Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 20.3s

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: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-*r* 99.5%

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

   3. div-sub 99.5%

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

   4. metadata-eval 99.5%

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

   5. div-inv 99.6%

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

   6. pow-sub 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

   1. associate-*r* 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-*l* 99.7%

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

   4. associate-*r* 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l* 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.05e-82)
   (/ (sqrt (* n (* PI 2.0))) (sqrt k))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.05e-82) {
		tmp = sqrt((n * (((double) M_PI) * 2.0))) / 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 <= 1.05e-82) {
		tmp = Math.sqrt((n * (Math.PI * 2.0))) / 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 <= 1.05e-82:
		tmp = math.sqrt((n * (math.pi * 2.0))) / 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 <= 1.05e-82)
		tmp = Float64(sqrt(Float64(n * Float64(pi * 2.0))) / 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 <= 1.05e-82)
		tmp = sqrt((n * (pi * 2.0))) / 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, 1.05e-82], N[(N[Sqrt[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e-82

   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. add-sqr-sqrt 98.9%

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

      2. pow2 98.9%

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

   3. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{{\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{1 - k}{4}\right)}}{{k}^{0.25}}\right)}^{2}} \]

   4. Taylor expanded in k around 0 66.8%

      \[\leadsto {\color{blue}{\left({\left(2 \cdot \frac{n \cdot \pi}{k}\right)}^{0.25}\right)}}^{2} \]
   5. Step-by-step derivation

      1. associate-*r/ 66.8%

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

      2. *-commutative 66.8%

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

      3. associate-*r* 66.8%

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

      4. *-commutative 66.8%

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

      5. rem-exp-log 66.7%

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

      6. *-commutative 66.7%

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

      7. rem-exp-log 62.6%

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

      8. exp-sum 62.4%

         \[\leadsto {\left({\left(\frac{\color{blue}{e^{\log \left(2 \cdot \pi\right) + \log n}}}{k}\right)}^{0.25}\right)}^{2} \]

      9. +-commutative 62.4%

         \[\leadsto {\left({\left(\frac{e^{\color{blue}{\log n + \log \left(2 \cdot \pi\right)}}}{k}\right)}^{0.25}\right)}^{2} \]

      10. exp-sum 62.6%

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

      11. rem-exp-log 66.7%

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

      12. *-commutative 66.7%

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

      13. rem-exp-log 66.8%

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

      14. *-commutative 66.8%

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

   6. Simplified 66.8%

      \[\leadsto {\color{blue}{\left({\left(\frac{n \cdot \left(2 \cdot \pi\right)}{k}\right)}^{0.25}\right)}}^{2} \]
   7. Step-by-step derivation

      1. pow-pow 67.2%

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

      2. metadata-eval 67.2%

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

      3. pow1/2 67.2%

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

      4. *-commutative 67.2%

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

      5. associate-*r* 67.2%

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

      6. sqrt-div 99.5%

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

      7. associate-*r* 99.5%

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

      8. *-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

   if 1.05e-82 < k

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 99.6%

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

      2. sqrt-unprod 99.6%

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

      3. *-commutative 99.6%

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

      4. div-inv 99.6%

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

      5. *-commutative 99.6%

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

      6. div-inv 99.6%

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

      7. frac-times 99.6%

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

   3. Applied egg-rr 99.6%

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

   5. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

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(2.0 * Float64(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[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(0.5 - N[(k / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. sqr-pow 99.3%

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

   4. pow-sqr 99.6%

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

   5. associate-*l* 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*l/ 99.6%

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

   8. associate-/l* 99.6%

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

   9. metadata-eval 99.6%

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

   10. /-rgt-identity 99.6%

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

   11. div-sub 99.6%

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

   12. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 4: 50.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. pow2 99.3%

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

3. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{{\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{1 - k}{4}\right)}}{{k}^{0.25}}\right)}^{2}} \]

4. Taylor expanded in k around 0 39.6%

   \[\leadsto {\color{blue}{\left({\left(2 \cdot \frac{n \cdot \pi}{k}\right)}^{0.25}\right)}}^{2} \]
5. Step-by-step derivation

   1. associate-*r/ 39.6%

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

   2. *-commutative 39.6%

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

   3. associate-*r* 39.6%

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

   4. *-commutative 39.6%

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

   5. rem-exp-log 39.5%

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

   6. *-commutative 39.5%

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

   7. rem-exp-log 37.2%

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

   8. exp-sum 37.0%

      \[\leadsto {\left({\left(\frac{\color{blue}{e^{\log \left(2 \cdot \pi\right) + \log n}}}{k}\right)}^{0.25}\right)}^{2} \]

   9. +-commutative 37.0%

      \[\leadsto {\left({\left(\frac{e^{\color{blue}{\log n + \log \left(2 \cdot \pi\right)}}}{k}\right)}^{0.25}\right)}^{2} \]

   10. exp-sum 37.2%

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

   11. rem-exp-log 39.5%

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

   12. *-commutative 39.5%

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

   13. rem-exp-log 39.6%

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

   14. *-commutative 39.6%

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

6. Simplified 39.6%

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

7. Taylor expanded in n around 0 48.1%

   \[\leadsto \color{blue}{{\left(e^{0.25 \cdot \left(\log n + \log \left(2 \cdot \frac{\pi}{k}\right)\right)}\right)}^{2}} \]

9. Simplified 39.7%

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

    1. associate-*r* 39.7%

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

    2. sqrt-prod 52.4%

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

11. Applied egg-rr 52.4%

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

12. Final simplification 52.4%

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

Alternative 5: 50.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. pow2 99.3%

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

3. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{{\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{1 - k}{4}\right)}}{{k}^{0.25}}\right)}^{2}} \]

4. Taylor expanded in k around 0 39.6%

   \[\leadsto {\color{blue}{\left({\left(2 \cdot \frac{n \cdot \pi}{k}\right)}^{0.25}\right)}}^{2} \]
5. Step-by-step derivation

   1. associate-*r/ 39.6%

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

   2. *-commutative 39.6%

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

   3. associate-*r* 39.6%

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

   4. *-commutative 39.6%

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

   5. rem-exp-log 39.5%

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

   6. *-commutative 39.5%

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

   7. rem-exp-log 37.2%

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

   8. exp-sum 37.0%

      \[\leadsto {\left({\left(\frac{\color{blue}{e^{\log \left(2 \cdot \pi\right) + \log n}}}{k}\right)}^{0.25}\right)}^{2} \]

   9. +-commutative 37.0%

      \[\leadsto {\left({\left(\frac{e^{\color{blue}{\log n + \log \left(2 \cdot \pi\right)}}}{k}\right)}^{0.25}\right)}^{2} \]

   10. exp-sum 37.2%

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

   11. rem-exp-log 39.5%

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

   12. *-commutative 39.5%

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

   13. rem-exp-log 39.6%

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

   14. *-commutative 39.6%

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

6. Simplified 39.6%

   \[\leadsto {\color{blue}{\left({\left(\frac{n \cdot \left(2 \cdot \pi\right)}{k}\right)}^{0.25}\right)}}^{2} \]
7. Step-by-step derivation

   1. pow-pow 39.7%

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

   2. metadata-eval 39.7%

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

   3. pow1/2 39.7%

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

   4. associate-/l* 39.7%

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

   5. sqrt-div 52.4%

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

8. Applied egg-rr 52.4%

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

9. Final simplification 52.4%

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

Alternative 6: 50.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. pow2 99.3%

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

3. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{{\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{1 - k}{4}\right)}}{{k}^{0.25}}\right)}^{2}} \]

4. Taylor expanded in k around 0 39.6%

   \[\leadsto {\color{blue}{\left({\left(2 \cdot \frac{n \cdot \pi}{k}\right)}^{0.25}\right)}}^{2} \]
5. Step-by-step derivation

   1. associate-*r/ 39.6%

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

   2. *-commutative 39.6%

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

   3. associate-*r* 39.6%

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

   4. *-commutative 39.6%

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

   5. rem-exp-log 39.5%

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

   6. *-commutative 39.5%

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

   7. rem-exp-log 37.2%

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

   8. exp-sum 37.0%

      \[\leadsto {\left({\left(\frac{\color{blue}{e^{\log \left(2 \cdot \pi\right) + \log n}}}{k}\right)}^{0.25}\right)}^{2} \]

   9. +-commutative 37.0%

      \[\leadsto {\left({\left(\frac{e^{\color{blue}{\log n + \log \left(2 \cdot \pi\right)}}}{k}\right)}^{0.25}\right)}^{2} \]

   10. exp-sum 37.2%

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

   11. rem-exp-log 39.5%

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

   12. *-commutative 39.5%

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

   13. rem-exp-log 39.6%

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

   14. *-commutative 39.6%

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

6. Simplified 39.6%

   \[\leadsto {\color{blue}{\left({\left(\frac{n \cdot \left(2 \cdot \pi\right)}{k}\right)}^{0.25}\right)}}^{2} \]
7. Step-by-step derivation

   1. pow-pow 39.7%

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

   2. metadata-eval 39.7%

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

   3. pow1/2 39.7%

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

   4. *-commutative 39.7%

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

   5. associate-*r* 39.7%

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

   6. sqrt-div 52.5%

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

   7. associate-*r* 52.5%

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

   8. *-commutative 52.5%

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

8. Applied egg-rr 52.5%

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

9. Final simplification 52.5%

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

Alternative 7: 39.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. pow2 99.3%

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

3. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{{\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{1 - k}{4}\right)}}{{k}^{0.25}}\right)}^{2}} \]

4. Taylor expanded in k around 0 39.6%

   \[\leadsto {\color{blue}{\left({\left(2 \cdot \frac{n \cdot \pi}{k}\right)}^{0.25}\right)}}^{2} \]
5. Step-by-step derivation

   1. associate-*r/ 39.6%

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

   2. *-commutative 39.6%

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

   3. associate-*r* 39.6%

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

   4. *-commutative 39.6%

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

   5. rem-exp-log 39.5%

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

   6. *-commutative 39.5%

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

   7. rem-exp-log 37.2%

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

   8. exp-sum 37.0%

      \[\leadsto {\left({\left(\frac{\color{blue}{e^{\log \left(2 \cdot \pi\right) + \log n}}}{k}\right)}^{0.25}\right)}^{2} \]

   9. +-commutative 37.0%

      \[\leadsto {\left({\left(\frac{e^{\color{blue}{\log n + \log \left(2 \cdot \pi\right)}}}{k}\right)}^{0.25}\right)}^{2} \]

   10. exp-sum 37.2%

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

   11. rem-exp-log 39.5%

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

   12. *-commutative 39.5%

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

   13. rem-exp-log 39.6%

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

   14. *-commutative 39.6%

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

6. Simplified 39.6%

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

7. Taylor expanded in n around 0 48.1%

   \[\leadsto \color{blue}{{\left(e^{0.25 \cdot \left(\log n + \log \left(2 \cdot \frac{\pi}{k}\right)\right)}\right)}^{2}} \]

9. Simplified 39.7%

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

    1. associate-*l/ 39.7%

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

    2. *-commutative 39.7%

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

    3. associate-*r/ 39.7%

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

    4. *-commutative 39.7%

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

    5. associate-*r* 39.7%

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

    6. *-commutative 39.7%

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

    7. sqrt-undiv 52.5%

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

    8. clear-num 52.4%

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

    9. sqrt-div 39.7%

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

    10. inv-pow 39.7%

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

    11. sqrt-pow2 39.8%

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

    12. div-inv 39.8%

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

    13. *-commutative 39.8%

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

    14. associate-*r* 39.8%

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

    15. *-commutative 39.8%

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

    16. associate-/r* 39.8%

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

    17. metadata-eval 39.8%

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

    18. *-commutative 39.8%

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

    19. metadata-eval 39.8%

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

11. Applied egg-rr 39.8%

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

12. Final simplification 39.8%

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

Alternative 8: 38.6% accurate, 1.5× 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 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. pow2 99.3%

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

3. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{{\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{1 - k}{4}\right)}}{{k}^{0.25}}\right)}^{2}} \]

4. Taylor expanded in k around 0 39.6%

   \[\leadsto {\color{blue}{\left({\left(2 \cdot \frac{n \cdot \pi}{k}\right)}^{0.25}\right)}}^{2} \]
5. Step-by-step derivation

   1. associate-*r/ 39.6%

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

   2. *-commutative 39.6%

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

   3. associate-*r* 39.6%

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

   4. *-commutative 39.6%

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

   5. rem-exp-log 39.5%

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

   6. *-commutative 39.5%

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

   7. rem-exp-log 37.2%

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

   8. exp-sum 37.0%

      \[\leadsto {\left({\left(\frac{\color{blue}{e^{\log \left(2 \cdot \pi\right) + \log n}}}{k}\right)}^{0.25}\right)}^{2} \]

   9. +-commutative 37.0%

      \[\leadsto {\left({\left(\frac{e^{\color{blue}{\log n + \log \left(2 \cdot \pi\right)}}}{k}\right)}^{0.25}\right)}^{2} \]

   10. exp-sum 37.2%

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

   11. rem-exp-log 39.5%

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

   12. *-commutative 39.5%

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

   13. rem-exp-log 39.6%

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

   14. *-commutative 39.6%

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

6. Simplified 39.6%

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

7. Taylor expanded in n around 0 48.1%

   \[\leadsto \color{blue}{{\left(e^{0.25 \cdot \left(\log n + \log \left(2 \cdot \frac{\pi}{k}\right)\right)}\right)}^{2}} \]

9. Simplified 39.7%

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

10. Final simplification 39.7%

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

Alternative 9: 38.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. pow2 99.3%

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

3. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{{\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{1 - k}{4}\right)}}{{k}^{0.25}}\right)}^{2}} \]

4. Taylor expanded in k around 0 39.6%

   \[\leadsto {\color{blue}{\left({\left(2 \cdot \frac{n \cdot \pi}{k}\right)}^{0.25}\right)}}^{2} \]
5. Step-by-step derivation

   1. associate-*r/ 39.6%

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

   2. *-commutative 39.6%

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

   3. associate-*r* 39.6%

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

   4. *-commutative 39.6%

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

   5. rem-exp-log 39.5%

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

   6. *-commutative 39.5%

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

   7. rem-exp-log 37.2%

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

   8. exp-sum 37.0%

      \[\leadsto {\left({\left(\frac{\color{blue}{e^{\log \left(2 \cdot \pi\right) + \log n}}}{k}\right)}^{0.25}\right)}^{2} \]

   9. +-commutative 37.0%

      \[\leadsto {\left({\left(\frac{e^{\color{blue}{\log n + \log \left(2 \cdot \pi\right)}}}{k}\right)}^{0.25}\right)}^{2} \]

   10. exp-sum 37.2%

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

   11. rem-exp-log 39.5%

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

   12. *-commutative 39.5%

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

   13. rem-exp-log 39.6%

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

   14. *-commutative 39.6%

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

6. Simplified 39.6%

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

7. Taylor expanded in n around 0 48.1%

   \[\leadsto \color{blue}{{\left(e^{0.25 \cdot \left(\log n + \log \left(2 \cdot \frac{\pi}{k}\right)\right)}\right)}^{2}} \]

9. Simplified 39.7%

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

    1. associate-*l/ 39.7%

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

11. Applied egg-rr 39.7%

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

12. Final simplification 39.7%

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

Reproduce

herbie shell --seed 2023308 
(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))))