Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 21.8s

Alternatives: 15

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, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{k}{\pi}}}{\sqrt{n \cdot 2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6.8e-18)
   (/ 1.0 (/ (sqrt (/ k PI)) (sqrt (* n 2.0))))
   (sqrt (/ (pow (* PI (* n 2.0)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6.8e-18) {
		tmp = 1.0 / (sqrt((k / ((double) M_PI))) / sqrt((n * 2.0)));
	} else {
		tmp = sqrt((pow((((double) M_PI) * (n * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 6.8e-18) {
		tmp = 1.0 / (Math.sqrt((k / Math.PI)) / Math.sqrt((n * 2.0)));
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (n * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 6.8e-18:
		tmp = 1.0 / (math.sqrt((k / math.pi)) / math.sqrt((n * 2.0)))
	else:
		tmp = math.sqrt((math.pow((math.pi * (n * 2.0)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 6.8e-18)
		tmp = Float64(1.0 / Float64(sqrt(Float64(k / pi)) / sqrt(Float64(n * 2.0))));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(n * 2.0)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 6.8e-18)
		tmp = 1.0 / (sqrt((k / pi)) / sqrt((n * 2.0)));
	else
		tmp = sqrt((((pi * (n * 2.0)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6.8e-18], N[(1.0 / N[(N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.80000000000000002e-18

   1. Initial program 98.4%

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

   3. Taylor expanded in k around 0 71.4%

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

      1. *-commutative 71.4%

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

      2. associate-/l* 71.5%

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

   5. Simplified 71.5%

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

      1. pow1 71.5%

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

      2. sqrt-unprod 71.7%

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

      3. associate-*r/ 71.7%

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

      4. *-commutative 71.7%

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

      5. associate-*r/ 71.7%

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

      6. *-commutative 71.7%

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

      7. associate-*r* 71.7%

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

   7. Applied egg-rr 71.7%

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

      1. unpow1 71.7%

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

      2. associate-/l* 71.7%

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

      3. associate-/l* 71.6%

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

   9. Simplified 71.6%

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

       1. associate-*r/ 71.7%

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

       2. associate-/l* 71.7%

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

       3. clear-num 71.6%

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

       4. sqrt-div 72.2%

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

       5. metadata-eval 72.2%

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

       6. associate-*r* 72.2%

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

       7. *-commutative 72.2%

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

       8. *-commutative 72.2%

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

       9. *-commutative 72.2%

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

   11. Applied egg-rr 72.2%

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

       1. *-commutative 72.2%

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

       2. associate-/r* 72.2%

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

       3. associate-/r* 72.2%

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

   13. Simplified 72.2%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{\frac{\frac{k}{\pi}}{n}}{2}}}} \]
   14. Step-by-step derivation

       1. associate-/l/ 72.2%

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

       2. sqrt-div 99.2%

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

       3. *-commutative 99.2%

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

   15. Applied egg-rr 99.2%

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

   if 6.80000000000000002e-18 < k

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r* 99.8%

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

      5. div-sub 99.8%

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

      6. metadata-eval 99.8%

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

      7. div-inv 99.8%

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

      8. *-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   6. Simplified 99.8%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. associate-*l/ 99.2%

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

   2. *-un-lft-identity 99.2%

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

   3. associate-*r* 99.2%

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

   4. div-sub 99.2%

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

   5. metadata-eval 99.2%

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

   6. pow-div 99.3%

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

   7. pow1/2 99.3%

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

   8. associate-/l/ 99.3%

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

   9. div-inv 99.3%

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

   10. metadata-eval 99.3%

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

4. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*l* 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*l* 99.3%

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

   6. *-commutative 99.3%

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

6. Simplified 99.3%

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

Alternative 3: 60.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{k}{\pi}}}{\sqrt{n \cdot 2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot \left(-1 + \mathsf{fma}\left(2, \frac{n}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5.2e+67)
   (/ 1.0 (/ (sqrt (/ k PI)) (sqrt (* n 2.0))))
   (sqrt (* PI (+ -1.0 (fma 2.0 (/ n k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5.2e+67) {
		tmp = 1.0 / (sqrt((k / ((double) M_PI))) / sqrt((n * 2.0)));
	} else {
		tmp = sqrt((((double) M_PI) * (-1.0 + fma(2.0, (n / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5.2e+67)
		tmp = Float64(1.0 / Float64(sqrt(Float64(k / pi)) / sqrt(Float64(n * 2.0))));
	else
		tmp = sqrt(Float64(pi * Float64(-1.0 + fma(2.0, Float64(n / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5.2e+67], N[(1.0 / N[(N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(Pi * N[(-1.0 + N[(2.0 * N[(n / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.2000000000000001e67

   1. Initial program 98.5%

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

   3. Taylor expanded in k around 0 58.6%

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

      1. *-commutative 58.6%

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

      2. associate-/l* 58.6%

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

   5. Simplified 58.6%

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

      1. pow1 58.6%

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

      2. sqrt-unprod 58.8%

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

      3. associate-*r/ 58.8%

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

      4. *-commutative 58.8%

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

      5. associate-*r/ 58.8%

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

      6. *-commutative 58.8%

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

      7. associate-*r* 58.8%

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

   7. Applied egg-rr 58.8%

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

      1. unpow1 58.8%

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

      2. associate-/l* 58.8%

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

      3. associate-/l* 58.7%

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

   9. Simplified 58.7%

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

       1. associate-*r/ 58.8%

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

       2. associate-/l* 58.8%

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

       3. clear-num 58.7%

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

       4. sqrt-div 59.2%

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

       5. metadata-eval 59.2%

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

       6. associate-*r* 59.2%

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

       7. *-commutative 59.2%

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

       8. *-commutative 59.2%

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

       9. *-commutative 59.2%

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

   11. Applied egg-rr 59.2%

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

       1. *-commutative 59.2%

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

       2. associate-/r* 59.2%

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

       3. associate-/r* 59.2%

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

   13. Simplified 59.2%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{\frac{\frac{k}{\pi}}{n}}{2}}}} \]
   14. Step-by-step derivation

       1. associate-/l/ 59.2%

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

       2. sqrt-div 80.6%

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

       3. *-commutative 80.6%

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

   15. Applied egg-rr 80.6%

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

   if 5.2000000000000001e67 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.7%

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

      1. *-commutative 2.7%

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

      2. associate-/l* 2.7%

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

   5. Simplified 2.7%

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

      1. pow1 2.7%

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

      2. sqrt-unprod 2.7%

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

      3. associate-*r/ 2.7%

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

      4. *-commutative 2.7%

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

      5. associate-*r/ 2.7%

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

      6. *-commutative 2.7%

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

      7. associate-*r* 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. associate-/l* 2.7%

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

      3. associate-/l* 2.7%

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

   9. Simplified 2.7%

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

       1. expm1-log1p-u 2.7%

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

       2. expm1-undefine 35.6%

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

       3. associate-*r/ 35.6%

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

       4. *-commutative 35.6%

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

       5. associate-/l* 35.6%

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

   11. Applied egg-rr 35.6%

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

       1. sub-neg 35.6%

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

       2. metadata-eval 35.6%

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

       3. +-commutative 35.6%

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

       4. log1p-undefine 35.6%

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

       5. rem-exp-log 35.6%

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

       6. +-commutative 35.6%

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

       7. fma-define 35.6%

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

   13. Simplified 35.6%

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

Alternative 4: 60.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{+67}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot \left(-1 + \mathsf{fma}\left(2, \frac{n}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.6e+67)
   (/ (sqrt (* n (* PI 2.0))) (sqrt k))
   (sqrt (* PI (+ -1.0 (fma 2.0 (/ n k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.6e+67) {
		tmp = sqrt((n * (((double) M_PI) * 2.0))) / sqrt(k);
	} else {
		tmp = sqrt((((double) M_PI) * (-1.0 + fma(2.0, (n / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.6e+67)
		tmp = Float64(sqrt(Float64(n * Float64(pi * 2.0))) / sqrt(k));
	else
		tmp = sqrt(Float64(pi * Float64(-1.0 + fma(2.0, Float64(n / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.6e+67], N[(N[Sqrt[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(Pi * N[(-1.0 + N[(2.0 * N[(n / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.5999999999999997e67

   1. Initial program 98.5%

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

   3. Taylor expanded in k around 0 58.6%

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

      1. *-commutative 58.6%

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

      2. associate-/l* 58.6%

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

   5. Simplified 58.6%

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

      1. pow1 58.6%

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

      2. sqrt-unprod 58.8%

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

      3. associate-*r/ 58.8%

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

      4. *-commutative 58.8%

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

      5. associate-*r/ 58.8%

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

      6. *-commutative 58.8%

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

      7. associate-*r* 58.8%

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

   7. Applied egg-rr 58.8%

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

      1. unpow1 58.8%

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

      2. associate-/l* 58.8%

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

      3. associate-/l* 58.7%

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

   9. Simplified 58.7%

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

       1. associate-*r/ 58.8%

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

       2. associate-/l* 58.8%

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

       3. sqrt-div 80.1%

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

       4. add-sqr-sqrt 79.8%

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

       5. associate-/l* 79.8%

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

       6. pow1/2 79.8%

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

       7. sqrt-pow1 79.9%

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

       8. associate-*r* 79.9%

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

       9. *-commutative 79.9%

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

       10. *-commutative 79.9%

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

       11. *-commutative 79.9%

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

       12. metadata-eval 79.9%

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

   11. Applied egg-rr 79.8%

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

       1. associate-*r/ 79.8%

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

       2. pow-sqr 80.1%

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

       3. metadata-eval 80.1%

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

       4. unpow1/2 80.1%

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

       5. associate-*r* 80.1%

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

       6. *-commutative 80.1%

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

       7. *-commutative 80.1%

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

   13. Simplified 80.1%

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

   if 4.5999999999999997e67 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.7%

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

      1. *-commutative 2.7%

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

      2. associate-/l* 2.7%

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

   5. Simplified 2.7%

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

      1. pow1 2.7%

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

      2. sqrt-unprod 2.7%

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

      3. associate-*r/ 2.7%

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

      4. *-commutative 2.7%

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

      5. associate-*r/ 2.7%

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

      6. *-commutative 2.7%

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

      7. associate-*r* 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. associate-/l* 2.7%

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

      3. associate-/l* 2.7%

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

   9. Simplified 2.7%

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

       1. expm1-log1p-u 2.7%

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

       2. expm1-undefine 35.6%

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

       3. associate-*r/ 35.6%

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

       4. *-commutative 35.6%

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

       5. associate-/l* 35.6%

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

   11. Applied egg-rr 35.6%

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

       1. sub-neg 35.6%

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

       2. metadata-eval 35.6%

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

       3. +-commutative 35.6%

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

       4. log1p-undefine 35.6%

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

       5. rem-exp-log 35.6%

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

       6. +-commutative 35.6%

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

       7. fma-define 35.6%

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

   13. Simplified 35.6%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. associate-*l/ 99.2%

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

   2. *-lft-identity 99.2%

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

   3. associate-*l* 99.2%

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

   4. div-sub 99.2%

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

   5. metadata-eval 99.2%

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

3. Simplified 99.2%

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

Alternative 6: 49.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in k around 0 35.0%

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

   1. *-commutative 35.0%

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

   2. associate-/l* 35.0%

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

5. Simplified 35.0%

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

   1. pow1 35.0%

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

   2. sqrt-unprod 35.1%

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

   3. associate-*r/ 35.1%

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

   4. *-commutative 35.1%

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

   5. associate-*r/ 35.1%

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

   6. *-commutative 35.1%

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

   7. associate-*r* 35.1%

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

7. Applied egg-rr 35.1%

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

   1. unpow1 35.1%

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

   2. associate-/l* 35.1%

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

   3. associate-/l* 35.1%

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

9. Simplified 35.1%

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

    1. associate-*r/ 35.1%

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

    2. associate-/l* 35.1%

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

    3. sqrt-div 47.5%

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

    4. add-sqr-sqrt 47.3%

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

    5. associate-/l* 47.3%

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

    6. pow1/2 47.3%

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

    7. sqrt-pow1 47.3%

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

    8. associate-*r* 47.3%

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

    9. *-commutative 47.3%

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

    10. *-commutative 47.3%

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

    11. *-commutative 47.3%

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

    12. metadata-eval 47.3%

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

11. Applied egg-rr 47.3%

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

    1. associate-*r/ 47.3%

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

    2. pow-sqr 47.5%

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

    3. metadata-eval 47.5%

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

    4. unpow1/2 47.5%

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

    5. associate-*r* 47.5%

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

    6. *-commutative 47.5%

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

    7. *-commutative 47.5%

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

13. Simplified 47.5%

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

Alternative 7: 49.5% 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 99.2%

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

3. Taylor expanded in k around 0 35.0%

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

   1. *-commutative 35.0%

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

   2. associate-/l* 35.0%

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

5. Simplified 35.0%

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

   1. sqrt-unprod 35.1%

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

   2. associate-*r/ 35.1%

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

   3. *-commutative 35.1%

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

   4. associate-*r/ 35.1%

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

   5. *-commutative 35.1%

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

   6. associate-*r* 35.1%

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

7. Applied egg-rr 35.1%

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

   1. *-commutative 35.1%

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

   2. associate-/l* 35.1%

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

   3. *-commutative 35.1%

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

9. Applied egg-rr 35.1%

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

    1. *-commutative 35.1%

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

    2. sqrt-prod 47.5%

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

    3. *-commutative 47.5%

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

11. Applied egg-rr 47.5%

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

12. Final simplification 47.5%

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

Alternative 8: 49.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in k around 0 35.0%

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

   1. *-commutative 35.0%

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

   2. associate-/l* 35.0%

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

5. Simplified 35.0%

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

   1. sqrt-unprod 35.1%

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

   2. associate-*r/ 35.1%

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

   3. *-commutative 35.1%

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

   4. associate-*r/ 35.1%

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

   5. *-commutative 35.1%

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

   6. associate-*r* 35.1%

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

7. Applied egg-rr 35.1%

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

   1. *-commutative 35.1%

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

   2. associate-/l* 35.1%

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

   3. *-commutative 35.1%

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

9. Applied egg-rr 35.1%

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

    1. *-commutative 35.1%

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

    2. associate-*l* 35.1%

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

    3. metadata-eval 35.1%

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

    4. times-frac 35.1%

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

    5. *-commutative 35.1%

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

    6. *-un-lft-identity 35.1%

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

    7. associate-*r/ 35.1%

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

    8. *-commutative 35.1%

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

    9. sqrt-prod 47.4%

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

11. Applied egg-rr 47.4%

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

Alternative 9: 49.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in k around 0 35.0%

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

   1. *-commutative 35.0%

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

   2. associate-/l* 35.0%

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

5. Simplified 35.0%

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

   1. sqrt-unprod 35.1%

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

   2. associate-*r/ 35.1%

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

   3. *-commutative 35.1%

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

   4. associate-*r/ 35.1%

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

   5. *-commutative 35.1%

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

   6. associate-*r* 35.1%

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

7. Applied egg-rr 35.1%

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

   1. *-commutative 35.1%

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

   2. associate-/l* 35.1%

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

   3. *-commutative 35.1%

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

9. Applied egg-rr 35.1%

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

    1. *-commutative 35.1%

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

    2. associate-*l* 35.1%

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

    3. metadata-eval 35.1%

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

    4. times-frac 35.1%

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

    5. *-commutative 35.1%

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

    6. *-un-lft-identity 35.1%

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

    7. associate-*r/ 35.1%

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

    8. sqrt-prod 47.4%

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

11. Applied egg-rr 47.4%

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

    1. associate-*r/ 47.4%

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

13. Simplified 47.4%

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

Alternative 10: 38.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in k around 0 35.0%

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

   1. *-commutative 35.0%

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

   2. associate-/l* 35.0%

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

5. Simplified 35.0%

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

   1. pow1 35.0%

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

   2. sqrt-unprod 35.1%

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

   3. associate-*r/ 35.1%

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

   4. *-commutative 35.1%

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

   5. associate-*r/ 35.1%

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

   6. *-commutative 35.1%

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

   7. associate-*r* 35.1%

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

7. Applied egg-rr 35.1%

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

   1. unpow1 35.1%

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

   2. associate-/l* 35.1%

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

   3. associate-/l* 35.1%

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

9. Simplified 35.1%

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

    1. associate-*r/ 35.1%

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

    2. associate-/l* 35.1%

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

    3. clear-num 35.1%

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

    4. sqrt-div 35.4%

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

    5. metadata-eval 35.4%

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

    6. associate-*r* 35.4%

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

    7. *-commutative 35.4%

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

    8. *-commutative 35.4%

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

    9. *-commutative 35.4%

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

11. Applied egg-rr 35.4%

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

    1. *-commutative 35.4%

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

    2. associate-/r* 35.3%

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

    3. associate-/r* 35.3%

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

13. Simplified 35.3%

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{\frac{\frac{k}{\pi}}{n}}{2}}}} \]
14. Step-by-step derivation

    1. div-inv 35.4%

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

15. Applied egg-rr 35.4%

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

Alternative 11: 38.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in k around 0 35.0%

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

   1. *-commutative 35.0%

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

   2. associate-/l* 35.0%

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

5. Simplified 35.0%

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

   1. pow1 35.0%

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

   2. sqrt-unprod 35.1%

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

   3. associate-*r/ 35.1%

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

   4. *-commutative 35.1%

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

   5. associate-*r/ 35.1%

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

   6. *-commutative 35.1%

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

   7. associate-*r* 35.1%

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

7. Applied egg-rr 35.1%

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

   1. unpow1 35.1%

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

   2. associate-/l* 35.1%

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

   3. associate-/l* 35.1%

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

9. Simplified 35.1%

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

    1. associate-*r/ 35.1%

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

    2. associate-/l* 35.1%

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

    3. clear-num 35.1%

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

    4. sqrt-div 35.4%

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

    5. metadata-eval 35.4%

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

    6. associate-*r* 35.4%

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

    7. *-commutative 35.4%

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

    8. *-commutative 35.4%

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

    9. *-commutative 35.4%

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

11. Applied egg-rr 35.4%

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

    1. associate-/r* 35.4%

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

    2. *-commutative 35.4%

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

13. Simplified 35.4%

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

14. Final simplification 35.4%

    \[\leadsto \frac{1}{\sqrt{\frac{\frac{k}{2}}{\pi \cdot n}}} \]
15. Add Preprocessing

Alternative 12: 37.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in k around 0 35.0%

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

   1. *-commutative 35.0%

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

   2. associate-/l* 35.0%

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

5. Simplified 35.0%

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

   1. sqrt-unprod 35.1%

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

   2. associate-*r/ 35.1%

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

   3. *-commutative 35.1%

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

   4. associate-*r/ 35.1%

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

   5. *-commutative 35.1%

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

   6. associate-*r* 35.1%

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

7. Applied egg-rr 35.1%

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

   1. *-commutative 35.1%

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

   2. associate-/l* 35.1%

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

   3. *-commutative 35.1%

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

9. Applied egg-rr 35.1%

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

    1. clear-num 35.1%

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

    2. un-div-inv 35.1%

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

    3. *-commutative 35.1%

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

11. Applied egg-rr 35.1%

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

Alternative 13: 37.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in k around 0 35.0%

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

   1. *-commutative 35.0%

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

   2. associate-/l* 35.0%

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

5. Simplified 35.0%

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

   1. sqrt-unprod 35.1%

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

   2. associate-*r/ 35.1%

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

   3. *-commutative 35.1%

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

   4. associate-*r/ 35.1%

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

   5. *-commutative 35.1%

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

   6. associate-*r* 35.1%

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

7. Applied egg-rr 35.1%

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

   1. *-commutative 35.1%

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

   2. associate-/l* 35.1%

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

   3. *-commutative 35.1%

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

9. Applied egg-rr 35.1%

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

10. Final simplification 35.1%

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

Alternative 14: 37.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in k around 0 35.0%

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

   1. *-commutative 35.0%

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

   2. associate-/l* 35.0%

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

5. Simplified 35.0%

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

   1. pow1 35.0%

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

   2. sqrt-unprod 35.1%

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

   3. associate-*r/ 35.1%

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

   4. *-commutative 35.1%

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

   5. associate-*r/ 35.1%

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

   6. *-commutative 35.1%

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

   7. associate-*r* 35.1%

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

7. Applied egg-rr 35.1%

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

   1. unpow1 35.1%

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

   2. associate-/l* 35.1%

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

9. Simplified 35.1%

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

Alternative 15: 37.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in k around 0 35.0%

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

   1. *-commutative 35.0%

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

   2. associate-/l* 35.0%

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

5. Simplified 35.0%

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

   1. pow1 35.0%

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

   2. sqrt-unprod 35.1%

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

   3. associate-*r/ 35.1%

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

   4. *-commutative 35.1%

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

   5. associate-*r/ 35.1%

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

   6. *-commutative 35.1%

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

   7. associate-*r* 35.1%

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

7. Applied egg-rr 35.1%

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

   1. unpow1 35.1%

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

   2. associate-/l* 35.1%

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

   3. associate-/l* 35.1%

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

9. Simplified 35.1%

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

Reproduce

herbie shell --seed 2024101 
(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))))