Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 17.6s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l/ 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*r* 99.3%

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

   5. div-sub 99.3%

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

   6. metadata-eval 99.3%

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

   7. pow-sub 99.7%

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

   8. pow1/2 99.7%

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

   9. associate-/l/ 99.6%

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

   10. associate-*r* 99.6%

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

   11. *-commutative 99.6%

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

   12. associate-*l* 99.6%

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

4. Applied egg-rr 99.6%

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

   1. associate-/r* 99.7%

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

   2. associate-*r* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*r* 99.7%

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

   5. *-commutative 99.7%

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

   6. *-commutative 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l/ 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*r* 99.3%

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

   5. div-sub 99.3%

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

   6. metadata-eval 99.3%

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

   7. pow-sub 99.7%

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

   8. pow1/2 99.7%

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

   9. associate-/l/ 99.6%

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

   10. associate-*r* 99.6%

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

   11. *-commutative 99.6%

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

   12. associate-*l* 99.6%

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

4. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*r* 99.6%

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

   5. *-commutative 99.6%

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

   6. *-commutative 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{n \cdot 2}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{k \cdot {\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(k + -1\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.8e-17)
   (/ (sqrt (* n 2.0)) (sqrt (/ k PI)))
   (/ 1.0 (sqrt (* k (pow (* PI (* n 2.0)) (+ k -1.0)))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.8e-17) {
		tmp = sqrt((n * 2.0)) / sqrt((k / ((double) M_PI)));
	} else {
		tmp = 1.0 / sqrt((k * pow((((double) M_PI) * (n * 2.0)), (k + -1.0))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.8e-17) {
		tmp = Math.sqrt((n * 2.0)) / Math.sqrt((k / Math.PI));
	} else {
		tmp = 1.0 / Math.sqrt((k * Math.pow((Math.PI * (n * 2.0)), (k + -1.0))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.8e-17:
		tmp = math.sqrt((n * 2.0)) / math.sqrt((k / math.pi))
	else:
		tmp = 1.0 / math.sqrt((k * math.pow((math.pi * (n * 2.0)), (k + -1.0))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.8e-17)
		tmp = Float64(sqrt(Float64(n * 2.0)) / sqrt(Float64(k / pi)));
	else
		tmp = Float64(1.0 / sqrt(Float64(k * (Float64(pi * Float64(n * 2.0)) ^ Float64(k + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.8e-17)
		tmp = sqrt((n * 2.0)) / sqrt((k / pi));
	else
		tmp = 1.0 / sqrt((k * ((pi * (n * 2.0)) ^ (k + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.8e-17], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(k * N[Power[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision], N[(k + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.79999999999999997e-17

   1. Initial program 99.2%

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

      1. add-sqr-sqrt 98.9%

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

      2. pow2 98.9%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in k around 0 80.4%

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

      1. associate-*r/ 80.4%

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

      2. associate-*r* 80.4%

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

      3. *-commutative 80.4%

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

      4. associate-/l* 80.3%

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

      5. *-commutative 80.3%

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

   7. Simplified 80.3%

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

      1. pow-pow 80.8%

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

      2. metadata-eval 80.8%

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

      3. pow1/2 80.8%

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

      4. sqrt-div 99.3%

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

   9. Applied egg-rr 99.3%

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

   if 1.79999999999999997e-17 < k

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*r* 99.3%

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

      3. div-sub 99.3%

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

      4. metadata-eval 99.3%

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

      5. sub-neg 99.3%

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

      6. unpow-prod-up 99.9%

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

      7. pow1/2 99.9%

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

      8. associate-*r* 99.9%

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

      9. *-commutative 99.9%

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

      10. associate-*l* 99.9%

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

      11. associate-*r* 99.9%

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

      12. *-commutative 99.9%

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

      13. associate-*l* 99.9%

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

      14. div-inv 99.9%

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

      15. metadata-eval 99.9%

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

      16. distribute-rgt-neg-in 99.9%

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

      17. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.9%

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

   6. Applied egg-rr 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{n \cdot 2}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{k \cdot {\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(k + -1\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{n \cdot 2}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.4e-21)
   (/ (sqrt (* n 2.0)) (sqrt (/ k PI)))
   (sqrt (/ (pow (* n (* 2.0 PI)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.4e-21) {
		tmp = sqrt((n * 2.0)) / sqrt((k / ((double) M_PI)));
	} else {
		tmp = sqrt((pow((n * (2.0 * ((double) M_PI))), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.4e-21) {
		tmp = Math.sqrt((n * 2.0)) / Math.sqrt((k / Math.PI));
	} else {
		tmp = Math.sqrt((Math.pow((n * (2.0 * Math.PI)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.4e-21:
		tmp = math.sqrt((n * 2.0)) / math.sqrt((k / math.pi))
	else:
		tmp = math.sqrt((math.pow((n * (2.0 * math.pi)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.4e-21)
		tmp = Float64(sqrt(Float64(n * 2.0)) / sqrt(Float64(k / pi)));
	else
		tmp = sqrt(Float64((Float64(n * Float64(2.0 * pi)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.4e-21)
		tmp = sqrt((n * 2.0)) / sqrt((k / pi));
	else
		tmp = sqrt((((n * (2.0 * pi)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.4e-21], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.40000000000000002e-21

   1. Initial program 99.3%

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

      1. add-sqr-sqrt 98.9%

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

      2. pow2 98.9%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in k around 0 79.9%

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

      1. associate-*r/ 79.9%

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

      2. associate-*r* 79.9%

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

      3. *-commutative 79.9%

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

      4. associate-/l* 79.9%

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

      5. *-commutative 79.9%

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

   7. Simplified 79.9%

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

      1. pow-pow 80.3%

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

      2. metadata-eval 80.3%

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

      3. pow1/2 80.3%

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

      4. sqrt-div 99.4%

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

   9. Applied egg-rr 99.4%

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

   if 1.40000000000000002e-21 < k

   1. Initial program 99.3%

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

      1. add-sqr-sqrt 99.3%

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

      2. sqrt-unprod 99.3%

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

      3. *-commutative 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-*r* 99.3%

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

      6. div-sub 99.3%

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

      7. metadata-eval 99.3%

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

      8. div-inv 99.3%

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

      9. *-commutative 99.3%

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

   4. Applied egg-rr 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{n \cdot 2}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+54}:\\ \;\;\;\;\frac{\sqrt{n \cdot 2}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.95e+54)
   (/ (sqrt (* n 2.0)) (sqrt (/ k PI)))
   (sqrt (* 2.0 (+ -1.0 (fma n (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.95e+54) {
		tmp = sqrt((n * 2.0)) / sqrt((k / ((double) M_PI)));
	} else {
		tmp = sqrt((2.0 * (-1.0 + fma(n, (((double) M_PI) / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.95e+54)
		tmp = Float64(sqrt(Float64(n * 2.0)) / sqrt(Float64(k / pi)));
	else
		tmp = sqrt(Float64(2.0 * Float64(-1.0 + fma(n, Float64(pi / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.95e+54], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(-1.0 + N[(n * N[(Pi / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9500000000000001e54

   1. Initial program 98.8%

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

      1. add-sqr-sqrt 98.5%

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

      2. pow2 98.5%

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in k around 0 68.3%

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

      1. associate-*r/ 68.3%

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

      2. associate-*r* 68.3%

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

      3. *-commutative 68.3%

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

      4. associate-/l* 68.3%

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

      5. *-commutative 68.3%

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

   7. Simplified 68.3%

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

      1. pow-pow 68.6%

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

      2. metadata-eval 68.6%

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

      3. pow1/2 68.6%

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

      4. sqrt-div 83.4%

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

   9. Applied egg-rr 83.4%

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

   if 1.9500000000000001e54 < 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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. pow2 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in k around 0 2.6%

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

      1. associate-*r/ 2.6%

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

      2. associate-*r* 2.6%

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

      3. *-commutative 2.6%

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

      4. associate-/l* 2.6%

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

      5. *-commutative 2.6%

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

   7. Simplified 2.6%

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

      1. pow-pow 2.6%

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

      2. metadata-eval 2.6%

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

      3. pow1/2 2.6%

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

      4. *-un-lft-identity 2.6%

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

      5. times-frac 2.6%

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

      6. metadata-eval 2.6%

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

   9. Applied egg-rr 2.6%

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

       1. expm1-log1p-u 2.6%

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

       2. expm1-undefine 36.5%

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

       3. associate-/r/ 36.5%

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

       4. *-commutative 36.5%

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

   11. Applied egg-rr 36.5%

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

       1. sub-neg 36.5%

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

       2. metadata-eval 36.5%

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

       3. +-commutative 36.5%

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

       4. log1p-undefine 36.5%

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

       5. rem-exp-log 36.5%

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

       6. +-commutative 36.5%

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

       7. associate-*r/ 36.5%

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

       8. associate-*l/ 36.5%

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

       9. *-commutative 36.5%

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

       10. fma-define 36.5%

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

   13. Simplified 36.5%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+54}:\\ \;\;\;\;\frac{\sqrt{n \cdot 2}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-/r/ 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*r* 99.3%

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

   4. div-sub 99.3%

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

   5. metadata-eval 99.3%

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

   6. associate-*r* 99.3%

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

   7. *-commutative 99.3%

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

   8. associate-*l* 99.3%

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

   9. div-inv 99.3%

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

   10. metadata-eval 99.3%

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

4. Applied egg-rr 99.3%

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

5. Final simplification 99.3%

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

Alternative 7: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l/ 99.3%

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

   2. *-lft-identity 99.3%

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

   3. sqr-pow 99.2%

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

   4. pow-sqr 99.3%

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

   5. *-commutative 99.3%

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

   6. associate-*l* 99.3%

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

   7. associate-*r/ 99.3%

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

   8. *-commutative 99.3%

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

   9. associate-/l* 99.3%

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

   10. metadata-eval 99.3%

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

   11. /-rgt-identity 99.3%

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

   12. div-sub 99.3%

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

   13. metadata-eval 99.3%

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 8: 49.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. add-sqr-sqrt 99.1%

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

   2. pow2 99.1%

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

4. Applied egg-rr 99.0%

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

5. Taylor expanded in k around 0 41.4%

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

   1. associate-*r/ 41.4%

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

   2. associate-*r* 41.4%

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

   3. *-commutative 41.4%

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

   4. associate-/l* 41.4%

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

   5. *-commutative 41.4%

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

7. Simplified 41.4%

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

   1. pow-pow 41.6%

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

   2. div-inv 41.6%

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

   3. unpow-prod-down 50.3%

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

   4. metadata-eval 50.3%

      \[\leadsto {\left(2 \cdot n\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{1}{\frac{k}{\pi}}\right)}^{\left(0.25 \cdot 2\right)} \]

   5. pow1/2 50.3%

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

   6. clear-num 50.3%

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

   7. metadata-eval 50.3%

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

9. Applied egg-rr 50.3%

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

    1. unpow1/2 50.3%

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

11. Simplified 50.3%

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

12. Final simplification 50.3%

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

Alternative 9: 49.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. add-sqr-sqrt 99.1%

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

   2. pow2 99.1%

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

4. Applied egg-rr 99.0%

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

5. Taylor expanded in k around 0 41.4%

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

   1. associate-*r/ 41.4%

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

   2. associate-*r* 41.4%

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

   3. *-commutative 41.4%

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

   4. associate-/l* 41.4%

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

   5. *-commutative 41.4%

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

7. Simplified 41.4%

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

   1. pow-pow 41.6%

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

   2. metadata-eval 41.6%

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

   3. pow1/2 41.6%

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

   4. sqrt-div 50.3%

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

9. Applied egg-rr 50.3%

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

10. Final simplification 50.3%

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

Alternative 10: 39.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in k around 0 50.2%

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

   1. *-commutative 50.2%

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

   2. sqrt-unprod 50.3%

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

   3. *-commutative 50.3%

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

   4. *-commutative 50.3%

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

   5. associate-*r* 50.3%

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

   6. *-commutative 50.3%

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

   7. div-inv 50.3%

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

   8. clear-num 50.3%

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

   9. sqrt-undiv 42.5%

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

   10. associate-*r* 42.5%

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

5. Applied egg-rr 42.5%

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

   1. inv-pow 42.5%

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

   2. sqrt-pow2 42.6%

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

   3. associate-/r* 42.6%

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

   4. *-un-lft-identity 42.6%

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

   5. *-commutative 42.6%

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

   6. times-frac 42.7%

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

   7. metadata-eval 42.7%

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

   8. metadata-eval 42.7%

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

7. Applied egg-rr 42.7%

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

8. Final simplification 42.7%

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

Alternative 11: 38.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. add-sqr-sqrt 99.1%

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

   2. pow2 99.1%

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

4. Applied egg-rr 99.0%

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

5. Taylor expanded in k around 0 41.4%

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

   1. associate-*r/ 41.4%

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

   2. associate-*r* 41.4%

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

   3. *-commutative 41.4%

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

   4. associate-/l* 41.4%

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

   5. *-commutative 41.4%

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

7. Simplified 41.4%

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

   1. pow-pow 41.6%

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

   2. metadata-eval 41.6%

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

   3. pow1/2 41.6%

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

   4. *-un-lft-identity 41.6%

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

   5. times-frac 41.6%

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

   6. metadata-eval 41.6%

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

9. Applied egg-rr 41.6%

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

    1. associate-/r/ 41.6%

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

11. Applied egg-rr 41.6%

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

12. Final simplification 41.6%

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

Reproduce

herbie shell --seed 2024034 
(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))))