Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.3%

Time: 22.5s

Alternatives: 9

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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{k}}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(k + -1\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5e-23)
   (* (sqrt (* 2.0 (/ PI k))) (sqrt n))
   (sqrt (/ (/ 1.0 k) (pow (* n (* 2.0 PI)) (+ k -1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5e-23) {
		tmp = sqrt((2.0 * (((double) M_PI) / k))) * sqrt(n);
	} else {
		tmp = sqrt(((1.0 / k) / pow((n * (2.0 * ((double) M_PI))), (k + -1.0))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 5e-23) {
		tmp = Math.sqrt((2.0 * (Math.PI / k))) * Math.sqrt(n);
	} else {
		tmp = Math.sqrt(((1.0 / k) / Math.pow((n * (2.0 * Math.PI)), (k + -1.0))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 5e-23:
		tmp = math.sqrt((2.0 * (math.pi / k))) * math.sqrt(n)
	else:
		tmp = math.sqrt(((1.0 / k) / math.pow((n * (2.0 * math.pi)), (k + -1.0))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5e-23)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi / k))) * sqrt(n));
	else
		tmp = sqrt(Float64(Float64(1.0 / k) / (Float64(n * Float64(2.0 * pi)) ^ Float64(k + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 5e-23)
		tmp = sqrt((2.0 * (pi / k))) * sqrt(n);
	else
		tmp = sqrt(((1.0 / k) / ((n * (2.0 * pi)) ^ (k + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5e-23], N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(1.0 / k), $MachinePrecision] / N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(k + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0000000000000002e-23

   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 99.2%

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

      1. pow1 99.2%

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

      2. associate-*l/ 99.2%

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

      3. *-un-lft-identity 99.2%

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

      4. sqrt-unprod 99.4%

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

      5. *-commutative 99.4%

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

      6. *-commutative 99.4%

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

      7. associate-*r* 99.4%

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

      8. sqrt-undiv 73.8%

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

      9. *-commutative 73.8%

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

      10. associate-*r* 73.8%

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

   5. Applied egg-rr 73.8%

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

      1. unpow1 73.8%

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

      2. associate-/l* 73.8%

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

      3. *-commutative 73.8%

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

   7. Simplified 73.8%

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

      1. clear-num 73.8%

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

      2. un-div-inv 73.8%

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

      3. *-un-lft-identity 73.8%

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

      4. *-commutative 73.8%

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

      5. times-frac 73.8%

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

      6. metadata-eval 73.8%

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

   9. Applied egg-rr 73.8%

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

       1. associate-*r/ 73.8%

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

       2. *-commutative 73.8%

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

       3. associate-/r/ 73.8%

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

   11. Simplified 73.8%

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

       1. sqrt-prod 99.4%

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

       2. metadata-eval 99.4%

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

       3. div-inv 99.4%

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

       4. div-inv 99.4%

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

       5. clear-num 99.4%

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

   13. Applied egg-rr 99.4%

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

       1. associate-*r/ 99.4%

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

       2. associate-*l/ 99.4%

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

       3. *-commutative 99.4%

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

   15. Simplified 99.4%

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

   if 5.0000000000000002e-23 < k

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. *-commutative 99.8%

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

      4. div-sub 99.8%

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

      5. metadata-eval 99.8%

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

      6. div-inv 99.8%

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

      7. *-commutative 99.8%

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

      8. div-sub 99.8%

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

      9. metadata-eval 99.8%

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

      10. div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. distribute-lft-in 99.8%

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

      2. metadata-eval 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r* 99.8%

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

      5. metadata-eval 99.8%

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

      6. mul-1-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. associate-*r* 99.8%

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

      9. *-commutative 99.8%

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

      10. associate-*l* 99.8%

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

   6. Simplified 99.8%

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

      1. pow-sub 100.0%

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

      2. pow1 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. div-inv 100.0%

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

      4. clear-num 100.0%

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

      5. pow1 100.0%

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

      6. pow-div 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. unpow-1 99.8%

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

       2. associate-/r* 99.8%

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

       3. *-commutative 99.8%

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

       4. *-commutative 99.8%

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

       5. associate-*l* 99.8%

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

       6. sub-neg 99.8%

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

       7. metadata-eval 99.8%

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

   12. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{k}}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(k + -1\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.2e-24)
   (* (sqrt (* 2.0 (/ PI k))) (sqrt n))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.2e-24) {
		tmp = sqrt((2.0 * (((double) M_PI) / k))) * sqrt(n);
	} else {
		tmp = sqrt((pow((((double) M_PI) * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.2e-24) {
		tmp = Math.sqrt((2.0 * (Math.PI / k))) * Math.sqrt(n);
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.2e-24:
		tmp = math.sqrt((2.0 * (math.pi / k))) * math.sqrt(n)
	else:
		tmp = math.sqrt((math.pow((math.pi * (2.0 * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.2e-24)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi / k))) * sqrt(n));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(2.0 * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.2e-24)
		tmp = sqrt((2.0 * (pi / k))) * sqrt(n);
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.2e-24], N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.20000000000000002e-24

   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 99.2%

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

      1. pow1 99.2%

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

      2. associate-*l/ 99.2%

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

      3. *-un-lft-identity 99.2%

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

      4. sqrt-unprod 99.4%

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

      5. *-commutative 99.4%

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

      6. *-commutative 99.4%

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

      7. associate-*r* 99.4%

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

      8. sqrt-undiv 73.8%

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

      9. *-commutative 73.8%

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

      10. associate-*r* 73.8%

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

   5. Applied egg-rr 73.8%

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

      1. unpow1 73.8%

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

      2. associate-/l* 73.8%

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

      3. *-commutative 73.8%

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

   7. Simplified 73.8%

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

      1. clear-num 73.8%

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

      2. un-div-inv 73.8%

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

      3. *-un-lft-identity 73.8%

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

      4. *-commutative 73.8%

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

      5. times-frac 73.8%

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

      6. metadata-eval 73.8%

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

   9. Applied egg-rr 73.8%

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

       1. associate-*r/ 73.8%

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

       2. *-commutative 73.8%

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

       3. associate-/r/ 73.8%

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

   11. Simplified 73.8%

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

       1. sqrt-prod 99.4%

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

       2. metadata-eval 99.4%

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

       3. div-inv 99.4%

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

       4. div-inv 99.4%

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

       5. clear-num 99.4%

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

   13. Applied egg-rr 99.4%

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

       1. associate-*r/ 99.4%

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

       2. associate-*l/ 99.4%

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

       3. *-commutative 99.4%

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

   15. Simplified 99.4%

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

   if 2.20000000000000002e-24 < k

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. *-commutative 99.8%

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

      4. div-sub 99.8%

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

      5. metadata-eval 99.8%

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

      6. div-inv 99.8%

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

      7. *-commutative 99.8%

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

      8. div-sub 99.8%

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

      9. metadata-eval 99.8%

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

      10. div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. distribute-lft-in 99.8%

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

      2. metadata-eval 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r* 99.8%

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

      5. metadata-eval 99.8%

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

      6. mul-1-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. associate-*r* 99.8%

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

      9. *-commutative 99.8%

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

      10. associate-*l* 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(2, n \cdot \frac{\pi}{k}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5e+71)
   (* (sqrt (* 2.0 (/ PI k))) (sqrt n))
   (sqrt (+ -1.0 (fma 2.0 (* n (/ PI k)) 1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5e+71) {
		tmp = sqrt((2.0 * (((double) M_PI) / k))) * sqrt(n);
	} else {
		tmp = sqrt((-1.0 + fma(2.0, (n * (((double) M_PI) / k)), 1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5e+71)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi / k))) * sqrt(n));
	else
		tmp = sqrt(Float64(-1.0 + fma(2.0, Float64(n * Float64(pi / k)), 1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5e+71], N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-1.0 + N[(2.0 * N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.99999999999999972e71

   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 80.1%

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

      1. pow1 80.1%

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

      2. associate-*l/ 80.2%

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

      3. *-un-lft-identity 80.2%

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

      4. sqrt-unprod 80.3%

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

      5. *-commutative 80.3%

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

      6. *-commutative 80.3%

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

      7. associate-*r* 80.3%

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

      8. sqrt-undiv 60.4%

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

      9. *-commutative 60.4%

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

      10. associate-*r* 60.4%

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

   5. Applied egg-rr 60.4%

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

      1. unpow1 60.4%

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

      2. associate-/l* 60.4%

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

      3. *-commutative 60.4%

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

   7. Simplified 60.4%

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

      1. clear-num 60.3%

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

      2. un-div-inv 60.4%

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

      3. *-un-lft-identity 60.4%

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

      4. *-commutative 60.4%

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

      5. times-frac 60.4%

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

      6. metadata-eval 60.4%

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

   9. Applied egg-rr 60.4%

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

       1. associate-*r/ 60.4%

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

       2. *-commutative 60.4%

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

       3. associate-/r/ 60.4%

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

   11. Simplified 60.4%

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

       1. sqrt-prod 80.3%

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

       2. metadata-eval 80.3%

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

       3. div-inv 80.3%

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

       4. div-inv 80.3%

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

       5. clear-num 80.3%

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

   13. Applied egg-rr 80.3%

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

       1. associate-*r/ 80.3%

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

       2. associate-*l/ 80.3%

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

       3. *-commutative 80.3%

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

   15. Simplified 80.3%

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

   if 4.99999999999999972e71 < 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%

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

      1. pow1 2.6%

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

      2. associate-*l/ 2.6%

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

      3. *-un-lft-identity 2.6%

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

      4. sqrt-unprod 2.6%

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

      5. *-commutative 2.6%

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

      6. *-commutative 2.6%

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

      7. associate-*r* 2.6%

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

      8. sqrt-undiv 2.5%

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

      9. *-commutative 2.5%

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

      10. associate-*r* 2.5%

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

   5. Applied egg-rr 2.5%

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

      1. unpow1 2.5%

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

      2. associate-/l* 2.5%

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

      3. *-commutative 2.5%

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

   7. Simplified 2.5%

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

      1. expm1-log1p-u 2.5%

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

      2. expm1-undefine 26.0%

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

      3. associate-*r/ 26.0%

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

      4. *-commutative 26.0%

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

      5. *-commutative 26.0%

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

      6. associate-*r* 26.0%

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

      7. *-un-lft-identity 26.0%

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

      8. times-frac 26.0%

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

      9. metadata-eval 26.0%

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

      10. *-commutative 26.0%

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

   9. Applied egg-rr 26.0%

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

       1. sub-neg 26.0%

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

       2. metadata-eval 26.0%

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

       3. +-commutative 26.0%

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

       4. log1p-undefine 26.0%

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

       5. rem-exp-log 26.0%

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

       6. +-commutative 26.0%

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

       7. fma-define 26.0%

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

       8. *-commutative 26.0%

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

       9. associate-/l* 26.0%

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

   11. Simplified 26.0%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(2, n \cdot \frac{\pi}{k}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% 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 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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. div-sub 99.6%

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

   4. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. metadata-eval 99.5%

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

   3. div-sub 99.5%

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

   4. unpow-prod-down 72.8%

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

   5. unpow-prod-down 99.5%

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

   6. associate-*l* 99.5%

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

   7. div-sub 99.5%

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

   8. metadata-eval 99.5%

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

   9. sub-neg 99.5%

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

   10. div-inv 99.5%

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

   11. metadata-eval 99.5%

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

   12. distribute-rgt-neg-in 99.5%

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

   13. metadata-eval 99.5%

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

   14. pow1/2 99.5%

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

   15. pow-flip 99.6%

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

   16. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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. div-sub 99.6%

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

   4. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 6: 49.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 50.4%

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

   1. pow1 50.4%

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

   2. associate-*l/ 50.5%

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

   3. *-un-lft-identity 50.5%

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

   4. sqrt-unprod 50.6%

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

   5. *-commutative 50.6%

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

   6. *-commutative 50.6%

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

   7. associate-*r* 50.6%

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

   8. sqrt-undiv 38.2%

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

   9. *-commutative 38.2%

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

   10. associate-*r* 38.2%

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

5. Applied egg-rr 38.2%

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

   1. unpow1 38.2%

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

   2. associate-/l* 38.2%

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

   3. *-commutative 38.2%

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

7. Simplified 38.2%

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

   1. clear-num 38.2%

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

   2. un-div-inv 38.2%

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

   3. *-un-lft-identity 38.2%

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

   4. *-commutative 38.2%

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

   5. times-frac 38.2%

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

   6. metadata-eval 38.2%

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

9. Applied egg-rr 38.2%

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

    1. associate-*r/ 38.2%

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

    2. *-commutative 38.2%

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

    3. associate-/r/ 38.2%

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

11. Simplified 38.2%

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

    1. sqrt-prod 50.6%

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

    2. metadata-eval 50.6%

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

    3. div-inv 50.6%

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

    4. div-inv 50.5%

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

    5. clear-num 50.5%

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

13. Applied egg-rr 50.5%

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

    1. associate-*r/ 50.6%

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

    2. associate-*l/ 50.6%

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

    3. *-commutative 50.6%

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

15. Simplified 50.6%

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

16. Final simplification 50.6%

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

Alternative 7: 39.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 50.4%

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

   1. pow1 50.4%

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

   2. associate-*l/ 50.5%

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

   3. *-un-lft-identity 50.5%

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

   4. sqrt-unprod 50.6%

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

   5. *-commutative 50.6%

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

   6. *-commutative 50.6%

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

   7. associate-*r* 50.6%

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

   8. sqrt-undiv 38.2%

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

   9. *-commutative 38.2%

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

   10. associate-*r* 38.2%

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

5. Applied egg-rr 38.2%

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

   1. unpow1 38.2%

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

   2. associate-/l* 38.2%

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

   3. *-commutative 38.2%

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

7. Simplified 38.2%

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

   1. clear-num 38.2%

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

   2. un-div-inv 38.2%

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

   3. *-un-lft-identity 38.2%

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

   4. *-commutative 38.2%

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

   5. times-frac 38.2%

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

   6. metadata-eval 38.2%

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

9. Applied egg-rr 38.2%

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

    1. associate-*r/ 38.2%

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

    2. *-commutative 38.2%

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

    3. associate-/r/ 38.2%

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

11. Simplified 38.2%

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

    1. metadata-eval 38.2%

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

    2. div-inv 38.2%

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

    3. associate-*l/ 38.2%

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

    4. clear-num 38.2%

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

    5. metadata-eval 38.2%

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

    6. *-commutative 38.2%

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

    7. add-sqr-sqrt 38.1%

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

    8. frac-times 38.1%

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

    9. sqrt-unprod 38.7%

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

    10. add-sqr-sqrt 38.8%

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

    11. inv-pow 38.8%

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

    12. sqrt-pow2 38.9%

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

13. Applied egg-rr 38.8%

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

    1. associate-/l* 38.8%

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

    2. associate-/l/ 38.9%

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

15. Simplified 38.9%

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

16. Final simplification 38.9%

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

Alternative 8: 38.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 50.4%

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

   1. pow1 50.4%

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

   2. associate-*l/ 50.5%

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

   3. *-un-lft-identity 50.5%

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

   4. sqrt-unprod 50.6%

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

   5. *-commutative 50.6%

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

   6. *-commutative 50.6%

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

   7. associate-*r* 50.6%

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

   8. sqrt-undiv 38.2%

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

   9. *-commutative 38.2%

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

   10. associate-*r* 38.2%

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

5. Applied egg-rr 38.2%

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

   1. unpow1 38.2%

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

   2. associate-/l* 38.2%

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

   3. *-commutative 38.2%

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

7. Simplified 38.2%

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

   1. associate-*r/ 38.2%

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

   2. *-commutative 38.2%

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

   3. sqrt-undiv 50.6%

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

   4. clear-num 50.5%

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

   5. sqrt-undiv 38.8%

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

   6. associate-*r* 38.8%

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

   7. *-commutative 38.8%

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

   8. associate-*l* 38.8%

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

9. Applied egg-rr 38.8%

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

    1. associate-/r* 38.8%

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

    2. *-commutative 38.8%

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

11. Simplified 38.8%

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

    1. add-sqr-sqrt 38.7%

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

    2. sqrt-unprod 38.1%

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

    3. frac-times 38.1%

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

    4. metadata-eval 38.1%

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

    5. add-sqr-sqrt 38.2%

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

    6. *-commutative 38.2%

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

    7. clear-num 38.2%

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

    8. associate-*l/ 38.2%

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

    9. div-inv 38.2%

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

    10. clear-num 38.2%

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

    11. associate-*l* 38.2%

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

13. Applied egg-rr 38.2%

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

14. Final simplification 38.2%

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

Alternative 9: 38.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 50.4%

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

   1. pow1 50.4%

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

   2. associate-*l/ 50.5%

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

   3. *-un-lft-identity 50.5%

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

   4. sqrt-unprod 50.6%

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

   5. *-commutative 50.6%

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

   6. *-commutative 50.6%

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

   7. associate-*r* 50.6%

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

   8. sqrt-undiv 38.2%

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

   9. *-commutative 38.2%

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

   10. associate-*r* 38.2%

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

5. Applied egg-rr 38.2%

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

   1. unpow1 38.2%

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

   2. associate-/l* 38.2%

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

   3. *-commutative 38.2%

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

7. Simplified 38.2%

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

   1. clear-num 38.2%

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

   2. un-div-inv 38.2%

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

   3. *-un-lft-identity 38.2%

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

   4. *-commutative 38.2%

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

   5. times-frac 38.2%

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

   6. metadata-eval 38.2%

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

9. Applied egg-rr 38.2%

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

    1. associate-*r/ 38.2%

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

    2. *-commutative 38.2%

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

    3. associate-/r/ 38.2%

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

11. Simplified 38.2%

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

12. Taylor expanded in k around 0 38.2%

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

    1. associate-*r/ 38.2%

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

    2. associate-*l/ 38.2%

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

    3. associate-/r/ 38.2%

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

14. Simplified 38.2%

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

15. Final simplification 38.2%

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

Reproduce

herbie shell --seed 2024042 
(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))))