

double code(double k, double n) {
	double t_0 = (2.0 * ((double) M_PI)) * n;
	double tmp;
	if (t_0 <= 1e-278) {
		tmp = sqrt((pow(fabs((2.0 * (((double) M_PI) * n))), (1.0 - k)) / k));
	} else {
		tmp = pow(t_0, (0.5 - (k / 2.0))) / sqrt(k);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = (2.0 * Math.PI) * n;
	double tmp;
	if (t_0 <= 1e-278) {
		tmp = Math.sqrt((Math.pow(Math.abs((2.0 * (Math.PI * n))), (1.0 - k)) / k));
	} else {
		tmp = Math.pow(t_0, (0.5 - (k / 2.0))) / Math.sqrt(k);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = (2.0 * math.pi) * n
	tmp = 0
	if t_0 <= 1e-278:
		tmp = math.sqrt((math.pow(math.fabs((2.0 * (math.pi * n))), (1.0 - k)) / k))
	else:
		tmp = math.pow(t_0, (0.5 - (k / 2.0))) / math.sqrt(k)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(Float64(2.0 * pi) * n)
	tmp = 0.0
	if (t_0 <= 1e-278)
		tmp = sqrt(Float64((abs(Float64(2.0 * Float64(pi * n))) ^ Float64(1.0 - k)) / k));
	else
		tmp = Float64((t_0 ^ Float64(0.5 - Float64(k / 2.0))) / sqrt(k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = (2.0 * pi) * n;
	tmp = 0.0;
	if (t_0 <= 1e-278)
		tmp = sqrt(((abs((2.0 * (pi * n))) ^ (1.0 - k)) / k));
	else
		tmp = (t_0 ^ (0.5 - (k / 2.0))) / sqrt(k);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-278], N[Sqrt[N[(N[Power[N[Abs[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision], N[(N[Power[t$95$0, N[(0.5 - N[(k / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 2 (PI.f64)) n) < 9.99999999999999938e-279

   1. Initial program 94.0%

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

      1. add-sqr-sqrt 93.9%

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

      2. sqrt-unprod 94.0%

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

      3. *-commutative 94.0%

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

      4. div-sub 94.0%

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

      5. metadata-eval 94.0%

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

      6. div-inv 94.0%

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

      7. *-commutative 94.0%

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

      8. div-sub 94.0%

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

      9. metadata-eval 94.0%

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

      10. div-inv 94.0%

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

   4. Applied egg-rr 94.0%

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

      1. sqr-pow 94.0%

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

      2. pow-sqr 94.0%

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

      3. associate-*r* 94.0%

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

      4. associate-/l* 94.0%

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

      5. metadata-eval 94.0%

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

      6. associate-*r* 94.0%

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

      7. *-commutative 94.0%

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

      8. associate-*r* 94.0%

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

      9. metadata-eval 94.0%

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

      10. associate-*r* 94.0%

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

      11. metadata-eval 94.0%

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

      12. *-lft-identity 94.0%

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

      13. *-commutative 94.0%

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

      14. associate-*l* 94.0%

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

   6. Simplified 94.0%

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

      1. associate-*r* 94.0%

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

      2. *-commutative 94.0%

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

      3. add-sqr-sqrt 11.9%

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

      4. pow1/2 11.9%

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

      5. pow1/2 11.9%

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

      6. pow-prod-down 95.7%

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

      7. pow2 95.7%

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

      8. *-commutative 95.7%

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

      9. associate-*r* 95.7%

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

      10. *-commutative 95.7%

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

   8. Applied egg-rr 95.7%

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

      1. unpow1/2 95.7%

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

      2. unpow2 95.7%

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

      3. rem-sqrt-square 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

   10. Simplified 100.0%

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

   if 9.99999999999999938e-279 < (*.f64 (*.f64 2 (PI.f64)) n)

   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. Step-by-step derivation

      1. associate-*l/ 99.4%

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

      2. *-lft-identity 99.4%

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

      3. div-sub 99.4%

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

      4. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot n \leq 10^{-278}:\\ \;\;\;\;\sqrt{\frac{{\left(\left|2 \cdot \left(\pi \cdot n\right)\right|\right)}^{\left(1 - k\right)}}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.2e-36)
   (/ (sqrt (* (* 2.0 PI) n)) (sqrt k))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.2e-36) {
		tmp = sqrt(((2.0 * ((double) M_PI)) * n)) / 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 <= 1.2e-36) {
		tmp = Math.sqrt(((2.0 * Math.PI) * n)) / 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 <= 1.2e-36:
		tmp = math.sqrt(((2.0 * math.pi) * n)) / 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 <= 1.2e-36)
		tmp = Float64(sqrt(Float64(Float64(2.0 * pi) * n)) / 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 <= 1.2e-36)
		tmp = sqrt(((2.0 * pi) * n)) / 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, 1.2e-36], N[(N[Sqrt[N[(N[(2.0 * Pi), $MachinePrecision] * n), $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 < 1.2e-36

   1. Initial program 99.2%

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

      1. add-sqr-sqrt 99.0%

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

      2. sqrt-unprod 74.1%

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

      3. *-commutative 74.1%

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

      4. div-sub 74.1%

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

      5. metadata-eval 74.1%

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

      6. div-inv 74.2%

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

      7. *-commutative 74.2%

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

      8. div-sub 74.2%

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

      9. metadata-eval 74.2%

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

      10. div-inv 74.2%

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

   4. Applied egg-rr 74.5%

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

      1. sqr-pow 74.2%

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

      2. pow-sqr 74.5%

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

      3. associate-*r* 74.5%

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

      4. associate-/l* 74.5%

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

      5. metadata-eval 74.5%

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

      6. associate-*r* 74.5%

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

      7. *-commutative 74.5%

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

      8. associate-*r* 74.5%

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

      9. metadata-eval 74.5%

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

      10. associate-*r* 74.5%

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

      11. metadata-eval 74.5%

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

      12. *-lft-identity 74.5%

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

      13. *-commutative 74.5%

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

      14. associate-*l* 74.5%

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

   6. Simplified 74.5%

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

   7. Taylor expanded in k around 0 74.5%

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

      1. sqrt-div 99.4%

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

      2. div-inv 99.2%

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

      3. *-commutative 99.2%

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

   9. Applied egg-rr 99.2%

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

       1. associate-*r/ 99.4%

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

       2. *-rgt-identity 99.4%

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

       3. associate-*r* 99.4%

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

       4. *-commutative 99.4%

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

   11. Simplified 99.4%

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

   if 1.2e-36 < k

   1. Initial program 97.2%

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

      1. add-sqr-sqrt 97.2%

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

      2. sqrt-unprod 96.7%

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

      3. *-commutative 96.7%

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

      4. div-sub 96.7%

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

      5. metadata-eval 96.7%

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

      6. div-inv 96.7%

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

      7. *-commutative 96.7%

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

      8. div-sub 96.7%

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

      9. metadata-eval 96.7%

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

      10. div-inv 96.7%

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

   4. Applied egg-rr 96.7%

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

      1. sqr-pow 96.7%

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

      2. pow-sqr 96.7%

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

      3. associate-*r* 96.7%

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

      4. associate-/l* 96.7%

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

      5. metadata-eval 96.7%

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

      6. associate-*r* 96.7%

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

      7. *-commutative 96.7%

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

      8. associate-*r* 96.7%

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

      9. metadata-eval 96.7%

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

      10. associate-*r* 96.7%

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

      11. metadata-eval 96.7%

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

      12. *-lft-identity 96.7%

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

      13. *-commutative 96.7%

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

      14. associate-*l* 96.7%

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

   6. Simplified 96.7%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 98.0%

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

   2. *-lft-identity 98.0%

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

   3. div-sub 98.0%

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

   4. metadata-eval 98.0%

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

3. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 5: 30.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

   1. add-sqr-sqrt 97.8%

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

   2. sqrt-unprod 88.8%

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

   3. *-commutative 88.8%

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

   4. div-sub 88.8%

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

   5. metadata-eval 88.8%

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

   6. div-inv 88.9%

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

   7. *-commutative 88.9%

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

   8. div-sub 88.9%

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

   9. metadata-eval 88.9%

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

   10. div-inv 88.9%

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

4. Applied egg-rr 89.0%

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

   1. sqr-pow 88.9%

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

   2. pow-sqr 89.0%

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

   3. associate-*r* 89.0%

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

   4. associate-/l* 89.0%

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

   5. metadata-eval 89.0%

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

   6. associate-*r* 89.0%

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

   7. *-commutative 89.0%

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

   8. associate-*r* 89.0%

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

   9. metadata-eval 89.0%

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

   10. associate-*r* 89.0%

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

   11. metadata-eval 89.0%

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

   12. *-lft-identity 89.0%

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

   13. *-commutative 89.0%

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

   14. associate-*l* 89.0%

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

6. Simplified 89.0%

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

7. Taylor expanded in k around 0 30.7%

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

   1. add-sqr-sqrt 30.7%

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

   2. pow1/2 30.7%

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

   3. pow1/2 30.7%

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

   4. pow-prod-down 28.1%

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

   5. pow2 28.1%

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

   6. associate-/l* 28.1%

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

   7. *-commutative 28.1%

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

9. Applied egg-rr 28.1%

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

    1. unpow1/2 28.1%

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

    2. unpow2 28.1%

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

    3. rem-sqrt-square 31.4%

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

    4. associate-*r/ 31.4%

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

    5. associate-*r* 31.4%

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

    6. associate-*l/ 31.4%

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

    7. *-commutative 31.4%

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

    8. associate-/l* 31.4%

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

11. Simplified 31.4%

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

12. Final simplification 31.4%

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

Alternative 6: 39.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

   1. add-sqr-sqrt 97.8%

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

   2. sqrt-unprod 88.8%

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

   3. *-commutative 88.8%

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

   4. div-sub 88.8%

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

   5. metadata-eval 88.8%

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

   6. div-inv 88.9%

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

   7. *-commutative 88.9%

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

   8. div-sub 88.9%

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

   9. metadata-eval 88.9%

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

   10. div-inv 88.9%

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

4. Applied egg-rr 89.0%

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

   1. sqr-pow 88.9%

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

   2. pow-sqr 89.0%

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

   3. associate-*r* 89.0%

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

   4. associate-/l* 89.0%

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

   5. metadata-eval 89.0%

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

   6. associate-*r* 89.0%

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

   7. *-commutative 89.0%

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

   8. associate-*r* 89.0%

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

   9. metadata-eval 89.0%

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

   10. associate-*r* 89.0%

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

   11. metadata-eval 89.0%

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

   12. *-lft-identity 89.0%

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

   13. *-commutative 89.0%

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

   14. associate-*l* 89.0%

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

6. Simplified 89.0%

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

7. Taylor expanded in k around 0 30.7%

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

   1. sqrt-div 39.4%

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

   2. div-inv 39.3%

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

   3. *-commutative 39.3%

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

9. Applied egg-rr 39.3%

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

    1. associate-*r/ 39.4%

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

    2. *-rgt-identity 39.4%

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

    3. associate-*r* 39.4%

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

    4. *-commutative 39.4%

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

11. Simplified 39.4%

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

12. Final simplification 39.4%

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

Alternative 7: 30.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

   1. add-sqr-sqrt 97.8%

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

   2. sqrt-unprod 88.8%

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

   3. *-commutative 88.8%

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

   4. div-sub 88.8%

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

   5. metadata-eval 88.8%

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

   6. div-inv 88.9%

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

   7. *-commutative 88.9%

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

   8. div-sub 88.9%

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

   9. metadata-eval 88.9%

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

   10. div-inv 88.9%

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

4. Applied egg-rr 89.0%

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

   1. sqr-pow 88.9%

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

   2. pow-sqr 89.0%

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

   3. associate-*r* 89.0%

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

   4. associate-/l* 89.0%

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

   5. metadata-eval 89.0%

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

   6. associate-*r* 89.0%

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

   7. *-commutative 89.0%

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

   8. associate-*r* 89.0%

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

   9. metadata-eval 89.0%

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

   10. associate-*r* 89.0%

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

   11. metadata-eval 89.0%

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

   12. *-lft-identity 89.0%

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

   13. *-commutative 89.0%

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

   14. associate-*l* 89.0%

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

6. Simplified 89.0%

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

7. Taylor expanded in k around 0 30.7%

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

   1. associate-/l* 30.7%

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

   2. *-commutative 30.7%

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

9. Applied egg-rr 30.7%

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

    1. sqrt-prod 30.5%

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

    2. sqrt-div 39.2%

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

    3. associate-/l* 39.3%

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

    4. sqrt-prod 39.4%

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

    5. clear-num 39.3%

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

    6. inv-pow 39.3%

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

    7. sqrt-undiv 31.3%

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

    8. associate-*r* 31.3%

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

    9. *-commutative 31.3%

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

    10. sqrt-pow2 31.4%

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

    11. *-commutative 31.4%

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

    12. associate-*r* 31.4%

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

    13. *-un-lft-identity 31.4%

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

    14. times-frac 31.3%

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

    15. metadata-eval 31.3%

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

    16. *-commutative 31.3%

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

    17. metadata-eval 31.3%

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

11. Applied egg-rr 31.3%

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

12. Final simplification 31.3%

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

Alternative 8: 30.1% accurate, 2.0× 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 (* n (/ PI k)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

   1. add-sqr-sqrt 97.8%

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

   2. sqrt-unprod 88.8%

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

   3. *-commutative 88.8%

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

   4. div-sub 88.8%

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

   5. metadata-eval 88.8%

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

   6. div-inv 88.9%

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

   7. *-commutative 88.9%

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

   8. div-sub 88.9%

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

   9. metadata-eval 88.9%

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

   10. div-inv 88.9%

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

4. Applied egg-rr 89.0%

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

   1. sqr-pow 88.9%

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

   2. pow-sqr 89.0%

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

   3. associate-*r* 89.0%

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

   4. associate-/l* 89.0%

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

   5. metadata-eval 89.0%

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

   6. associate-*r* 89.0%

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

   7. *-commutative 89.0%

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

   8. associate-*r* 89.0%

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

   9. metadata-eval 89.0%

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

   10. associate-*r* 89.0%

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

   11. metadata-eval 89.0%

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

   12. *-lft-identity 89.0%

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

   13. *-commutative 89.0%

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

   14. associate-*l* 89.0%

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

6. Simplified 89.0%

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

7. Taylor expanded in k around 0 30.7%

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

   1. associate-/l* 30.7%

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

   2. *-commutative 30.7%

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

9. Applied egg-rr 30.7%

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

10. Taylor expanded in n around 0 30.7%

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

    1. associate-*r/ 30.7%

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

12. Simplified 30.7%

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

13. Final simplification 30.7%

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

Reproduce

herbie shell --seed 2024067 
(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))))