Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 13.1s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. k = 1.2213493850468635e+179
  2. n = -6.029197545504138e-23

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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. div-sub N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

      \[\leadsto {\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{k}{2}\right)\right)\right)} \cdot \frac{1}{\sqrt{k}} \]

   6. distribute-neg-frac2 N/A

      \[\leadsto {\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{1}{2} + \frac{k}{\mathsf{neg}\left(2\right)}\right)} \cdot \frac{1}{\sqrt{k}} \]

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. unpow-prod-up N/A

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

   10. associate-*l* N/A

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

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{-2}\right)}\right), \color{blue}{\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}} \cdot \frac{1}{\sqrt{k}}\right)}\right) \]

4. Applied egg-rr 99.7%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing
3. Step-by-step derivation

   1. associate-*r* N/A

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

   2. div-sub N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. metadata-eval N/A

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

   7. associate-/r/ N/A

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

   8. clear-num N/A

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

   9. unpow-prod-up N/A

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

   10. associate-/l* N/A

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

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{-2}\right)}}{\sqrt{k}}\right)}\right) \]

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 61.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.9 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{2 \cdot \left(\pi \cdot n\right)} \cdot {k}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot n}}{{\left(\frac{k \cdot k}{4}\right)}^{0.25}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.9e-45)
   (* (sqrt (* 2.0 (* PI n))) (pow k -0.5))
   (/ (sqrt (* PI n)) (pow (/ (* k k) 4.0) 0.25))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.9e-45) {
		tmp = sqrt((2.0 * (((double) M_PI) * n))) * pow(k, -0.5);
	} else {
		tmp = sqrt((((double) M_PI) * n)) / pow(((k * k) / 4.0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.9e-45) {
		tmp = Math.sqrt((2.0 * (Math.PI * n))) * Math.pow(k, -0.5);
	} else {
		tmp = Math.sqrt((Math.PI * n)) / Math.pow(((k * k) / 4.0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 4.9e-45:
		tmp = math.sqrt((2.0 * (math.pi * n))) * math.pow(k, -0.5)
	else:
		tmp = math.sqrt((math.pi * n)) / math.pow(((k * k) / 4.0), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.9e-45)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi * n))) * (k ^ -0.5));
	else
		tmp = Float64(sqrt(Float64(pi * n)) / (Float64(Float64(k * k) / 4.0) ^ 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 4.9e-45)
		tmp = sqrt((2.0 * (pi * n))) * (k ^ -0.5);
	else
		tmp = sqrt((pi * n)) / (((k * k) / 4.0) ^ 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.9e-45], N[(N[Sqrt[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[k, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(Pi * n), $MachinePrecision]], $MachinePrecision] / N[Power[N[(N[(k * k), $MachinePrecision] / 4.0), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.8999999999999998e-45

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 75.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 75.7%

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

      1. sqrt-unprod N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), k\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right), k\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      11. *-lowering-*.f64 75.9%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right), k\right)\right) \]

   7. Applied egg-rr 75.9%

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

      1. pow1/2 N/A

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

      2. clear-num N/A

         \[\leadsto {\left(\frac{1}{\frac{k}{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}}\right)}^{\frac{1}{2}} \]

      3. associate-/r/ N/A

         \[\leadsto {\left(\frac{1}{k} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right)\right)}^{\frac{1}{2}} \]

      4. unpow-prod-down N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({k}^{\frac{-1}{2}}\right), \color{blue}{\left({\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right)}^{\frac{1}{2}}\right)}\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \left({\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right)}}^{\frac{1}{2}}\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \left(\sqrt{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right)\right) \]

      18. PI-lowering-PI.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right) \]

   9. Applied egg-rr 99.5%

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

   if 4.8999999999999998e-45 < k

   1. Initial program 99.7%

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

   3. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 13.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 13.8%

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

      1. sqrt-unprod N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), k\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right), k\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      11. *-lowering-*.f64 13.8%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right), k\right)\right) \]

   7. Applied egg-rr 13.8%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. associate-/l/ N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. sqrt-unprod N/A

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

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\left(n \cdot \mathsf{PI}\left(\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{k}\right)}\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{n \cdot \mathsf{PI}\left(\right)}\right), \left(\color{blue}{\sqrt{\frac{1}{2}}} \cdot \sqrt{k}\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{\sqrt{\frac{1}{2}}} \cdot \sqrt{k}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{PI}\left(\right) \cdot n\right)\right), \left(\sqrt{\color{blue}{\frac{1}{2}}} \cdot \sqrt{k}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right), \left(\sqrt{\color{blue}{\frac{1}{2}}} \cdot \sqrt{k}\right)\right) \]

      15. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\sqrt{\frac{1}{2}} \cdot \sqrt{k}\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\sqrt{k} \cdot \color{blue}{\sqrt{\frac{1}{2}}}\right)\right) \]

      17. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left({k}^{\frac{1}{2}} \cdot \sqrt{\color{blue}{\frac{1}{2}}}\right)\right) \]

      18. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left({k}^{\frac{1}{2}} \cdot {\frac{1}{2}}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      19. pow-prod-down N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left({\left(k \cdot \frac{1}{2}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left({\left(k \cdot \frac{1}{2}\right)}^{\frac{1}{2}}\right)\right) \]

      21. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left({\left(\frac{k}{2}\right)}^{\frac{1}{2}}\right)\right) \]

      22. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{pow.f64}\left(\left(\frac{k}{2}\right), \color{blue}{\frac{1}{2}}\right)\right) \]

      23. /-lowering-/.f64 15.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, 2\right), \frac{1}{2}\right)\right) \]

   9. Applied egg-rr 15.1%

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

       1. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left({\left(\frac{k}{2}\right)}^{\left(2 \cdot \color{blue}{\frac{1}{4}}\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left({\left(\frac{k}{2}\right)}^{\left(2 \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{2}}\right)\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left({\left(\frac{k}{2}\right)}^{\left(\frac{1}{2} \cdot \frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{k}{2}\right)}^{\left(\frac{1}{2} \cdot \frac{1}{2}\right)}}\right)\right) \]

       4. pow-prod-down N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left({\left(\frac{k}{2} \cdot \frac{k}{2}\right)}^{\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{2}\right)}}\right)\right) \]

       5. frac-times N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left({\left(\frac{k \cdot k}{2 \cdot 2}\right)}^{\left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{2}\right)}\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left({\left(\frac{k \cdot k}{4}\right)}^{\left(\frac{1}{2} \cdot \frac{1}{2}\right)}\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left({\left(\frac{k \cdot k}{-2 \cdot -2}\right)}^{\left(\frac{1}{2} \cdot \frac{1}{2}\right)}\right)\right) \]

       8. frac-times N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left({\left(\frac{k}{-2} \cdot \frac{k}{-2}\right)}^{\left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{2}\right)}\right)\right) \]

       9. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{pow.f64}\left(\left(\frac{k}{-2} \cdot \frac{k}{-2}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{2}\right)}\right)\right) \]

       10. frac-times N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{pow.f64}\left(\left(\frac{k \cdot k}{-2 \cdot -2}\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{2}\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(k \cdot k\right), \left(-2 \cdot -2\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{2}\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(-2 \cdot -2\right)\right), \left(\frac{1}{2} \cdot \frac{1}{2}\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), 4\right), \left(\frac{1}{2} \cdot \frac{1}{2}\right)\right)\right) \]

       14. metadata-eval 29.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), 4\right), \frac{1}{4}\right)\right) \]

   11. Applied egg-rr 29.5%

       \[\leadsto \frac{\sqrt{\pi \cdot n}}{\color{blue}{{\left(\frac{k \cdot k}{4}\right)}^{0.25}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.9 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{2 \cdot \left(\pi \cdot n\right)} \cdot {k}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot n}}{{\left(\frac{k \cdot k}{4}\right)}^{0.25}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\frac{k}{\pi \cdot n}}\\ \mathbf{if}\;k \leq 3.1 \cdot 10^{+230}:\\ \;\;\;\;\sqrt{2 \cdot \left(\pi \cdot n\right)} \cdot {k}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(t\_0 \cdot t\_0\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (/ 2.0 (/ k (* PI n)))))
   (if (<= k 3.1e+230)
     (* (sqrt (* 2.0 (* PI n))) (pow k -0.5))
     (pow (* t_0 t_0) 0.25))))

C

double code(double k, double n) {
	double t_0 = 2.0 / (k / (((double) M_PI) * n));
	double tmp;
	if (k <= 3.1e+230) {
		tmp = sqrt((2.0 * (((double) M_PI) * n))) * pow(k, -0.5);
	} else {
		tmp = pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = 2.0 / (k / (Math.PI * n));
	double tmp;
	if (k <= 3.1e+230) {
		tmp = Math.sqrt((2.0 * (Math.PI * n))) * Math.pow(k, -0.5);
	} else {
		tmp = Math.pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = 2.0 / (k / (math.pi * n))
	tmp = 0
	if k <= 3.1e+230:
		tmp = math.sqrt((2.0 * (math.pi * n))) * math.pow(k, -0.5)
	else:
		tmp = math.pow((t_0 * t_0), 0.25)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(2.0 / Float64(k / Float64(pi * n)))
	tmp = 0.0
	if (k <= 3.1e+230)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi * n))) * (k ^ -0.5));
	else
		tmp = Float64(t_0 * t_0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = 2.0 / (k / (pi * n));
	tmp = 0.0;
	if (k <= 3.1e+230)
		tmp = sqrt((2.0 * (pi * n))) * (k ^ -0.5);
	else
		tmp = (t_0 * t_0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 / N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 3.1e+230], N[(N[Sqrt[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[k, -0.5], $MachinePrecision]), $MachinePrecision], N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 0.25], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.09999999999999981e230

   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. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 42.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 42.4%

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

      1. sqrt-unprod N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), k\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right), k\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      11. *-lowering-*.f64 42.5%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right), k\right)\right) \]

   7. Applied egg-rr 42.5%

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

      1. pow1/2 N/A

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

      2. clear-num N/A

         \[\leadsto {\left(\frac{1}{\frac{k}{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}}\right)}^{\frac{1}{2}} \]

      3. associate-/r/ N/A

         \[\leadsto {\left(\frac{1}{k} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right)\right)}^{\frac{1}{2}} \]

      4. unpow-prod-down N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({k}^{\frac{-1}{2}}\right), \color{blue}{\left({\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right)}^{\frac{1}{2}}\right)}\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \left({\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right)}}^{\frac{1}{2}}\right)\right) \]

      10. unpow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \left(\sqrt{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}\right)\right) \]

      11. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(n \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right)\right) \]

      18. PI-lowering-PI.f64 53.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right)\right) \]

   9. Applied egg-rr 53.8%

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

   if 3.09999999999999981e230 < 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

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 2.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 2.7%

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

      1. sqrt-unprod N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), k\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right), k\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      11. *-lowering-*.f64 2.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right), k\right)\right) \]

   7. Applied egg-rr 2.7%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}{k} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}{k}\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

   9. Applied egg-rr 22.6%

      \[\leadsto \color{blue}{{\left(\frac{2}{\frac{k}{\pi \cdot n}} \cdot \frac{2}{\frac{k}{\pi \cdot n}}\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1 \cdot 10^{+230}:\\ \;\;\;\;\sqrt{2 \cdot \left(\pi \cdot n\right)} \cdot {k}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{2}{\frac{k}{\pi \cdot n}} \cdot \frac{2}{\frac{k}{\pi \cdot n}}\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right), \color{blue}{\left(\frac{1}{\sqrt{k}}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}\right), \left(\frac{1}{\sqrt{k}}\right)\right) \]

   4. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\frac{\color{blue}{1}}{\sqrt{k}}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\frac{1}{\sqrt{k}}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\frac{1}{\sqrt{k}}\right)\right) \]

   7. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\frac{1}{\sqrt{k}}\right)\right) \]

   8. div-sub N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\frac{1}{2} - \frac{k}{2}\right)\right), \left(\frac{1}{\sqrt{k}}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\frac{1}{2} - \frac{k}{2}\right)\right), \left(\frac{1}{\sqrt{k}}\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{k}{2}\right)\right)\right)\right), \left(\frac{1}{\sqrt{k}}\right)\right) \]

   11. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\frac{1}{2} + \frac{k}{\mathsf{neg}\left(2\right)}\right)\right), \left(\frac{1}{\sqrt{k}}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\frac{1}{2} + \frac{k}{-2}\right)\right), \left(\frac{1}{\sqrt{k}}\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{k}{-2}\right)\right)\right), \left(\frac{1}{\sqrt{k}}\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, -2\right)\right)\right), \left(\frac{1}{\sqrt{k}}\right)\right) \]

   15. pow1/2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, -2\right)\right)\right), \left(\frac{1}{{k}^{\color{blue}{\frac{1}{2}}}}\right)\right) \]

   16. pow-flip N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, -2\right)\right)\right), \left({k}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, -2\right)\right)\right), \left({k}^{\frac{-1}{2}}\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, -2\right)\right)\right), \left({k}^{\left(\frac{1}{\color{blue}{-2}}\right)}\right)\right) \]

   19. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, -2\right)\right)\right), \mathsf{pow.f64}\left(k, \color{blue}{\left(\frac{1}{-2}\right)}\right)\right) \]

   20. metadata-eval 99.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, -2\right)\right)\right), \mathsf{pow.f64}\left(k, \frac{-1}{2}\right)\right) \]

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 6: 51.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\frac{k}{\pi \cdot n}}\\ \mathbf{if}\;k \leq 8.8 \cdot 10^{+230}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(t\_0 \cdot t\_0\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (/ 2.0 (/ k (* PI n)))))
   (if (<= k 8.8e+230)
     (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
     (pow (* t_0 t_0) 0.25))))

C

double code(double k, double n) {
	double t_0 = 2.0 / (k / (((double) M_PI) * n));
	double tmp;
	if (k <= 8.8e+230) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = 2.0 / (k / (Math.PI * n));
	double tmp;
	if (k <= 8.8e+230) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	} else {
		tmp = Math.pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = 2.0 / (k / (math.pi * n))
	tmp = 0
	if k <= 8.8e+230:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (math.pi / k)))
	else:
		tmp = math.pow((t_0 * t_0), 0.25)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(2.0 / Float64(k / Float64(pi * n)))
	tmp = 0.0
	if (k <= 8.8e+230)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = Float64(t_0 * t_0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = 2.0 / (k / (pi * n));
	tmp = 0.0;
	if (k <= 8.8e+230)
		tmp = sqrt(n) * sqrt((2.0 * (pi / k)));
	else
		tmp = (t_0 * t_0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 / N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 8.8e+230], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 0.25], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 8.8000000000000004e230

   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. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 42.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 42.4%

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

      1. sqrt-unprod N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({n}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{\frac{\mathsf{PI}\left(\right)}{k} \cdot 2}\right)}\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{n}\right), \left(\sqrt{\color{blue}{\frac{\mathsf{PI}\left(\right)}{k} \cdot 2}}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \left(\sqrt{\color{blue}{\frac{\mathsf{PI}\left(\right)}{k} \cdot 2}}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{k} \cdot 2\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{k}\right), 2\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), k\right), 2\right)\right)\right) \]

      12. PI-lowering-PI.f64 53.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), k\right), 2\right)\right)\right) \]

   7. Applied egg-rr 53.8%

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

   if 8.8000000000000004e230 < 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

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 2.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 2.7%

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

      1. sqrt-unprod N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), k\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right), k\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      11. *-lowering-*.f64 2.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right), k\right)\right) \]

   7. Applied egg-rr 2.7%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}{k} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}{k}\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

   9. Applied egg-rr 22.6%

      \[\leadsto \color{blue}{{\left(\frac{2}{\frac{k}{\pi \cdot n}} \cdot \frac{2}{\frac{k}{\pi \cdot n}}\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{+230}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{2}{\frac{k}{\pi \cdot n}} \cdot \frac{2}{\frac{k}{\pi \cdot n}}\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. associate-*l/ N/A

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

   2. *-lft-identity N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right), \color{blue}{\left(\sqrt{k}\right)}\right) \]

   4. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{\color{blue}{k}}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   8. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   9. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\frac{1}{2} - \frac{k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{k}{2}\right)\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\mathsf{neg}\left(\frac{k}{2}\right)\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{k}{2}\right)\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   13. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{k}{\mathsf{neg}\left(2\right)}\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, -2\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   16. sqrt-lowering-sqrt.f64 99.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, -2\right)\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

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. Add Preprocessing

5. Final simplification 99.6%

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

Alternative 8: 40.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\frac{k}{\pi \cdot n}}\\ \mathbf{if}\;k \leq 3.6 \cdot 10^{+220}:\\ \;\;\;\;{\left(\frac{k}{2 \cdot \left(\pi \cdot n\right)}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(t\_0 \cdot t\_0\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (/ 2.0 (/ k (* PI n)))))
   (if (<= k 3.6e+220)
     (pow (/ k (* 2.0 (* PI n))) -0.5)
     (pow (* t_0 t_0) 0.25))))

C

double code(double k, double n) {
	double t_0 = 2.0 / (k / (((double) M_PI) * n));
	double tmp;
	if (k <= 3.6e+220) {
		tmp = pow((k / (2.0 * (((double) M_PI) * n))), -0.5);
	} else {
		tmp = pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = 2.0 / (k / (Math.PI * n));
	double tmp;
	if (k <= 3.6e+220) {
		tmp = Math.pow((k / (2.0 * (Math.PI * n))), -0.5);
	} else {
		tmp = Math.pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = 2.0 / (k / (math.pi * n))
	tmp = 0
	if k <= 3.6e+220:
		tmp = math.pow((k / (2.0 * (math.pi * n))), -0.5)
	else:
		tmp = math.pow((t_0 * t_0), 0.25)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(2.0 / Float64(k / Float64(pi * n)))
	tmp = 0.0
	if (k <= 3.6e+220)
		tmp = Float64(k / Float64(2.0 * Float64(pi * n))) ^ -0.5;
	else
		tmp = Float64(t_0 * t_0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = 2.0 / (k / (pi * n));
	tmp = 0.0;
	if (k <= 3.6e+220)
		tmp = (k / (2.0 * (pi * n))) ^ -0.5;
	else
		tmp = (t_0 * t_0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 / N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 3.6e+220], N[Power[N[(k / N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 0.25], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.60000000000000019e220

   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. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 43.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 43.7%

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

      1. sqrt-unprod N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), k\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right), k\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      11. *-lowering-*.f64 43.8%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right), k\right)\right) \]

   7. Applied egg-rr 43.8%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. sqrt-pow1 N/A

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

      4. metadata-eval N/A

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

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{k}{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}\right), \color{blue}{\frac{-1}{2}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right)\right), \frac{-1}{2}\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right)\right), \frac{-1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right), \frac{-1}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)\right), \frac{-1}{2}\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(2, \left(n \cdot \mathsf{PI}\left(\right)\right)\right)\right), \frac{-1}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \frac{-1}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right), \frac{-1}{2}\right) \]

      13. PI-lowering-PI.f64 44.4%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \frac{-1}{2}\right) \]

   9. Applied egg-rr 44.4%

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

   if 3.60000000000000019e220 < 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

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 2.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 2.6%

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

      1. sqrt-unprod N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), k\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right), k\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      11. *-lowering-*.f64 2.6%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right), k\right)\right) \]

   7. Applied egg-rr 2.6%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-down N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}{k} \cdot \frac{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}{k}\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

   9. Applied egg-rr 18.0%

      \[\leadsto \color{blue}{{\left(\frac{2}{\frac{k}{\pi \cdot n}} \cdot \frac{2}{\frac{k}{\pi \cdot n}}\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 38.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in k around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 38.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 38.7%

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

   1. sqrt-unprod N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), k\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right), k\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), k\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

   11. *-lowering-*.f64 38.8%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right), k\right)\right) \]

7. Applied egg-rr 38.8%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. sqrt-pow1 N/A

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

   4. metadata-eval N/A

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

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{k}{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}\right), \color{blue}{\frac{-1}{2}}\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right)\right), \frac{-1}{2}\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right)\right), \frac{-1}{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right), \frac{-1}{2}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)\right), \frac{-1}{2}\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(2, \left(n \cdot \mathsf{PI}\left(\right)\right)\right)\right), \frac{-1}{2}\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \frac{-1}{2}\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right), \frac{-1}{2}\right) \]

   13. PI-lowering-PI.f64 39.3%

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \frac{-1}{2}\right) \]

9. Applied egg-rr 39.3%

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

Alternative 10: 38.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in k around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 38.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 38.7%

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

   1. sqrt-unprod N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), k\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right), k\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), k\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

   11. *-lowering-*.f64 38.8%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right), k\right)\right) \]

7. Applied egg-rr 38.8%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(n \cdot 2\right) \cdot \mathsf{PI}\left(\right)}{k}\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \frac{\mathsf{PI}\left(\right)}{k}\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot 2\right) \cdot \frac{1}{\frac{k}{\mathsf{PI}\left(\right)}}\right)\right) \]

   4. un-div-inv N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot 2}{\frac{k}{\mathsf{PI}\left(\right)}}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot 2\right), \left(\frac{k}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot n\right), \left(\frac{k}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, n\right), \left(\frac{k}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, n\right), \mathsf{/.f64}\left(k, \mathsf{PI}\left(\right)\right)\right)\right) \]

   9. PI-lowering-PI.f64 38.8%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, n\right), \mathsf{/.f64}\left(k, \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

9. Applied egg-rr 38.8%

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

Alternative 11: 38.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in k around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 38.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 38.7%

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

   1. sqrt-unprod N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), k\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right), k\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), k\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

   11. *-lowering-*.f64 38.8%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right), k\right)\right) \]

7. Applied egg-rr 38.8%

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

   1. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{n \cdot 2}{k}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot 2}{k} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(n \cdot \frac{2}{k}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(n \cdot \left(\frac{2}{k} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \left(\frac{2}{k} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\left(\frac{2}{k}\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. PI-lowering-PI.f64 38.8%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

9. Applied egg-rr 38.8%

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

10. Final simplification 38.8%

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

Reproduce

herbie shell --seed 2024192 
(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))))