Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 16.2s

Alternatives: 7

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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.1 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.1e-16)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (sqrt (/ (pow (* n (* 2.0 PI)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.1e-16) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = sqrt((pow((n * (2.0 * ((double) M_PI))), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.1e-16) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((2.0 * n));
	} else {
		tmp = Math.sqrt((Math.pow((n * (2.0 * Math.PI)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 4.1e-16:
		tmp = math.sqrt((math.pi / k)) * math.sqrt((2.0 * n))
	else:
		tmp = math.sqrt((math.pow((n * (2.0 * math.pi)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.1e-16)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = sqrt(Float64((Float64(n * Float64(2.0 * pi)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 4.1e-16)
		tmp = sqrt((pi / k)) * sqrt((2.0 * n));
	else
		tmp = sqrt((((n * (2.0 * pi)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.1e-16], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.10000000000000006e-16

   1. Initial program 98.8%

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

      1. add-sqr-sqrt 98.2%

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

      2. sqrt-unprod 77.9%

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

      3. *-commutative 77.9%

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

      4. associate-*r* 77.9%

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

      5. div-sub 77.9%

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

      6. metadata-eval 77.9%

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

      7. div-inv 77.9%

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

      8. *-commutative 77.9%

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

   4. Applied egg-rr 78.0%

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

   6. Simplified 78.1%

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

   7. Taylor expanded in k around 0 78.1%

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

      1. associate-*r/ 78.1%

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

   9. Simplified 78.1%

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

       1. pow1/2 78.1%

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

       2. associate-*r* 78.1%

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

       3. unpow-prod-down 99.3%

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

       4. pow1/2 99.4%

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

   11. Applied egg-rr 99.4%

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

       1. *-commutative 99.4%

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

       2. unpow1/2 99.4%

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

       3. *-commutative 99.4%

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

   13. Simplified 99.4%

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

   if 4.10000000000000006e-16 < 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. Step-by-step derivation

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

      3. associate-*l* 99.8%

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

      4. div-sub 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. div-inv 99.8%

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

      2. metadata-eval 99.8%

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

      3. div-sub 99.8%

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

      4. sqr-pow 99.8%

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

      5. associate-*l* 99.8%

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

      6. div-inv 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/l* 99.8%

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

      9. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\left(1 - k\right) \cdot 0.25\right)} \cdot \left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\left(1 - k\right) \cdot 0.25\right)} \cdot {k}^{-0.5}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 99.8%

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

      2. pow-sqr 99.8%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*r* 99.8%

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

      7. metadata-eval 99.8%

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

   8. Simplified 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. swap-sqr 99.8%

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

      4. pow-unpow 99.8%

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

      5. pow1/2 99.8%

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

      6. pow-unpow 99.8%

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

      7. pow1/2 99.8%

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

      8. pow-prod-down 99.8%

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

      9. add-sqr-sqrt 99.8%

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

      10. pow-prod-up 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*r/ 99.8%

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

       2. *-rgt-identity 99.8%

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

   12. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.1 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, \pi \cdot \frac{2}{k}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5e+68)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (sqrt (+ -1.0 (fma n (* PI (/ 2.0 k)) 1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5e+68) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = sqrt((-1.0 + fma(n, (((double) M_PI) * (2.0 / k)), 1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5e+68)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = sqrt(Float64(-1.0 + fma(n, Float64(pi * Float64(2.0 / k)), 1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5e+68], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-1.0 + N[(n * N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0000000000000004e68

   1. Initial program 98.8%

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

      1. add-sqr-sqrt 98.4%

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

      2. sqrt-unprod 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-*r* 81.0%

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

      5. div-sub 81.0%

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

      6. metadata-eval 81.0%

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

      7. div-inv 81.0%

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

      8. *-commutative 81.0%

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

   4. Applied egg-rr 81.1%

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

   6. Simplified 68.0%

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

   7. Taylor expanded in k around 0 68.2%

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

      1. associate-*r/ 68.2%

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

   9. Simplified 68.2%

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

       1. pow1/2 68.2%

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

       2. associate-*r* 68.2%

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

       3. unpow-prod-down 86.3%

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

       4. pow1/2 86.3%

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

   11. Applied egg-rr 86.3%

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

       1. *-commutative 86.3%

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

       2. unpow1/2 86.3%

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

       3. *-commutative 86.3%

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

   13. Simplified 86.3%

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

   if 5.0000000000000004e68 < 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 1.6%

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

   7. Taylor expanded in k around 0 2.7%

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

      1. associate-*r/ 2.7%

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

   9. Simplified 2.7%

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

       1. expm1-log1p-u 2.7%

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

       2. expm1-undefine 32.5%

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

   11. Applied egg-rr 32.5%

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

       1. sub-neg 32.5%

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

       2. metadata-eval 32.5%

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

       3. +-commutative 32.5%

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

       4. log1p-undefine 32.5%

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

       5. rem-exp-log 32.5%

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

       6. +-commutative 32.5%

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

       7. associate-*r/ 32.5%

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

       8. associate-*l/ 32.5%

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

       9. associate-/r/ 32.5%

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

       10. associate-/l* 32.5%

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

       11. *-commutative 32.5%

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

       12. associate-/l* 32.5%

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

       13. fma-define 32.5%

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

       14. associate-/r/ 32.5%

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

   13. Simplified 32.5%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, \pi \cdot \frac{2}{k}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.8e+68)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (sqrt (* 2.0 (+ -1.0 (fma n (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.8e+68) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = sqrt((2.0 * (-1.0 + fma(n, (((double) M_PI) / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.8e+68)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = sqrt(Float64(2.0 * Float64(-1.0 + fma(n, Float64(pi / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.8e+68], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(-1.0 + N[(n * N[(Pi / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.80000000000000016e68

   1. Initial program 98.8%

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

      1. add-sqr-sqrt 98.4%

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

      2. sqrt-unprod 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-*r* 81.0%

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

      5. div-sub 81.0%

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

      6. metadata-eval 81.0%

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

      7. div-inv 81.0%

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

      8. *-commutative 81.0%

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

   4. Applied egg-rr 81.1%

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

   6. Simplified 68.0%

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

   7. Taylor expanded in k around 0 68.2%

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

      1. associate-*r/ 68.2%

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

   9. Simplified 68.2%

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

       1. pow1/2 68.2%

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

       2. associate-*r* 68.2%

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

       3. unpow-prod-down 86.3%

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

       4. pow1/2 86.3%

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

   11. Applied egg-rr 86.3%

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

       1. *-commutative 86.3%

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

       2. unpow1/2 86.3%

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

       3. *-commutative 86.3%

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

   13. Simplified 86.3%

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

   if 4.80000000000000016e68 < 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 1.6%

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

   7. Taylor expanded in k around 0 2.7%

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

      1. associate-*r/ 2.7%

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

   9. Simplified 2.7%

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

       1. clear-num 2.7%

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

       2. un-div-inv 2.7%

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

   11. Applied egg-rr 2.7%

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

       1. expm1-log1p-u 2.7%

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

       2. expm1-undefine 32.5%

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

       3. div-inv 32.5%

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

       4. clear-num 32.5%

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

   13. Applied egg-rr 32.5%

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

       1. sub-neg 32.5%

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

       2. metadata-eval 32.5%

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

       3. +-commutative 32.5%

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

       4. log1p-undefine 32.5%

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

       5. rem-exp-log 32.5%

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

       6. +-commutative 32.5%

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

       7. fma-define 32.5%

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

   15. Simplified 32.5%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. associate-*l/ 99.3%

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

   2. *-lft-identity 99.3%

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

   3. associate-*l* 99.3%

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

   4. div-sub 99.3%

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

   5. metadata-eval 99.3%

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

3. Simplified 99.3%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. associate-*l/ 99.3%

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

   2. *-lft-identity 99.3%

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

   3. associate-*l* 99.3%

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

   4. div-sub 99.3%

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

   5. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. associate-*r* 99.3%

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

   2. metadata-eval 99.3%

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

   3. div-sub 99.3%

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

   4. unpow-prod-down 75.9%

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

   5. associate-/l* 75.9%

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

   6. div-sub 75.9%

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

   7. metadata-eval 75.9%

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

   8. div-inv 75.9%

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

   9. metadata-eval 75.9%

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

   10. div-sub 75.9%

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

   11. metadata-eval 75.9%

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

   12. div-inv 75.9%

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

   13. metadata-eval 75.9%

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

6. Applied egg-rr 75.9%

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

   1. clear-num 75.9%

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

   2. un-div-inv 76.0%

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

   3. *-commutative 76.0%

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

   4. *-commutative 76.0%

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

8. Applied egg-rr 76.0%

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

   1. associate-/r/ 75.9%

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

   2. cancel-sign-sub-inv 75.9%

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

   3. metadata-eval 75.9%

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

   4. *-commutative 75.9%

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

   5. cancel-sign-sub-inv 75.9%

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

   6. metadata-eval 75.9%

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

   7. *-commutative 75.9%

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

10. Simplified 75.9%

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

11. Taylor expanded in k around 0 99.0%

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

    1. pow1 99.0%

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

    2. sqrt-unprod 99.1%

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

13. Applied egg-rr 99.1%

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

    1. unpow1 99.1%

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

15. Simplified 99.1%

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

16. Final simplification 99.1%

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

Alternative 6: 48.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. add-sqr-sqrt 99.0%

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

   2. sqrt-unprod 87.9%

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

   3. *-commutative 87.9%

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

   4. associate-*r* 87.9%

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

   5. div-sub 87.9%

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

   6. metadata-eval 87.9%

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

   7. div-inv 87.9%

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

   8. *-commutative 87.9%

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

4. Applied egg-rr 88.0%

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

6. Simplified 43.9%

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

7. Taylor expanded in k around 0 44.4%

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

   1. associate-*r/ 44.4%

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

9. Simplified 44.4%

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

    1. pow1/2 44.4%

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

    2. associate-*r* 44.4%

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

    3. unpow-prod-down 56.0%

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

    4. pow1/2 56.0%

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

11. Applied egg-rr 56.0%

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

    1. *-commutative 56.0%

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

    2. unpow1/2 56.0%

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

    3. *-commutative 56.0%

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

13. Simplified 56.0%

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

14. Final simplification 56.0%

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

Alternative 7: 37.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. add-sqr-sqrt 99.0%

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

   2. sqrt-unprod 87.9%

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

   3. *-commutative 87.9%

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

   4. associate-*r* 87.9%

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

   5. div-sub 87.9%

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

   6. metadata-eval 87.9%

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

   7. div-inv 87.9%

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

   8. *-commutative 87.9%

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

4. Applied egg-rr 88.0%

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

6. Simplified 43.9%

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

7. Taylor expanded in k around 0 44.4%

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

   1. associate-*r/ 44.4%

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

9. Simplified 44.4%

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

Reproduce

herbie shell --seed 2024114 
(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))))