Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.2%

Time: 13.4s

Alternatives: 8

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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.1e-56)
   (/ (sqrt (* 2.0 n)) (sqrt (/ k PI)))
   (sqrt (/ (pow (* n (* 2.0 PI)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.1e-56) {
		tmp = sqrt((2.0 * n)) / 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 <= 4.1e-56) {
		tmp = Math.sqrt((2.0 * n)) / 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 <= 4.1e-56:
		tmp = math.sqrt((2.0 * n)) / 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 <= 4.1e-56)
		tmp = Float64(sqrt(Float64(2.0 * n)) / 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 <= 4.1e-56)
		tmp = sqrt((2.0 * n)) / 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, 4.1e-56], N[(N[Sqrt[N[(2.0 * n), $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 < 4.1000000000000001e-56

   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 67.4%

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

      1. *-commutative 67.4%

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

      2. associate-/l* 67.4%

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

   5. Simplified 67.4%

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

      1. pow1 67.4%

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

      2. sqrt-unprod 67.6%

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

      3. associate-*r/ 67.6%

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

      4. *-commutative 67.6%

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

      5. associate-/l* 67.6%

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

   7. Applied egg-rr 67.6%

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

      1. unpow1 67.6%

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

      2. *-commutative 67.6%

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

      3. associate-/r/ 67.6%

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

   9. Simplified 67.6%

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

       1. associate-*r/ 67.6%

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

       2. *-commutative 67.6%

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

       3. sqrt-div 99.4%

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

   11. Applied egg-rr 99.4%

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

   if 4.1000000000000001e-56 < k

   1. Initial program 99.4%

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

      1. associate-*l/ 99.4%

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

      2. *-lft-identity 99.4%

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

      3. associate-*l* 99.4%

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

      4. div-sub 99.4%

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

      5. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. /-rgt-identity 99.4%

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

      2. clear-num 99.4%

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

      3. inv-pow 99.4%

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

      4. sqrt-pow2 99.3%

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

      5. metadata-eval 99.3%

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

   6. Applied egg-rr 99.3%

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

   8. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. neg-mul-1 99.4%

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

   10. Simplified 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. associate-/r/ 99.0%

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

   2. associate-*r* 99.0%

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

   3. div-sub 99.0%

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

   4. metadata-eval 99.0%

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

   5. div-inv 99.0%

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

   6. metadata-eval 99.0%

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

4. Applied egg-rr 99.0%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Final simplification 99.0%

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

Alternative 4: 51.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{+266}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot \frac{2 \cdot \pi}{k}\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.5e+266)
   (/ (sqrt (* 2.0 n)) (sqrt (/ k PI)))
   (cbrt (pow (* n (/ (* 2.0 PI) k)) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.5e+266) {
		tmp = sqrt((2.0 * n)) / sqrt((k / ((double) M_PI)));
	} else {
		tmp = cbrt(pow((n * ((2.0 * ((double) M_PI)) / k)), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.5e+266) {
		tmp = Math.sqrt((2.0 * n)) / Math.sqrt((k / Math.PI));
	} else {
		tmp = Math.cbrt(Math.pow((n * ((2.0 * Math.PI) / k)), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.5e+266)
		tmp = Float64(sqrt(Float64(2.0 * n)) / sqrt(Float64(k / pi)));
	else
		tmp = cbrt((Float64(n * Float64(Float64(2.0 * pi) / k)) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.5e+266], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(n * N[(N[(2.0 * Pi), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4999999999999999e266

   1. Initial program 98.9%

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

   3. Taylor expanded in k around 0 40.1%

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

      1. *-commutative 40.1%

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

      2. associate-/l* 40.1%

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

   5. Simplified 40.1%

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

      1. pow1 40.1%

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

      2. sqrt-unprod 40.2%

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

      3. associate-*r/ 40.2%

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

      4. *-commutative 40.2%

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

      5. associate-/l* 40.2%

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

   7. Applied egg-rr 40.2%

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

      1. unpow1 40.2%

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

      2. *-commutative 40.2%

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

      3. associate-/r/ 40.2%

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

   9. Simplified 40.2%

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

       1. associate-*r/ 40.2%

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

       2. *-commutative 40.2%

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

       3. sqrt-div 54.4%

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

   11. Applied egg-rr 54.4%

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

   if 2.4999999999999999e266 < 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 2.6%

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

      1. *-commutative 2.6%

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

      2. associate-/l* 2.6%

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

   5. Simplified 2.6%

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

      1. sqrt-unprod 2.6%

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

      2. associate-*r/ 2.6%

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

      3. *-commutative 2.6%

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

      4. associate-/l* 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. add-cbrt-cube 26.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{2 \cdot \left(\pi \cdot \frac{n}{k}\right)} \cdot \sqrt{2 \cdot \left(\pi \cdot \frac{n}{k}\right)}\right) \cdot \sqrt{2 \cdot \left(\pi \cdot \frac{n}{k}\right)}}} \]

      2. pow1/3 26.5%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{2 \cdot \left(\pi \cdot \frac{n}{k}\right)} \cdot \sqrt{2 \cdot \left(\pi \cdot \frac{n}{k}\right)}\right) \cdot \sqrt{2 \cdot \left(\pi \cdot \frac{n}{k}\right)}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 26.5%

         \[\leadsto {\left(\color{blue}{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)} \cdot \sqrt{2 \cdot \left(\pi \cdot \frac{n}{k}\right)}\right)}^{0.3333333333333333} \]

      4. pow1 26.5%

         \[\leadsto {\left(\color{blue}{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(\pi \cdot \frac{n}{k}\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 26.5%

         \[\leadsto {\left({\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 26.5%

         \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. associate-*r* 26.5%

         \[\leadsto {\left({\color{blue}{\left(\left(2 \cdot \pi\right) \cdot \frac{n}{k}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. *-commutative 26.5%

         \[\leadsto {\left({\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot \frac{n}{k}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 26.5%

         \[\leadsto {\left({\left(\left(\pi \cdot 2\right) \cdot \frac{n}{k}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 26.5%

      \[\leadsto \color{blue}{{\left({\left(\left(\pi \cdot 2\right) \cdot \frac{n}{k}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   10. Step-by-step derivation

       1. unpow1/3 26.5%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(\pi \cdot 2\right) \cdot \frac{n}{k}\right)}^{1.5}}} \]

       2. associate-*r/ 26.5%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{\left(\pi \cdot 2\right) \cdot n}{k}\right)}}^{1.5}} \]

       3. associate-*l/ 26.5%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{\pi \cdot 2}{k} \cdot n\right)}}^{1.5}} \]

       4. *-commutative 26.5%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(n \cdot \frac{\pi \cdot 2}{k}\right)}}^{1.5}} \]

       5. *-commutative 26.5%

          \[\leadsto \sqrt[3]{{\left(n \cdot \frac{\color{blue}{2 \cdot \pi}}{k}\right)}^{1.5}} \]

   11. Simplified 26.5%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(n \cdot \frac{2 \cdot \pi}{k}\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{+266}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot \frac{2 \cdot \pi}{k}\right)}^{1.5}}\\ \end{array} \]
5. 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 99.0%

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

   1. associate-*l/ 99.0%

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

   2. *-lft-identity 99.0%

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

   3. associate-*l* 99.0%

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

   4. div-sub 99.0%

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

   5. metadata-eval 99.0%

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

3. Simplified 99.0%

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

Alternative 6: 50.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in k around 0 37.2%

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

   1. *-commutative 37.2%

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

   2. associate-/l* 37.2%

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

5. Simplified 37.2%

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

   1. pow1 37.2%

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

   2. sqrt-unprod 37.3%

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

   3. associate-*r/ 37.3%

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

   4. *-commutative 37.3%

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

   5. associate-/l* 37.3%

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

7. Applied egg-rr 37.3%

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

   1. unpow1 37.3%

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

   2. *-commutative 37.3%

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

   3. associate-/r/ 37.3%

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

9. Simplified 37.3%

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

    1. associate-*r/ 37.3%

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

    2. *-commutative 37.3%

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

    3. sqrt-div 50.4%

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

11. Applied egg-rr 50.4%

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

12. Final simplification 50.4%

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

Alternative 7: 39.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in k around 0 37.2%

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

   1. *-commutative 37.2%

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

   2. associate-/l* 37.2%

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

5. Simplified 37.2%

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

   1. pow1 37.2%

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

   2. sqrt-unprod 37.3%

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

   3. associate-*r/ 37.3%

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

   4. *-commutative 37.3%

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

   5. associate-/l* 37.3%

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

7. Applied egg-rr 37.3%

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

   1. unpow1 37.3%

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

   2. *-commutative 37.3%

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

   3. associate-/r/ 37.3%

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

9. Simplified 37.3%

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

    1. *-commutative 37.3%

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

    2. sqrt-prod 37.2%

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

    3. clear-num 37.2%

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

    4. sqrt-div 37.4%

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

    5. metadata-eval 37.4%

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

    6. associate-/r/ 37.4%

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

    7. inv-pow 37.4%

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

    8. sqrt-undiv 37.4%

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

    9. sqrt-pow2 37.5%

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

    10. associate-/l/ 37.5%

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

    11. metadata-eval 37.5%

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

11. Applied egg-rr 37.5%

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

12. Final simplification 37.5%

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

Alternative 8: 38.6% 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 99.0%

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

3. Taylor expanded in k around 0 37.2%

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

   1. *-commutative 37.2%

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

   2. associate-/l* 37.2%

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

5. Simplified 37.2%

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

   1. sqrt-unprod 37.3%

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

   2. associate-*r/ 37.3%

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

   3. *-commutative 37.3%

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

   4. associate-/l* 37.3%

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

7. Applied egg-rr 37.3%

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

Reproduce

herbie shell --seed 2024132 
(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))))