Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 98.8%

Time: 21.5s

Alternatives: 13

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: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.85 \cdot 10^{-78}:\\ \;\;\;\;\frac{1}{\sqrt{k \cdot \frac{0.5}{\pi}}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.85e-78)
   (* (/ 1.0 (sqrt (* k (/ 0.5 PI)))) (sqrt n))
   (sqrt (/ (pow (* PI (* n 2.0)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.85e-78) {
		tmp = (1.0 / sqrt((k * (0.5 / ((double) M_PI))))) * sqrt(n);
	} 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 <= 2.85e-78) {
		tmp = (1.0 / Math.sqrt((k * (0.5 / Math.PI)))) * Math.sqrt(n);
	} 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 <= 2.85e-78:
		tmp = (1.0 / math.sqrt((k * (0.5 / math.pi)))) * math.sqrt(n)
	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 <= 2.85e-78)
		tmp = Float64(Float64(1.0 / sqrt(Float64(k * Float64(0.5 / pi)))) * sqrt(n));
	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 <= 2.85e-78)
		tmp = (1.0 / sqrt((k * (0.5 / pi)))) * sqrt(n);
	else
		tmp = sqrt((((pi * (n * 2.0)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.85e-78], N[(N[(1.0 / N[Sqrt[N[(k * N[(0.5 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[n], $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 < 2.8499999999999999e-78

   1. Initial program 98.4%

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

   3. Taylor expanded in k around 0 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-/l* 63.8%

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

   5. Simplified 63.8%

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

      1. pow1 63.8%

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

      2. sqrt-unprod 63.9%

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

      3. associate-*r* 63.9%

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

      4. *-commutative 63.9%

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

   7. Applied egg-rr 63.9%

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

      1. unpow1 63.9%

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

      2. associate-*l* 63.9%

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

      3. metadata-eval 63.9%

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

      4. times-frac 63.9%

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

      5. *-commutative 63.9%

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

      6. times-frac 63.8%

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

      7. /-rgt-identity 63.8%

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

   9. Simplified 63.8%

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

       1. *-commutative 63.8%

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

       2. sqrt-prod 98.5%

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

       3. clear-num 98.5%

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

       4. un-div-inv 98.4%

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

       5. div-inv 98.4%

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

       6. metadata-eval 98.4%

          \[\leadsto \sqrt{\frac{\pi}{k \cdot \color{blue}{0.5}}} \cdot \sqrt{n} \]

   11. Applied egg-rr 98.4%

       \[\leadsto \color{blue}{\sqrt{\frac{\pi}{k \cdot 0.5}} \cdot \sqrt{n}} \]
   12. Step-by-step derivation

       1. clear-num 98.4%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{k \cdot 0.5}{\pi}}}} \cdot \sqrt{n} \]

       2. sqrt-div 99.3%

          \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{k \cdot 0.5}{\pi}}}} \cdot \sqrt{n} \]

       3. metadata-eval 99.3%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{k \cdot 0.5}{\pi}}} \cdot \sqrt{n} \]

       4. associate-/l* 99.3%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{k \cdot \frac{0.5}{\pi}}}} \cdot \sqrt{n} \]

   13. Applied egg-rr 99.3%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{k \cdot \frac{0.5}{\pi}}}} \cdot \sqrt{n} \]

   if 2.8499999999999999e-78 < k

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 99.5%

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

      2. sqrt-unprod 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*r* 99.5%

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

      5. div-sub 99.5%

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

      6. metadata-eval 99.5%

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

      7. div-inv 99.5%

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

      8. *-commutative 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.6%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(n \cdot 2\right)\\ \frac{1}{\sqrt{k}} \cdot \left(\sqrt{t\_0} \cdot {t\_0}^{\left(k \cdot -0.5\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* n 2.0))))
   (* (/ 1.0 (sqrt k)) (* (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 (1.0 / sqrt(k)) * (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 (1.0 / Math.sqrt(k)) * (Math.sqrt(t_0) * Math.pow(t_0, (k * -0.5)));
}

Python

def code(k, n):
	t_0 = math.pi * (n * 2.0)
	return (1.0 / math.sqrt(k)) * (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(Float64(1.0 / sqrt(k)) * Float64(sqrt(t_0) * (t_0 ^ Float64(k * -0.5))))
end

MATLAB

function tmp = code(k, n)
	t_0 = pi * (n * 2.0);
	tmp = (1.0 / sqrt(k)) * (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[(1.0 / N[Sqrt[k], $MachinePrecision]), $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 99.1%

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

   1. associate-*r* 99.1%

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

   2. div-sub 99.1%

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

   3. metadata-eval 99.1%

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

   4. sub-neg 99.1%

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

   5. unpow-prod-up 99.3%

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

   6. pow1/2 99.3%

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

   7. div-inv 99.3%

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

   8. metadata-eval 99.3%

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

   9. distribute-rgt-neg-in 99.3%

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

   10. metadata-eval 99.3%

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

4. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*l* 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*l* 99.3%

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

6. Simplified 99.3%

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

Alternative 3: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(n \cdot 2\right)\\ \frac{\sqrt{t\_0}}{\sqrt{k} \cdot {t\_0}^{\left(k \cdot 0.5\right)}} \end{array} \end{array} \]

FPCore

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

C

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

Python

def code(k, n):
	t_0 = math.pi * (n * 2.0)
	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(n * 2.0))
	return Float64(sqrt(t_0) / Float64(sqrt(k) * (t_0 ^ Float64(k * 0.5))))
end

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(n * 2.0), $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 99.1%

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

   1. associate-*l/ 99.1%

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

   2. *-un-lft-identity 99.1%

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

   3. associate-*r* 99.1%

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

   4. div-sub 99.1%

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

   5. metadata-eval 99.1%

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

   6. pow-div 99.3%

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

   7. pow1/2 99.3%

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

   8. associate-/l/ 99.3%

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

   9. div-inv 99.3%

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

   10. metadata-eval 99.3%

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

4. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*l* 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*l* 99.3%

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

   6. *-commutative 99.3%

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

6. Simplified 99.3%

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

7. Final simplification 99.3%

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

Alternative 4: 58.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.75 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{\sqrt{k \cdot \frac{0.5}{\pi}}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(\pi, \frac{n}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.75e+72)
   (* (/ 1.0 (sqrt (* k (/ 0.5 PI)))) (sqrt n))
   (sqrt (* 2.0 (+ -1.0 (fma PI (/ n k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.75e+72) {
		tmp = (1.0 / sqrt((k * (0.5 / ((double) M_PI))))) * sqrt(n);
	} else {
		tmp = sqrt((2.0 * (-1.0 + fma(((double) M_PI), (n / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.75e+72)
		tmp = Float64(Float64(1.0 / sqrt(Float64(k * Float64(0.5 / pi)))) * sqrt(n));
	else
		tmp = sqrt(Float64(2.0 * Float64(-1.0 + fma(pi, Float64(n / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.75e+72], N[(N[(1.0 / N[Sqrt[N[(k * N[(0.5 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(-1.0 + N[(Pi * N[(n / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.75e72

   1. Initial program 98.5%

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

   3. Taylor expanded in k around 0 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-/l* 56.9%

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

   5. Simplified 56.9%

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

      1. pow1 56.9%

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

      2. sqrt-unprod 57.0%

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

      3. associate-*r* 57.0%

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

      4. *-commutative 57.0%

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

   7. Applied egg-rr 57.0%

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

      1. unpow1 57.0%

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

      2. associate-*l* 57.0%

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

      3. metadata-eval 57.0%

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

      4. times-frac 57.0%

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

      5. *-commutative 57.0%

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

      6. times-frac 56.9%

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

      7. /-rgt-identity 56.9%

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

   9. Simplified 56.9%

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

       1. *-commutative 56.9%

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

       2. sqrt-prod 77.9%

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

       3. clear-num 77.9%

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

       4. un-div-inv 77.8%

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

       5. div-inv 77.8%

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

       6. metadata-eval 77.8%

          \[\leadsto \sqrt{\frac{\pi}{k \cdot \color{blue}{0.5}}} \cdot \sqrt{n} \]

   11. Applied egg-rr 77.8%

       \[\leadsto \color{blue}{\sqrt{\frac{\pi}{k \cdot 0.5}} \cdot \sqrt{n}} \]
   12. Step-by-step derivation

       1. clear-num 77.8%

          \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{k \cdot 0.5}{\pi}}}} \cdot \sqrt{n} \]

       2. sqrt-div 78.4%

          \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{k \cdot 0.5}{\pi}}}} \cdot \sqrt{n} \]

       3. metadata-eval 78.4%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{k \cdot 0.5}{\pi}}} \cdot \sqrt{n} \]

       4. associate-/l* 78.4%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{k \cdot \frac{0.5}{\pi}}}} \cdot \sqrt{n} \]

   13. Applied egg-rr 78.4%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{k \cdot \frac{0.5}{\pi}}}} \cdot \sqrt{n} \]

   if 2.75e72 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.7%

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

      1. *-commutative 2.7%

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

      2. associate-/l* 2.7%

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

   5. Simplified 2.7%

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

      1. pow1 2.7%

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

      2. sqrt-unprod 2.7%

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

      3. associate-*r* 2.7%

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

      4. *-commutative 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. associate-*l* 2.7%

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

      3. metadata-eval 2.7%

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

      4. times-frac 2.7%

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

      5. *-commutative 2.7%

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

      6. times-frac 2.7%

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

      7. /-rgt-identity 2.7%

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

   9. Simplified 2.7%

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

   10. Taylor expanded in n around 0 2.7%

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

       1. associate-*r/ 2.7%

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

   12. Simplified 2.7%

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

       1. expm1-log1p-u 2.7%

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

       2. expm1-undefine 32.6%

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

       3. *-commutative 32.6%

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

       4. associate-*l/ 32.6%

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

   14. Applied egg-rr 32.6%

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

       1. sub-neg 32.6%

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

       2. metadata-eval 32.6%

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

       3. +-commutative 32.6%

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

       4. log1p-undefine 32.6%

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

       5. rem-exp-log 32.6%

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

       6. +-commutative 32.6%

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

       7. associate-/l* 32.6%

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

       8. fma-define 32.6%

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

   16. Simplified 32.6%

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

Alternative 5: 58.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.35 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{n \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(\pi, \frac{n}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.35e+72)
   (* (sqrt (/ PI k)) (sqrt (* n 2.0)))
   (sqrt (* 2.0 (+ -1.0 (fma PI (/ n k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.35e+72) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((n * 2.0));
	} else {
		tmp = sqrt((2.0 * (-1.0 + fma(((double) M_PI), (n / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.35e+72)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(n * 2.0)));
	else
		tmp = sqrt(Float64(2.0 * Float64(-1.0 + fma(pi, Float64(n / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.35e+72], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(-1.0 + N[(Pi * N[(n / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.3499999999999999e72

   1. Initial program 98.5%

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

   3. Taylor expanded in k around 0 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-/l* 56.9%

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

   5. Simplified 56.9%

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

      1. pow1 56.9%

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

      2. sqrt-unprod 57.0%

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

      3. associate-*r* 57.0%

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

      4. *-commutative 57.0%

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

   7. Applied egg-rr 57.0%

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

      1. unpow1 57.0%

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

      2. associate-*l* 57.0%

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

      3. metadata-eval 57.0%

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

      4. times-frac 57.0%

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

      5. *-commutative 57.0%

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

      6. times-frac 56.9%

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

      7. /-rgt-identity 56.9%

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

   9. Simplified 56.9%

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

   10. Taylor expanded in n around 0 57.0%

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

       1. associate-*r/ 57.0%

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

   12. Simplified 57.0%

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

       1. associate-*r* 57.0%

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

       2. *-commutative 57.0%

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

       3. sqrt-prod 77.9%

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

   14. Applied egg-rr 77.9%

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

       1. *-commutative 77.9%

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

   16. Simplified 77.9%

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

   if 3.3499999999999999e72 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.7%

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

      1. *-commutative 2.7%

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

      2. associate-/l* 2.7%

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

   5. Simplified 2.7%

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

      1. pow1 2.7%

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

      2. sqrt-unprod 2.7%

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

      3. associate-*r* 2.7%

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

      4. *-commutative 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. associate-*l* 2.7%

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

      3. metadata-eval 2.7%

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

      4. times-frac 2.7%

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

      5. *-commutative 2.7%

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

      6. times-frac 2.7%

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

      7. /-rgt-identity 2.7%

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

   9. Simplified 2.7%

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

   10. Taylor expanded in n around 0 2.7%

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

       1. associate-*r/ 2.7%

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

   12. Simplified 2.7%

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

       1. expm1-log1p-u 2.7%

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

       2. expm1-undefine 32.6%

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

       3. *-commutative 32.6%

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

       4. associate-*l/ 32.6%

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

   14. Applied egg-rr 32.6%

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

       1. sub-neg 32.6%

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

       2. metadata-eval 32.6%

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

       3. +-commutative 32.6%

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

       4. log1p-undefine 32.6%

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

       5. rem-exp-log 32.6%

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

       6. +-commutative 32.6%

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

       7. associate-/l* 32.6%

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

       8. fma-define 32.6%

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

   16. Simplified 32.6%

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

Alternative 6: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. associate-/r/ 99.1%

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

   2. associate-*r* 99.1%

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

   3. div-sub 99.1%

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

   4. metadata-eval 99.1%

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

   5. div-inv 99.1%

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

   6. metadata-eval 99.1%

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

4. Applied egg-rr 99.1%

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

Alternative 7: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 99.1%

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

4. Final simplification 99.1%

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

Alternative 8: 99.5% 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 99.1%

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

   1. associate-*l/ 99.1%

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

   2. *-lft-identity 99.1%

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

   3. associate-*l* 99.1%

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

   4. div-sub 99.1%

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

   5. metadata-eval 99.1%

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

3. Simplified 99.1%

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

Alternative 9: 48.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 36.5%

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

   1. *-commutative 36.5%

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

   2. associate-/l* 36.5%

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

5. Simplified 36.5%

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

   1. pow1 36.5%

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

   2. sqrt-unprod 36.7%

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

   3. associate-*r* 36.7%

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

   4. *-commutative 36.7%

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

7. Applied egg-rr 36.7%

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

   1. unpow1 36.7%

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

   2. associate-*l* 36.7%

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

   3. metadata-eval 36.7%

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

   4. times-frac 36.7%

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

   5. *-commutative 36.7%

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

   6. times-frac 36.6%

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

   7. /-rgt-identity 36.6%

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

9. Simplified 36.6%

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

10. Taylor expanded in n around 0 36.6%

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

    1. associate-*r/ 36.7%

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

12. Simplified 36.7%

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

    1. associate-*r* 36.7%

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

    2. *-commutative 36.7%

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

    3. sqrt-prod 49.7%

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

14. Applied egg-rr 49.7%

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

    1. *-commutative 49.7%

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

16. Simplified 49.7%

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

Alternative 10: 48.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 36.5%

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

   1. *-commutative 36.5%

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

   2. associate-/l* 36.5%

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

5. Simplified 36.5%

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

   1. pow1 36.5%

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

   2. sqrt-unprod 36.7%

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

   3. associate-*r* 36.7%

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

   4. *-commutative 36.7%

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

7. Applied egg-rr 36.7%

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

   1. unpow1 36.7%

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

   2. associate-*l* 36.7%

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

   3. metadata-eval 36.7%

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

   4. times-frac 36.7%

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

   5. *-commutative 36.7%

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

   6. times-frac 36.6%

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

   7. /-rgt-identity 36.6%

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

9. Simplified 36.6%

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

    1. *-commutative 36.6%

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

    2. sqrt-prod 49.7%

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

    3. clear-num 49.7%

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

    4. un-div-inv 49.7%

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

    5. div-inv 49.7%

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

    6. metadata-eval 49.7%

       \[\leadsto \sqrt{\frac{\pi}{k \cdot \color{blue}{0.5}}} \cdot \sqrt{n} \]

11. Applied egg-rr 49.7%

    \[\leadsto \color{blue}{\sqrt{\frac{\pi}{k \cdot 0.5}} \cdot \sqrt{n}} \]
12. Step-by-step derivation

    1. clear-num 49.7%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{k \cdot 0.5}{\pi}}}} \cdot \sqrt{n} \]

    2. associate-/r/ 49.7%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{k \cdot 0.5} \cdot \pi}} \cdot \sqrt{n} \]

    3. *-commutative 49.7%

       \[\leadsto \sqrt{\frac{1}{\color{blue}{0.5 \cdot k}} \cdot \pi} \cdot \sqrt{n} \]

    4. associate-/r* 49.7%

       \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{0.5}}{k}} \cdot \pi} \cdot \sqrt{n} \]

    5. metadata-eval 49.7%

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

13. Applied egg-rr 49.7%

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

14. Final simplification 49.7%

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

Alternative 11: 48.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 36.5%

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

   1. *-commutative 36.5%

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

   2. associate-/l* 36.5%

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

5. Simplified 36.5%

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

   1. pow1 36.5%

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

   2. sqrt-unprod 36.7%

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

   3. associate-*r* 36.7%

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

   4. *-commutative 36.7%

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

7. Applied egg-rr 36.7%

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

   1. unpow1 36.7%

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

   2. associate-*l* 36.7%

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

   3. metadata-eval 36.7%

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

   4. times-frac 36.7%

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

   5. *-commutative 36.7%

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

   6. times-frac 36.6%

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

   7. /-rgt-identity 36.6%

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

9. Simplified 36.6%

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

    1. associate-*r* 36.6%

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

    2. *-commutative 36.6%

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

    3. sqrt-prod 49.3%

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

    4. *-commutative 49.3%

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

11. Applied egg-rr 49.3%

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

12. Final simplification 49.3%

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

Alternative 12: 38.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 36.5%

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

   1. *-commutative 36.5%

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

   2. associate-/l* 36.5%

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

5. Simplified 36.5%

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

   1. pow1 36.5%

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

   2. sqrt-unprod 36.7%

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

   3. associate-*r* 36.7%

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

   4. *-commutative 36.7%

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

7. Applied egg-rr 36.7%

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

   1. unpow1 36.7%

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

   2. associate-*l* 36.7%

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

   3. metadata-eval 36.7%

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

   4. times-frac 36.7%

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

   5. *-commutative 36.7%

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

   6. times-frac 36.6%

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

   7. /-rgt-identity 36.6%

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

9. Simplified 36.6%

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

10. Taylor expanded in n around 0 36.6%

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

    1. associate-*r/ 36.7%

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

12. Simplified 36.7%

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

    1. *-commutative 36.7%

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

    2. associate-*r/ 36.6%

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

    3. associate-*l/ 36.6%

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

    4. associate-*r* 36.6%

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

    5. sqrt-undiv 49.7%

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

    6. clear-num 49.7%

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

    7. sqrt-undiv 36.7%

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

    8. *-commutative 36.7%

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

    9. associate-*r* 36.7%

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

    10. *-commutative 36.7%

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

14. Applied egg-rr 36.7%

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

Alternative 13: 37.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 36.5%

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

   1. *-commutative 36.5%

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

   2. associate-/l* 36.5%

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

5. Simplified 36.5%

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

   1. pow1 36.5%

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

   2. sqrt-unprod 36.7%

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

   3. associate-*r* 36.7%

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

   4. *-commutative 36.7%

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

7. Applied egg-rr 36.7%

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

   1. unpow1 36.7%

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

   2. associate-*l* 36.7%

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

   3. metadata-eval 36.7%

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

   4. times-frac 36.7%

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

   5. *-commutative 36.7%

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

   6. times-frac 36.6%

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

   7. /-rgt-identity 36.6%

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

9. Simplified 36.6%

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

10. Taylor expanded in n around 0 36.6%

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

    1. associate-*r/ 36.7%

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

12. Simplified 36.7%

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

Reproduce

herbie shell --seed 2024090 
(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))))