Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 19.5s

Alternatives: 14

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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{k \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(k + -1\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2e-29)
   (* (sqrt (* 2.0 n)) (sqrt (/ PI k)))
   (/ 1.0 (sqrt (* k (pow (* PI (* 2.0 n)) (+ k -1.0)))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2e-29) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	} else {
		tmp = 1.0 / sqrt((k * pow((((double) M_PI) * (2.0 * n)), (k + -1.0))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2e-29) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((Math.PI / k));
	} else {
		tmp = 1.0 / Math.sqrt((k * Math.pow((Math.PI * (2.0 * n)), (k + -1.0))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2e-29:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((math.pi / k))
	else:
		tmp = 1.0 / math.sqrt((k * math.pow((math.pi * (2.0 * n)), (k + -1.0))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2e-29)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	else
		tmp = Float64(1.0 / sqrt(Float64(k * (Float64(pi * Float64(2.0 * n)) ^ Float64(k + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2e-29)
		tmp = sqrt((2.0 * n)) * sqrt((pi / k));
	else
		tmp = 1.0 / sqrt((k * ((pi * (2.0 * n)) ^ (k + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2e-29], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(k * N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(k + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999989e-29

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 78.7%

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

      1. *-commutative 78.7%

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

      2. associate-/l* 78.7%

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

   5. Simplified 78.7%

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

      1. *-commutative 78.7%

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

      2. sqrt-unprod 79.0%

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

   7. Applied egg-rr 79.0%

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

      1. associate-*r/ 79.0%

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

      2. associate-*l/ 79.0%

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

      3. associate-*r* 79.0%

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

      4. *-commutative 79.0%

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

      5. associate-*r* 79.0%

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

      6. metadata-eval 79.0%

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

      7. div-inv 79.0%

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

      8. sqrt-prod 99.5%

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

      9. div-inv 99.5%

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

      10. metadata-eval 99.5%

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

   9. Applied egg-rr 99.5%

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

   if 1.99999999999999989e-29 < k

   1. Initial program 99.9%

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

   3. Taylor expanded in k around 0 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. div-inv 99.9%

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

      2. pow-flip 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. sub-neg 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. remove-double-neg 99.9%

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

      8. metadata-eval 99.9%

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

   9. Simplified 99.9%

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

       1. add-sqr-sqrt 99.9%

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

       2. unpow-prod-down 99.9%

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

       3. *-commutative 99.9%

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

       4. *-commutative 99.9%

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

       5. associate-*l* 99.9%

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

       6. *-commutative 99.9%

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

   11. Applied egg-rr 99.9%

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

       1. pow-sqr 99.9%

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

       2. metadata-eval 99.9%

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

       3. unpow-1 99.9%

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

   13. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{k \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(k + -1\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(k + -1\right)}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2e-33)
   (* (sqrt (* 2.0 n)) (sqrt (/ PI k)))
   (pow (* k (pow (* (* 2.0 PI) n) (+ k -1.0))) -0.5)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2e-33) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	} else {
		tmp = pow((k * pow(((2.0 * ((double) M_PI)) * n), (k + -1.0))), -0.5);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2e-33) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((Math.PI / k));
	} else {
		tmp = Math.pow((k * Math.pow(((2.0 * Math.PI) * n), (k + -1.0))), -0.5);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2e-33:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((math.pi / k))
	else:
		tmp = math.pow((k * math.pow(((2.0 * math.pi) * n), (k + -1.0))), -0.5)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2e-33)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	else
		tmp = Float64(k * (Float64(Float64(2.0 * pi) * n) ^ Float64(k + -1.0))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2e-33)
		tmp = sqrt((2.0 * n)) * sqrt((pi / k));
	else
		tmp = (k * (((2.0 * pi) * n) ^ (k + -1.0))) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2e-33], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(k * N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(k + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.0000000000000001e-33

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 78.7%

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

      1. *-commutative 78.7%

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

      2. associate-/l* 78.7%

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

   5. Simplified 78.7%

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

      1. *-commutative 78.7%

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

      2. sqrt-unprod 79.0%

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

   7. Applied egg-rr 79.0%

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

      1. associate-*r/ 79.0%

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

      2. associate-*l/ 79.0%

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

      3. associate-*r* 79.0%

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

      4. *-commutative 79.0%

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

      5. associate-*r* 79.0%

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

      6. metadata-eval 79.0%

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

      7. div-inv 79.0%

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

      8. sqrt-prod 99.5%

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

      9. div-inv 99.5%

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

      10. metadata-eval 99.5%

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

   9. Applied egg-rr 99.5%

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

   if 2.0000000000000001e-33 < k

   1. Initial program 99.9%

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

   3. Taylor expanded in k around 0 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. div-inv 99.9%

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

      2. pow-flip 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. sub-neg 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. remove-double-neg 99.9%

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

      8. metadata-eval 99.9%

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

   9. Simplified 99.9%

      \[\leadsto {\color{blue}{\left(k \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(k + -1\right)}\right)}}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(k + -1\right)}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4e-17)
   (* (sqrt (* 2.0 n)) (sqrt (/ PI k)))
   (sqrt (/ (pow (* (* 2.0 PI) n) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4e-17) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	} else {
		tmp = sqrt((pow(((2.0 * ((double) M_PI)) * n), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4e-17) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((Math.PI / k));
	} else {
		tmp = Math.sqrt((Math.pow(((2.0 * Math.PI) * n), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 4e-17:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((math.pi / k))
	else:
		tmp = math.sqrt((math.pow(((2.0 * math.pi) * n), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4e-17)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	else
		tmp = sqrt(Float64((Float64(Float64(2.0 * pi) * n) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 4e-17)
		tmp = sqrt((2.0 * n)) * sqrt((pi / k));
	else
		tmp = sqrt(((((2.0 * pi) * n) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4e-17], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.00000000000000029e-17

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 79.1%

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

      1. *-commutative 79.1%

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

      2. associate-/l* 79.0%

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

   5. Simplified 79.0%

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

      1. *-commutative 79.0%

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

      2. sqrt-unprod 79.4%

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

   7. Applied egg-rr 79.4%

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

      1. associate-*r/ 79.4%

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

      2. associate-*l/ 79.4%

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

      3. associate-*r* 79.4%

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

      4. *-commutative 79.4%

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

      5. associate-*r* 79.4%

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

      6. metadata-eval 79.4%

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

      7. div-inv 79.4%

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

      8. sqrt-prod 99.5%

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

      9. div-inv 99.5%

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

      10. metadata-eval 99.5%

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

   9. Applied egg-rr 99.5%

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

   if 4.00000000000000029e-17 < k

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. sqrt-unprod 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r* 99.9%

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

      5. div-sub 99.9%

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

      6. metadata-eval 99.9%

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

      7. div-inv 99.9%

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

      8. *-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.4 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \mathsf{fma}\left(\pi, 2 \cdot \frac{n}{k}, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5.4e+78)
   (* (sqrt (* 2.0 n)) (sqrt (/ PI k)))
   (sqrt (+ 1.0 (fma PI (* 2.0 (/ n k)) -1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5.4e+78) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	} else {
		tmp = sqrt((1.0 + fma(((double) M_PI), (2.0 * (n / k)), -1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5.4e+78)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	else
		tmp = sqrt(Float64(1.0 + fma(pi, Float64(2.0 * Float64(n / k)), -1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5.4e+78], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(1.0 + N[(Pi * N[(2.0 * N[(n / k), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.40000000000000009e78

   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. Add Preprocessing

   3. Taylor expanded in k around 0 59.0%

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

      1. *-commutative 59.0%

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

      2. associate-/l* 59.0%

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

   5. Simplified 59.0%

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

      1. *-commutative 59.0%

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

      2. sqrt-unprod 59.2%

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

   7. Applied egg-rr 59.2%

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

      1. associate-*r/ 59.2%

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

      2. associate-*l/ 59.2%

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

      3. associate-*r* 59.2%

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

      4. *-commutative 59.2%

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

      5. associate-*r* 59.2%

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

      6. metadata-eval 59.2%

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

      7. div-inv 59.2%

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

      8. sqrt-prod 74.0%

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

      9. div-inv 74.0%

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

      10. metadata-eval 74.0%

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

   9. Applied egg-rr 74.0%

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

   if 5.40000000000000009e78 < k

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. div-sub 100.0%

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

      6. metadata-eval 100.0%

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

      7. div-inv 100.0%

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

      8. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in k around 0 2.6%

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

      1. associate-*r/ 2.6%

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

      2. associate-*r* 2.6%

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

      3. *-commutative 2.6%

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

      4. associate-*l* 2.6%

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

      5. *-commutative 2.6%

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

      6. associate-/l* 2.6%

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

   9. Simplified 2.6%

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

       1. clear-num 2.6%

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

       2. un-div-inv 2.6%

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

       3. *-un-lft-identity 2.6%

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

       4. *-commutative 2.6%

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

       5. times-frac 2.6%

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

       6. metadata-eval 2.6%

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

   11. Applied egg-rr 2.6%

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

       1. associate-/r* 2.6%

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

   13. Simplified 2.6%

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

       1. associate-/r/ 2.6%

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

       2. associate-*l/ 2.6%

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

       3. div-inv 2.6%

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

       4. metadata-eval 2.6%

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

       5. associate-*r* 2.6%

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

       6. *-commutative 2.6%

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

       7. associate-*r/ 2.6%

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

       8. expm1-log1p-u 2.6%

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

       9. associate-*r* 2.6%

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

       10. expm1-undefine 27.5%

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

   15. Applied egg-rr 27.5%

       \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\pi \cdot \frac{n}{k \cdot 0.5}\right)} - 1}} \]
   16. Step-by-step derivation

       1. log1p-undefine 27.5%

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

       2. rem-exp-log 27.5%

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

       3. associate-+r- 27.5%

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

       4. fma-neg 27.5%

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

       5. metadata-eval 27.5%

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

       6. *-rgt-identity 27.5%

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

       7. times-frac 27.5%

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

       8. metadata-eval 27.5%

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

       9. *-commutative 27.5%

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

   17. Simplified 27.5%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.4 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \mathsf{fma}\left(\pi, 2 \cdot \frac{n}{k}, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 99.7%

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

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. associate-*l* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.6%

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

   2. div-inv 99.6%

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

   3. metadata-eval 99.6%

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

   4. pow1/2 99.6%

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

   5. pow-flip 99.7%

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

   6. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

Alternative 7: 99.5% accurate, 1.0× speedup

Math

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

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 99.6%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. associate-*l* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

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

Alternative 8: 48.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 38.7%

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

   1. *-commutative 38.7%

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

   2. associate-/l* 38.7%

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

5. Simplified 38.7%

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

   1. *-commutative 38.7%

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

   2. sqrt-unprod 38.9%

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

7. Applied egg-rr 38.9%

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

   1. associate-*r/ 38.9%

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

   2. associate-*l/ 38.9%

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

   3. associate-*r* 38.9%

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

   4. *-commutative 38.9%

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

   5. associate-*r* 38.9%

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

   6. metadata-eval 38.9%

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

   7. div-inv 38.9%

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

   8. sqrt-prod 48.3%

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

   9. div-inv 48.3%

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

   10. metadata-eval 48.3%

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

9. Applied egg-rr 48.3%

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

10. Final simplification 48.3%

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

Alternative 9: 48.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-unprod 90.1%

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

   3. *-commutative 90.1%

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

   4. associate-*r* 90.1%

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

   5. div-sub 90.1%

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

   6. metadata-eval 90.1%

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

   7. div-inv 90.1%

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

   8. *-commutative 90.1%

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

4. Applied egg-rr 90.2%

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

6. Simplified 90.3%

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

7. Taylor expanded in k around 0 38.9%

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

   1. associate-*r/ 38.9%

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

   2. associate-*r* 38.9%

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

   3. *-commutative 38.9%

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

   4. associate-*l* 38.9%

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

   5. *-commutative 38.9%

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

   6. associate-/l* 38.9%

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

9. Simplified 38.9%

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

    1. sqrt-prod 48.3%

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

    2. *-commutative 48.3%

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

    3. *-un-lft-identity 48.3%

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

    4. times-frac 48.3%

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

    5. metadata-eval 48.3%

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

11. Applied egg-rr 48.3%

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

    1. associate-*r/ 48.3%

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

    2. *-commutative 48.3%

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

    3. associate-*r/ 48.3%

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

13. Simplified 48.3%

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

Alternative 10: 37.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 99.7%

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

5. Applied egg-rr 91.1%

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

6. Taylor expanded in k around 0 39.6%

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

   1. associate-*r/ 39.6%

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

   2. *-commutative 39.6%

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

8. Simplified 39.6%

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

9. Final simplification 39.6%

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

Alternative 11: 37.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 99.7%

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

5. Applied egg-rr 91.1%

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

   1. div-inv 91.0%

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

   2. pow-flip 91.0%

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

   3. *-commutative 91.0%

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

   4. *-commutative 91.0%

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

7. Applied egg-rr 91.0%

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

   1. associate-*r* 91.0%

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

   2. *-commutative 91.0%

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

   3. *-commutative 91.0%

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

   4. sub-neg 91.0%

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

   5. +-commutative 91.0%

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

   6. distribute-neg-in 91.0%

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

   7. remove-double-neg 91.0%

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

   8. metadata-eval 91.0%

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

9. Simplified 91.0%

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

10. Taylor expanded in k around 0 39.6%

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

    1. associate-/r* 39.6%

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

12. Simplified 39.6%

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

Alternative 12: 37.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 99.7%

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

5. Applied egg-rr 91.1%

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

   1. div-inv 91.0%

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

   2. pow-flip 91.0%

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

   3. *-commutative 91.0%

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

   4. *-commutative 91.0%

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

7. Applied egg-rr 91.0%

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

   1. associate-*r* 91.0%

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

   2. *-commutative 91.0%

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

   3. *-commutative 91.0%

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

   4. sub-neg 91.0%

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

   5. +-commutative 91.0%

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

   6. distribute-neg-in 91.0%

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

   7. remove-double-neg 91.0%

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

   8. metadata-eval 91.0%

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

9. Simplified 91.0%

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

10. Taylor expanded in k around 0 39.6%

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

11. Final simplification 39.6%

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

Alternative 13: 36.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 38.7%

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

   1. *-commutative 38.7%

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

   2. associate-/l* 38.7%

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

5. Simplified 38.7%

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

   1. *-commutative 38.7%

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

   2. sqrt-unprod 38.9%

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

7. Applied egg-rr 38.9%

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

   1. clear-num 38.9%

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

   2. un-div-inv 38.9%

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

9. Applied egg-rr 38.9%

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

Alternative 14: 36.6% 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 99.6%

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

3. Taylor expanded in k around 0 38.7%

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

   1. *-commutative 38.7%

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

   2. associate-/l* 38.7%

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

5. Simplified 38.7%

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

   1. *-commutative 38.7%

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

   2. sqrt-unprod 38.9%

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

7. Applied egg-rr 38.9%

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

Reproduce

herbie shell --seed 2024097 
(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))))