Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.6%

Time: 15.9s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(n \cdot 2\right)\\ {k}^{-0.5} \cdot \frac{\sqrt{t_0}}{{t_0}^{\left(k \cdot 0.5\right)}} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(k, n):
	t_0 = math.pi * (n * 2.0)
	return math.pow(k, -0.5) * (math.sqrt(t_0) / math.pow(t_0, (k * 0.5)))

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(n * 2.0))
	return Float64((k ^ -0.5) * Float64(sqrt(t_0) / (t_0 ^ Float64(k * 0.5))))
end

MATLAB

function tmp = code(k, n)
	t_0 = pi * (n * 2.0);
	tmp = (k ^ -0.5) * (sqrt(t_0) / (t_0 ^ (k * 0.5)));
end

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. div-sub 99.3%

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

   2. metadata-eval 99.3%

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

   3. pow-sub 99.6%

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

   4. pow1/2 99.6%

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

   5. associate-*l* 99.6%

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

   6. associate-*l* 99.6%

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

   7. div-inv 99.6%

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

   8. metadata-eval 99.6%

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

3. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. *-commutative 99.6%

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

   4. *-commutative 99.6%

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

   5. associate-*r* 99.6%

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

   6. *-commutative 99.6%

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

   7. *-commutative 99.6%

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

5. Simplified 99.6%

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

   1. expm1-log1p-u 96.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{k}}\right)\right)} \cdot \frac{\sqrt{\left(n \cdot 2\right) \cdot \pi}}{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(0.5 \cdot k\right)}} \]

   2. expm1-udef 74.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{k}}\right)} - 1\right)} \cdot \frac{\sqrt{\left(n \cdot 2\right) \cdot \pi}}{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(0.5 \cdot k\right)}} \]

   3. inv-pow 74.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{k}\right)}^{-1}}\right)} - 1\right) \cdot \frac{\sqrt{\left(n \cdot 2\right) \cdot \pi}}{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(0.5 \cdot k\right)}} \]

   4. sqrt-pow2 74.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{k}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \cdot \frac{\sqrt{\left(n \cdot 2\right) \cdot \pi}}{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(0.5 \cdot k\right)}} \]

   5. metadata-eval 74.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left({k}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot \frac{\sqrt{\left(n \cdot 2\right) \cdot \pi}}{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(0.5 \cdot k\right)}} \]

7. Applied egg-rr 74.2%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-0.5}\right)} - 1\right)} \cdot \frac{\sqrt{\left(n \cdot 2\right) \cdot \pi}}{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(0.5 \cdot k\right)}} \]
8. Step-by-step derivation

   1. expm1-def 96.4%

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

   2. expm1-log1p 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.4e-18)
   (/ (sqrt (* n (* 2.0 PI))) (sqrt k))
   (/ 1.0 (sqrt (/ k (pow (* PI (* n 2.0)) (- 1.0 k)))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.4e-18) {
		tmp = sqrt((n * (2.0 * ((double) M_PI)))) / sqrt(k);
	} else {
		tmp = 1.0 / sqrt((k / pow((((double) M_PI) * (n * 2.0)), (1.0 - k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.4e-18) {
		tmp = Math.sqrt((n * (2.0 * Math.PI))) / Math.sqrt(k);
	} else {
		tmp = 1.0 / Math.sqrt((k / Math.pow((Math.PI * (n * 2.0)), (1.0 - k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.4e-18:
		tmp = math.sqrt((n * (2.0 * math.pi))) / math.sqrt(k)
	else:
		tmp = 1.0 / math.sqrt((k / math.pow((math.pi * (n * 2.0)), (1.0 - k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.4e-18)
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * pi))) / sqrt(k));
	else
		tmp = Float64(1.0 / sqrt(Float64(k / (Float64(pi * Float64(n * 2.0)) ^ Float64(1.0 - k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.4e-18)
		tmp = sqrt((n * (2.0 * pi))) / sqrt(k);
	else
		tmp = 1.0 / sqrt((k / ((pi * (n * 2.0)) ^ (1.0 - k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.4e-18], N[(N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(k / N[Power[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.39999999999999994e-18

   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. *-commutative 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*r* 99.3%

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

      4. div-inv 99.4%

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

      5. expm1-log1p-u 93.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

      6. expm1-udef 73.3%

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

   3. Applied egg-rr 43.7%

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

      1. expm1-def 64.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

      2. expm1-log1p 66.9%

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

      3. *-commutative 66.9%

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

      4. associate-*r* 66.9%

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

      5. *-commutative 66.9%

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

   5. Simplified 66.9%

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

      1. add-cbrt-cube 53.8%

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

      2. add-sqr-sqrt 53.8%

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

      3. *-commutative 53.8%

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

      4. *-commutative 53.8%

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

   7. Applied egg-rr 53.8%

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

   9. Simplified 53.7%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5}}} \]

   10. Taylor expanded in k around 0 49.8%

       \[\leadsto \sqrt[3]{\color{blue}{e^{1.5 \cdot \left(-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)\right)}}} \]
   11. Step-by-step derivation

       1. *-commutative 49.8%

          \[\leadsto \sqrt[3]{e^{\color{blue}{\left(-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)\right) \cdot 1.5}}} \]

       2. exp-prod 50.0%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)}\right)}^{1.5}}} \]

       3. +-commutative 50.0%

          \[\leadsto \sqrt[3]{{\left(e^{\color{blue}{\log \left(2 \cdot \left(n \cdot \pi\right)\right) + -1 \cdot \log k}}\right)}^{1.5}} \]

       4. mul-1-neg 50.0%

          \[\leadsto \sqrt[3]{{\left(e^{\log \left(2 \cdot \left(n \cdot \pi\right)\right) + \color{blue}{\left(-\log k\right)}}\right)}^{1.5}} \]

       5. unsub-neg 50.0%

          \[\leadsto \sqrt[3]{{\left(e^{\color{blue}{\log \left(2 \cdot \left(n \cdot \pi\right)\right) - \log k}}\right)}^{1.5}} \]

       6. exp-diff 50.2%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{e^{\log \left(2 \cdot \left(n \cdot \pi\right)\right)}}{e^{\log k}}\right)}}^{1.5}} \]

       7. associate-*r* 50.2%

          \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \color{blue}{\left(\left(2 \cdot n\right) \cdot \pi\right)}}}{e^{\log k}}\right)}^{1.5}} \]

       8. *-commutative 50.2%

          \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \left(\color{blue}{\left(n \cdot 2\right)} \cdot \pi\right)}}{e^{\log k}}\right)}^{1.5}} \]

       9. associate-*r* 50.2%

          \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \color{blue}{\left(n \cdot \left(2 \cdot \pi\right)\right)}}}{e^{\log k}}\right)}^{1.5}} \]

       10. rem-exp-log 50.5%

           \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{n \cdot \left(2 \cdot \pi\right)}}{e^{\log k}}\right)}^{1.5}} \]

       11. associate-*r* 50.5%

           \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(n \cdot 2\right) \cdot \pi}}{e^{\log k}}\right)}^{1.5}} \]

       12. *-commutative 50.5%

           \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\pi \cdot \left(n \cdot 2\right)}}{e^{\log k}}\right)}^{1.5}} \]

       13. rem-exp-log 53.7%

           \[\leadsto \sqrt[3]{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{\color{blue}{k}}\right)}^{1.5}} \]

   12. Simplified 53.7%

       \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{1.5}}} \]
   13. Step-by-step derivation

       1. pow1/3 50.5%

          \[\leadsto \color{blue}{{\left({\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       2. pow-pow 66.9%

          \[\leadsto \color{blue}{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       3. metadata-eval 66.9%

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

       4. pow1/2 66.9%

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

       5. sqrt-div 99.4%

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

       6. *-commutative 99.4%

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

       7. associate-*l* 99.4%

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

   14. Applied egg-rr 99.4%

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

   if 2.39999999999999994e-18 < k

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*r* 99.4%

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

      3. associate-/r/ 99.4%

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

      4. add-sqr-sqrt 99.3%

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

      5. sqrt-unprod 99.4%

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

      6. associate-*r* 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-*r* 99.4%

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

      9. *-commutative 99.4%

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

      10. pow-prod-up 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in n around 0 99.2%

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

      1. distribute-lft-in 99.2%

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

      2. remove-double-neg 99.2%

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

      3. log-rec 99.2%

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

      4. mul-1-neg 99.2%

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

      5. distribute-lft-in 99.2%

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

      6. *-commutative 99.2%

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

   6. Simplified 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double k, double n) {
	return Math.pow(k, -0.5) / Math.pow((2.0 * (n * Math.PI)), (-0.5 * (1.0 - k)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r* 99.3%

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

   3. associate-/r/ 99.3%

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

   4. add-sqr-sqrt 99.1%

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

   5. sqrt-unprod 99.3%

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

   6. associate-*r* 99.3%

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

   7. *-commutative 99.3%

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

   8. associate-*r* 99.3%

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

   9. *-commutative 99.3%

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

   10. pow-prod-up 99.3%

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

3. Applied egg-rr 99.3%

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

   1. div-inv 99.3%

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

   2. pow1/2 99.3%

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

   3. associate-*r* 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. pow-flip 99.4%

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

   7. associate-*r* 99.4%

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

   8. *-commutative 99.4%

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

   9. *-commutative 99.4%

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

   10. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

   1. associate-*r* 99.4%

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

7. Simplified 99.4%

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

   1. expm1-log1p-u 96.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{k} \cdot {\left({\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(1 - k\right)}\right)}^{-0.5}}\right)\right)} \]

   2. expm1-udef 86.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\sqrt{k} \cdot {\left({\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(1 - k\right)}\right)}^{-0.5}}\right)} - 1} \]

   3. associate-/r* 86.7%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{\sqrt{k}}}{{\left({\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(1 - k\right)}\right)}^{-0.5}}}\right)} - 1 \]

   4. pow1/2 86.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{\color{blue}{{k}^{0.5}}}}{{\left({\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(1 - k\right)}\right)}^{-0.5}}\right)} - 1 \]

   5. pow-flip 86.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{k}^{\left(-0.5\right)}}}{{\left({\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(1 - k\right)}\right)}^{-0.5}}\right)} - 1 \]

   6. metadata-eval 86.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{{k}^{\color{blue}{-0.5}}}{{\left({\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(1 - k\right)}\right)}^{-0.5}}\right)} - 1 \]

   7. pow-pow 86.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{{k}^{-0.5}}{\color{blue}{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(\left(1 - k\right) \cdot -0.5\right)}}}\right)} - 1 \]

   8. associate-*l* 86.7%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{{k}^{-0.5}}{{\color{blue}{\left(n \cdot \left(2 \cdot \pi\right)\right)}}^{\left(\left(1 - k\right) \cdot -0.5\right)}}\right)} - 1 \]

9. Applied egg-rr 86.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{k}^{-0.5}}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\left(1 - k\right) \cdot -0.5\right)}}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 96.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{-0.5}}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\left(1 - k\right) \cdot -0.5\right)}}\right)\right)} \]

    2. expm1-log1p 99.4%

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

    3. *-commutative 99.4%

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

    4. associate-*r* 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.5e-18)
   (/ (sqrt (* n (* 2.0 PI))) (sqrt k))
   (sqrt (/ (pow (* PI (* n 2.0)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.5e-18) {
		tmp = sqrt((n * (2.0 * ((double) M_PI)))) / sqrt(k);
	} else {
		tmp = sqrt((pow((((double) M_PI) * (n * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.5e-18) {
		tmp = Math.sqrt((n * (2.0 * Math.PI))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (n * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.5e-18:
		tmp = math.sqrt((n * (2.0 * math.pi))) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow((math.pi * (n * 2.0)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.5e-18)
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * pi))) / sqrt(k));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(n * 2.0)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.5e-18)
		tmp = sqrt((n * (2.0 * pi))) / sqrt(k);
	else
		tmp = sqrt((((pi * (n * 2.0)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.5e-18], 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[(n * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.50000000000000018e-18

   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. *-commutative 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*r* 99.3%

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

      4. div-inv 99.4%

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

      5. expm1-log1p-u 93.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

      6. expm1-udef 73.3%

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

   3. Applied egg-rr 43.7%

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

      1. expm1-def 64.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

      2. expm1-log1p 66.9%

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

      3. *-commutative 66.9%

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

      4. associate-*r* 66.9%

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

      5. *-commutative 66.9%

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

   5. Simplified 66.9%

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

      1. add-cbrt-cube 53.8%

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

      2. add-sqr-sqrt 53.8%

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

      3. *-commutative 53.8%

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

      4. *-commutative 53.8%

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

   7. Applied egg-rr 53.8%

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

   9. Simplified 53.7%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5}}} \]

   10. Taylor expanded in k around 0 49.8%

       \[\leadsto \sqrt[3]{\color{blue}{e^{1.5 \cdot \left(-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)\right)}}} \]
   11. Step-by-step derivation

       1. *-commutative 49.8%

          \[\leadsto \sqrt[3]{e^{\color{blue}{\left(-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)\right) \cdot 1.5}}} \]

       2. exp-prod 50.0%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)}\right)}^{1.5}}} \]

       3. +-commutative 50.0%

          \[\leadsto \sqrt[3]{{\left(e^{\color{blue}{\log \left(2 \cdot \left(n \cdot \pi\right)\right) + -1 \cdot \log k}}\right)}^{1.5}} \]

       4. mul-1-neg 50.0%

          \[\leadsto \sqrt[3]{{\left(e^{\log \left(2 \cdot \left(n \cdot \pi\right)\right) + \color{blue}{\left(-\log k\right)}}\right)}^{1.5}} \]

       5. unsub-neg 50.0%

          \[\leadsto \sqrt[3]{{\left(e^{\color{blue}{\log \left(2 \cdot \left(n \cdot \pi\right)\right) - \log k}}\right)}^{1.5}} \]

       6. exp-diff 50.2%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{e^{\log \left(2 \cdot \left(n \cdot \pi\right)\right)}}{e^{\log k}}\right)}}^{1.5}} \]

       7. associate-*r* 50.2%

          \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \color{blue}{\left(\left(2 \cdot n\right) \cdot \pi\right)}}}{e^{\log k}}\right)}^{1.5}} \]

       8. *-commutative 50.2%

          \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \left(\color{blue}{\left(n \cdot 2\right)} \cdot \pi\right)}}{e^{\log k}}\right)}^{1.5}} \]

       9. associate-*r* 50.2%

          \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \color{blue}{\left(n \cdot \left(2 \cdot \pi\right)\right)}}}{e^{\log k}}\right)}^{1.5}} \]

       10. rem-exp-log 50.5%

           \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{n \cdot \left(2 \cdot \pi\right)}}{e^{\log k}}\right)}^{1.5}} \]

       11. associate-*r* 50.5%

           \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(n \cdot 2\right) \cdot \pi}}{e^{\log k}}\right)}^{1.5}} \]

       12. *-commutative 50.5%

           \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\pi \cdot \left(n \cdot 2\right)}}{e^{\log k}}\right)}^{1.5}} \]

       13. rem-exp-log 53.7%

           \[\leadsto \sqrt[3]{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{\color{blue}{k}}\right)}^{1.5}} \]

   12. Simplified 53.7%

       \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{1.5}}} \]
   13. Step-by-step derivation

       1. pow1/3 50.5%

          \[\leadsto \color{blue}{{\left({\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       2. pow-pow 66.9%

          \[\leadsto \color{blue}{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       3. metadata-eval 66.9%

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

       4. pow1/2 66.9%

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

       5. sqrt-div 99.4%

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

       6. *-commutative 99.4%

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

       7. associate-*l* 99.4%

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

   14. Applied egg-rr 99.4%

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

   if 2.50000000000000018e-18 < k

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-*r* 99.4%

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

      4. div-inv 99.4%

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

      5. expm1-log1p-u 99.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

      6. expm1-udef 97.6%

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

   3. Applied egg-rr 97.6%

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

      1. expm1-def 99.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

      2. expm1-log1p 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-*r* 99.4%

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

      5. *-commutative 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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. *-commutative 99.4%

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

   4. associate-*l* 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 6: 38.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (k n) :precision binary64 (/ (sqrt PI) (sqrt (/ k (+ n n)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. *-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*r* 99.3%

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

   4. div-inv 99.4%

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

   5. expm1-log1p-u 96.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

   6. expm1-udef 86.7%

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

3. Applied egg-rr 73.3%

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

   1. expm1-def 83.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

   2. expm1-log1p 84.8%

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

   3. *-commutative 84.8%

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

   4. associate-*r* 84.8%

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

   5. *-commutative 84.8%

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

5. Simplified 84.8%

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

   1. add-cbrt-cube 78.6%

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

   2. add-sqr-sqrt 78.5%

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

   3. *-commutative 78.5%

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

   4. *-commutative 78.5%

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

7. Applied egg-rr 78.5%

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

9. Simplified 78.5%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5}}} \]

10. Taylor expanded in k around 0 26.3%

    \[\leadsto \sqrt[3]{\color{blue}{e^{1.5 \cdot \left(-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)\right)}}} \]
11. Step-by-step derivation

    1. *-commutative 26.3%

       \[\leadsto \sqrt[3]{e^{\color{blue}{\left(-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)\right) \cdot 1.5}}} \]

    2. exp-prod 26.4%

       \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)}\right)}^{1.5}}} \]

    3. +-commutative 26.4%

       \[\leadsto \sqrt[3]{{\left(e^{\color{blue}{\log \left(2 \cdot \left(n \cdot \pi\right)\right) + -1 \cdot \log k}}\right)}^{1.5}} \]

    4. mul-1-neg 26.4%

       \[\leadsto \sqrt[3]{{\left(e^{\log \left(2 \cdot \left(n \cdot \pi\right)\right) + \color{blue}{\left(-\log k\right)}}\right)}^{1.5}} \]

    5. unsub-neg 26.4%

       \[\leadsto \sqrt[3]{{\left(e^{\color{blue}{\log \left(2 \cdot \left(n \cdot \pi\right)\right) - \log k}}\right)}^{1.5}} \]

    6. exp-diff 26.5%

       \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{e^{\log \left(2 \cdot \left(n \cdot \pi\right)\right)}}{e^{\log k}}\right)}}^{1.5}} \]

    7. associate-*r* 26.5%

       \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \color{blue}{\left(\left(2 \cdot n\right) \cdot \pi\right)}}}{e^{\log k}}\right)}^{1.5}} \]

    8. *-commutative 26.5%

       \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \left(\color{blue}{\left(n \cdot 2\right)} \cdot \pi\right)}}{e^{\log k}}\right)}^{1.5}} \]

    9. associate-*r* 26.5%

       \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \color{blue}{\left(n \cdot \left(2 \cdot \pi\right)\right)}}}{e^{\log k}}\right)}^{1.5}} \]

    10. rem-exp-log 26.6%

        \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{n \cdot \left(2 \cdot \pi\right)}}{e^{\log k}}\right)}^{1.5}} \]

    11. associate-*r* 26.6%

        \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(n \cdot 2\right) \cdot \pi}}{e^{\log k}}\right)}^{1.5}} \]

    12. *-commutative 26.6%

        \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\pi \cdot \left(n \cdot 2\right)}}{e^{\log k}}\right)}^{1.5}} \]

    13. rem-exp-log 28.0%

        \[\leadsto \sqrt[3]{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{\color{blue}{k}}\right)}^{1.5}} \]

12. Simplified 28.0%

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{1.5}}} \]
13. Step-by-step derivation

    1. pow1/3 26.6%

       \[\leadsto \color{blue}{{\left({\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

    2. pow-pow 33.7%

       \[\leadsto \color{blue}{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

    3. metadata-eval 33.7%

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

    4. pow1/2 33.7%

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

    5. associate-/l* 33.7%

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

    6. sqrt-div 35.0%

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

14. Applied egg-rr 35.0%

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

    1. *-commutative 35.0%

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

    2. rem-log-exp 4.3%

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

    3. log-pow 4.3%

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

    4. unpow2 4.3%

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

    5. log-prod 4.3%

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

    6. rem-log-exp 7.9%

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

    7. rem-log-exp 35.0%

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

16. Simplified 35.0%

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

17. Final simplification 35.0%

    \[\leadsto \frac{\sqrt{\pi}}{\sqrt{\frac{k}{n + n}}} \]

Alternative 7: 49.2% 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 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. *-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*r* 99.3%

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

   4. div-inv 99.4%

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

   5. expm1-log1p-u 96.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

   6. expm1-udef 86.7%

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

3. Applied egg-rr 73.3%

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

   1. expm1-def 83.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

   2. expm1-log1p 84.8%

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

   3. *-commutative 84.8%

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

   4. associate-*r* 84.8%

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

   5. *-commutative 84.8%

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

5. Simplified 84.8%

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

   1. add-cbrt-cube 78.6%

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

   2. add-sqr-sqrt 78.5%

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

   3. *-commutative 78.5%

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

   4. *-commutative 78.5%

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

7. Applied egg-rr 78.5%

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

9. Simplified 78.5%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5}}} \]

10. Taylor expanded in k around 0 26.3%

    \[\leadsto \sqrt[3]{\color{blue}{e^{1.5 \cdot \left(-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)\right)}}} \]
11. Step-by-step derivation

    1. *-commutative 26.3%

       \[\leadsto \sqrt[3]{e^{\color{blue}{\left(-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)\right) \cdot 1.5}}} \]

    2. exp-prod 26.4%

       \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)}\right)}^{1.5}}} \]

    3. +-commutative 26.4%

       \[\leadsto \sqrt[3]{{\left(e^{\color{blue}{\log \left(2 \cdot \left(n \cdot \pi\right)\right) + -1 \cdot \log k}}\right)}^{1.5}} \]

    4. mul-1-neg 26.4%

       \[\leadsto \sqrt[3]{{\left(e^{\log \left(2 \cdot \left(n \cdot \pi\right)\right) + \color{blue}{\left(-\log k\right)}}\right)}^{1.5}} \]

    5. unsub-neg 26.4%

       \[\leadsto \sqrt[3]{{\left(e^{\color{blue}{\log \left(2 \cdot \left(n \cdot \pi\right)\right) - \log k}}\right)}^{1.5}} \]

    6. exp-diff 26.5%

       \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{e^{\log \left(2 \cdot \left(n \cdot \pi\right)\right)}}{e^{\log k}}\right)}}^{1.5}} \]

    7. associate-*r* 26.5%

       \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \color{blue}{\left(\left(2 \cdot n\right) \cdot \pi\right)}}}{e^{\log k}}\right)}^{1.5}} \]

    8. *-commutative 26.5%

       \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \left(\color{blue}{\left(n \cdot 2\right)} \cdot \pi\right)}}{e^{\log k}}\right)}^{1.5}} \]

    9. associate-*r* 26.5%

       \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \color{blue}{\left(n \cdot \left(2 \cdot \pi\right)\right)}}}{e^{\log k}}\right)}^{1.5}} \]

    10. rem-exp-log 26.6%

        \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{n \cdot \left(2 \cdot \pi\right)}}{e^{\log k}}\right)}^{1.5}} \]

    11. associate-*r* 26.6%

        \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(n \cdot 2\right) \cdot \pi}}{e^{\log k}}\right)}^{1.5}} \]

    12. *-commutative 26.6%

        \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\pi \cdot \left(n \cdot 2\right)}}{e^{\log k}}\right)}^{1.5}} \]

    13. rem-exp-log 28.0%

        \[\leadsto \sqrt[3]{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{\color{blue}{k}}\right)}^{1.5}} \]

12. Simplified 28.0%

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{1.5}}} \]
13. Step-by-step derivation

    1. pow1/3 26.6%

       \[\leadsto \color{blue}{{\left({\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

    2. pow-pow 33.7%

       \[\leadsto \color{blue}{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

    3. metadata-eval 33.7%

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

    4. pow1/2 33.7%

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

    5. sqrt-div 48.3%

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

    6. *-commutative 48.3%

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

    7. associate-*l* 48.3%

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

14. Applied egg-rr 48.3%

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

15. Final simplification 48.3%

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

Alternative 8: 38.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. *-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*r* 99.3%

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

   4. div-inv 99.4%

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

   5. expm1-log1p-u 96.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

   6. expm1-udef 86.7%

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

3. Applied egg-rr 73.3%

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

   1. expm1-def 83.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

   2. expm1-log1p 84.8%

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

   3. *-commutative 84.8%

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

   4. associate-*r* 84.8%

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

   5. *-commutative 84.8%

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

5. Simplified 84.8%

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

   1. add-cbrt-cube 78.6%

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

   2. add-sqr-sqrt 78.5%

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

   3. *-commutative 78.5%

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

   4. *-commutative 78.5%

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

7. Applied egg-rr 78.5%

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

9. Simplified 78.5%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5}}} \]

10. Taylor expanded in k around 0 26.3%

    \[\leadsto \sqrt[3]{\color{blue}{e^{1.5 \cdot \left(-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)\right)}}} \]
11. Step-by-step derivation

    1. *-commutative 26.3%

       \[\leadsto \sqrt[3]{e^{\color{blue}{\left(-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)\right) \cdot 1.5}}} \]

    2. exp-prod 26.4%

       \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)}\right)}^{1.5}}} \]

    3. +-commutative 26.4%

       \[\leadsto \sqrt[3]{{\left(e^{\color{blue}{\log \left(2 \cdot \left(n \cdot \pi\right)\right) + -1 \cdot \log k}}\right)}^{1.5}} \]

    4. mul-1-neg 26.4%

       \[\leadsto \sqrt[3]{{\left(e^{\log \left(2 \cdot \left(n \cdot \pi\right)\right) + \color{blue}{\left(-\log k\right)}}\right)}^{1.5}} \]

    5. unsub-neg 26.4%

       \[\leadsto \sqrt[3]{{\left(e^{\color{blue}{\log \left(2 \cdot \left(n \cdot \pi\right)\right) - \log k}}\right)}^{1.5}} \]

    6. exp-diff 26.5%

       \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{e^{\log \left(2 \cdot \left(n \cdot \pi\right)\right)}}{e^{\log k}}\right)}}^{1.5}} \]

    7. associate-*r* 26.5%

       \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \color{blue}{\left(\left(2 \cdot n\right) \cdot \pi\right)}}}{e^{\log k}}\right)}^{1.5}} \]

    8. *-commutative 26.5%

       \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \left(\color{blue}{\left(n \cdot 2\right)} \cdot \pi\right)}}{e^{\log k}}\right)}^{1.5}} \]

    9. associate-*r* 26.5%

       \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \color{blue}{\left(n \cdot \left(2 \cdot \pi\right)\right)}}}{e^{\log k}}\right)}^{1.5}} \]

    10. rem-exp-log 26.6%

        \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{n \cdot \left(2 \cdot \pi\right)}}{e^{\log k}}\right)}^{1.5}} \]

    11. associate-*r* 26.6%

        \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(n \cdot 2\right) \cdot \pi}}{e^{\log k}}\right)}^{1.5}} \]

    12. *-commutative 26.6%

        \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\pi \cdot \left(n \cdot 2\right)}}{e^{\log k}}\right)}^{1.5}} \]

    13. rem-exp-log 28.0%

        \[\leadsto \sqrt[3]{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{\color{blue}{k}}\right)}^{1.5}} \]

12. Simplified 28.0%

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{1.5}}} \]
13. Step-by-step derivation

    1. pow1/3 26.6%

       \[\leadsto \color{blue}{{\left({\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

    2. pow-pow 33.7%

       \[\leadsto \color{blue}{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

    3. metadata-eval 33.7%

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

    4. pow1/2 33.7%

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

    5. clear-num 33.7%

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

    6. sqrt-div 35.0%

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

    7. metadata-eval 35.0%

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

    8. *-commutative 35.0%

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

    9. associate-*l* 35.0%

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

14. Applied egg-rr 35.0%

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

15. Final simplification 35.0%

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

Alternative 9: 38.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. *-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*r* 99.3%

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

   4. div-inv 99.4%

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

   5. expm1-log1p-u 96.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

   6. expm1-udef 86.7%

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

3. Applied egg-rr 73.3%

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

   1. expm1-def 83.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

   2. expm1-log1p 84.8%

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

   3. *-commutative 84.8%

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

   4. associate-*r* 84.8%

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

   5. *-commutative 84.8%

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

5. Simplified 84.8%

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

   1. add-cbrt-cube 78.6%

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

   2. add-sqr-sqrt 78.5%

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

   3. *-commutative 78.5%

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

   4. *-commutative 78.5%

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

7. Applied egg-rr 78.5%

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

9. Simplified 78.5%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5}}} \]

10. Taylor expanded in k around 0 26.3%

    \[\leadsto \sqrt[3]{\color{blue}{e^{1.5 \cdot \left(-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)\right)}}} \]
11. Step-by-step derivation

    1. *-commutative 26.3%

       \[\leadsto \sqrt[3]{e^{\color{blue}{\left(-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)\right) \cdot 1.5}}} \]

    2. exp-prod 26.4%

       \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)}\right)}^{1.5}}} \]

    3. +-commutative 26.4%

       \[\leadsto \sqrt[3]{{\left(e^{\color{blue}{\log \left(2 \cdot \left(n \cdot \pi\right)\right) + -1 \cdot \log k}}\right)}^{1.5}} \]

    4. mul-1-neg 26.4%

       \[\leadsto \sqrt[3]{{\left(e^{\log \left(2 \cdot \left(n \cdot \pi\right)\right) + \color{blue}{\left(-\log k\right)}}\right)}^{1.5}} \]

    5. unsub-neg 26.4%

       \[\leadsto \sqrt[3]{{\left(e^{\color{blue}{\log \left(2 \cdot \left(n \cdot \pi\right)\right) - \log k}}\right)}^{1.5}} \]

    6. exp-diff 26.5%

       \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{e^{\log \left(2 \cdot \left(n \cdot \pi\right)\right)}}{e^{\log k}}\right)}}^{1.5}} \]

    7. associate-*r* 26.5%

       \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \color{blue}{\left(\left(2 \cdot n\right) \cdot \pi\right)}}}{e^{\log k}}\right)}^{1.5}} \]

    8. *-commutative 26.5%

       \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \left(\color{blue}{\left(n \cdot 2\right)} \cdot \pi\right)}}{e^{\log k}}\right)}^{1.5}} \]

    9. associate-*r* 26.5%

       \[\leadsto \sqrt[3]{{\left(\frac{e^{\log \color{blue}{\left(n \cdot \left(2 \cdot \pi\right)\right)}}}{e^{\log k}}\right)}^{1.5}} \]

    10. rem-exp-log 26.6%

        \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{n \cdot \left(2 \cdot \pi\right)}}{e^{\log k}}\right)}^{1.5}} \]

    11. associate-*r* 26.6%

        \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(n \cdot 2\right) \cdot \pi}}{e^{\log k}}\right)}^{1.5}} \]

    12. *-commutative 26.6%

        \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\pi \cdot \left(n \cdot 2\right)}}{e^{\log k}}\right)}^{1.5}} \]

    13. rem-exp-log 28.0%

        \[\leadsto \sqrt[3]{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{\color{blue}{k}}\right)}^{1.5}} \]

12. Simplified 28.0%

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{1.5}}} \]
13. Step-by-step derivation

    1. expm1-log1p-u 27.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{1.5}}\right)\right)} \]

    2. expm1-udef 25.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{1.5}}\right)} - 1} \]

    3. pow1/3 25.8%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{1.5}\right)}^{0.3333333333333333}}\right)} - 1 \]

    4. pow-pow 30.7%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} - 1 \]

    5. metadata-eval 30.7%

       \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{\color{blue}{0.5}}\right)} - 1 \]

    6. pow1/2 30.7%

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

    7. associate-/l* 30.7%

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

14. Applied egg-rr 30.7%

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

    1. expm1-def 32.5%

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

    2. expm1-log1p 33.7%

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

    3. associate-/r/ 33.7%

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

    4. *-commutative 33.7%

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

    5. rem-log-exp 4.3%

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

    6. log-pow 4.3%

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

    7. unpow2 4.3%

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

    8. log-prod 4.3%

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

    9. rem-log-exp 7.9%

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

    10. rem-log-exp 33.7%

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

16. Simplified 33.7%

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

17. Final simplification 33.7%

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

Reproduce

herbie shell --seed 2023207 
(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))))