Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 24.8s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(n * 2.0), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(k * N[Power[t$95$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. 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. *-un-lft-identity 99.6%

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

   3. associate-*r* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

   6. pow-div 99.8%

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

   7. pow1/2 99.8%

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

   8. associate-/l/ 99.8%

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

   9. div-inv 99.8%

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

   10. metadata-eval 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-*r* 99.8%

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

   2. associate-*r* 99.8%

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

6. Simplified 99.8%

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

   1. div-inv 99.7%

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

   2. associate-*r* 99.7%

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

   3. add-sqr-sqrt 99.5%

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

   4. sqrt-unprod 99.7%

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

   5. swap-sqr 99.7%

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

   6. add-sqr-sqrt 99.7%

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

   7. pow-unpow 99.7%

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

   8. unpow1/2 99.7%

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

   9. pow-unpow 99.7%

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

   10. unpow1/2 99.7%

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

   11. add-sqr-sqrt 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-*r* 99.8%

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

   5. *-commutative 99.8%

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

   6. *-commutative 99.8%

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

   7. associate-*r* 99.8%

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

   8. *-commutative 99.8%

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

10. Simplified 99.8%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5e-46)
   (* (sqrt (/ (* 2.0 PI) k)) (sqrt n))
   (sqrt (/ (pow (* n (* 2.0 PI)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5e-46) {
		tmp = sqrt(((2.0 * ((double) M_PI)) / k)) * sqrt(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 <= 5e-46) {
		tmp = Math.sqrt(((2.0 * Math.PI) / k)) * Math.sqrt(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 <= 5e-46:
		tmp = math.sqrt(((2.0 * math.pi) / k)) * math.sqrt(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 <= 5e-46)
		tmp = Float64(sqrt(Float64(Float64(2.0 * pi) / k)) * sqrt(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 <= 5e-46)
		tmp = sqrt(((2.0 * pi) / k)) * sqrt(n);
	else
		tmp = sqrt((((n * (2.0 * pi)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5e-46], N[(N[Sqrt[N[(N[(2.0 * Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $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 < 4.99999999999999992e-46

   1. Initial program 99.2%

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

   3. Taylor expanded in k around 0 76.2%

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

      1. associate-/l* 76.2%

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

   5. Simplified 76.2%

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

      1. pow1 76.2%

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

      2. *-commutative 76.2%

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

      3. sqrt-unprod 76.6%

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

   7. Applied egg-rr 76.6%

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

      1. unpow1 76.6%

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

   9. Simplified 76.6%

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

       1. clear-num 76.4%

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

       2. un-div-inv 76.5%

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

   11. Applied egg-rr 76.5%

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

       1. associate-*r/ 76.5%

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

       2. *-commutative 76.5%

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

       3. div-inv 76.4%

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

       4. clear-num 76.6%

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

       5. div-inv 76.4%

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

       6. associate-*r* 76.4%

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

       7. *-commutative 76.4%

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

       8. associate-*r* 76.4%

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

       9. *-commutative 76.4%

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

       10. *-commutative 76.4%

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

       11. associate-*r* 76.4%

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

       12. *-commutative 76.4%

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

       13. div-inv 76.4%

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

       14. associate-*r* 76.4%

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

       15. *-commutative 76.4%

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

       16. sqrt-prod 99.5%

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

       17. associate-*r/ 99.6%

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

   13. Applied egg-rr 99.6%

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

   if 4.99999999999999992e-46 < k

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 99.6%

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

      2. sqrt-unprod 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*r* 99.6%

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

      5. div-sub 99.6%

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

      6. metadata-eval 99.6%

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

      7. div-inv 99.7%

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

      8. *-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.7e+89)
   (* (sqrt (/ (* 2.0 PI) k)) (sqrt n))
   (sqrt (* 2.0 (+ -1.0 (fma n (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.7e+89) {
		tmp = sqrt(((2.0 * ((double) M_PI)) / k)) * sqrt(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 <= 4.7e+89)
		tmp = Float64(sqrt(Float64(Float64(2.0 * pi) / k)) * sqrt(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, 4.7e+89], N[(N[Sqrt[N[(N[(2.0 * Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $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 < 4.70000000000000022e89

   1. Initial program 99.2%

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

   3. Taylor expanded in k around 0 53.4%

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

      1. associate-/l* 53.4%

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

   5. Simplified 53.4%

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

      1. pow1 53.4%

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

      2. *-commutative 53.4%

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

      3. sqrt-unprod 53.6%

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

   7. Applied egg-rr 53.6%

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

      1. unpow1 53.6%

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

   9. Simplified 53.6%

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

       1. clear-num 53.5%

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

       2. un-div-inv 53.6%

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

   11. Applied egg-rr 53.6%

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

       1. associate-*r/ 53.6%

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

       2. *-commutative 53.6%

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

       3. div-inv 53.5%

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

       4. clear-num 53.6%

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

       5. div-inv 53.6%

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

       6. associate-*r* 53.6%

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

       7. *-commutative 53.6%

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

       8. associate-*r* 53.6%

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

       9. *-commutative 53.6%

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

       10. *-commutative 53.6%

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

       11. associate-*r* 53.6%

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

       12. *-commutative 53.6%

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

       13. div-inv 53.6%

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

       14. associate-*r* 53.6%

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

       15. *-commutative 53.6%

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

       16. sqrt-prod 66.9%

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

       17. associate-*r/ 67.0%

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

   13. Applied egg-rr 67.0%

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

   if 4.70000000000000022e89 < 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 2.9%

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

      1. associate-/l* 2.9%

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

   5. Simplified 2.9%

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

      1. pow1 2.9%

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

      2. *-commutative 2.9%

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

      3. sqrt-unprod 2.9%

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

   7. Applied egg-rr 2.9%

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

      1. unpow1 2.9%

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

   9. Simplified 2.9%

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

       1. expm1-log1p-u 2.9%

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

       2. expm1-undefine 41.7%

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

   11. Applied egg-rr 41.7%

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

       1. sub-neg 41.7%

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

       2. metadata-eval 41.7%

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

       3. +-commutative 41.7%

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

       4. log1p-undefine 41.7%

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

       5. rem-exp-log 41.7%

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

       6. +-commutative 41.7%

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

       7. fma-define 41.7%

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

   13. Simplified 41.7%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.55 \cdot 10^{+241}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.55e+241)
   (* (sqrt (/ (* 2.0 PI) k)) (sqrt n))
   (cbrt (pow (* n (* PI (/ 2.0 k))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.55e+241) {
		tmp = sqrt(((2.0 * ((double) M_PI)) / k)) * sqrt(n);
	} else {
		tmp = cbrt(pow((n * (((double) M_PI) * (2.0 / k))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.55e+241) {
		tmp = Math.sqrt(((2.0 * Math.PI) / k)) * Math.sqrt(n);
	} else {
		tmp = Math.cbrt(Math.pow((n * (Math.PI * (2.0 / k))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.55e+241)
		tmp = Float64(sqrt(Float64(Float64(2.0 * pi) / k)) * sqrt(n));
	else
		tmp = cbrt((Float64(n * Float64(pi * Float64(2.0 / k))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.55e+241], N[(N[Sqrt[N[(N[(2.0 * Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(n * N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.5500000000000001e241

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0 40.7%

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

      1. associate-/l* 40.7%

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

   5. Simplified 40.7%

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

      1. pow1 40.7%

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

      2. *-commutative 40.7%

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

      3. sqrt-unprod 40.9%

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

   7. Applied egg-rr 40.9%

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

      1. unpow1 40.9%

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

   9. Simplified 40.9%

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

       1. clear-num 40.8%

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

       2. un-div-inv 40.8%

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

   11. Applied egg-rr 40.8%

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

       1. associate-*r/ 40.8%

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

       2. *-commutative 40.8%

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

       3. div-inv 40.8%

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

       4. clear-num 40.9%

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

       5. div-inv 40.8%

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

       6. associate-*r* 40.8%

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

       7. *-commutative 40.8%

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

       8. associate-*r* 40.8%

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

       9. *-commutative 40.8%

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

       10. *-commutative 40.8%

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

       11. associate-*r* 40.8%

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

       12. *-commutative 40.8%

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

       13. div-inv 40.8%

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

       14. associate-*r* 40.8%

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

       15. *-commutative 40.8%

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

       16. sqrt-prod 50.9%

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

       17. associate-*r/ 50.9%

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

   13. Applied egg-rr 50.9%

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

   if 2.5500000000000001e241 < 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 2.9%

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

      1. associate-/l* 2.9%

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

   5. Simplified 2.9%

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

      1. pow1 2.9%

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

      2. *-commutative 2.9%

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

      3. sqrt-unprod 2.9%

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

   7. Applied egg-rr 2.9%

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

      1. unpow1 2.9%

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

   9. Simplified 2.9%

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

       1. add-cbrt-cube 20.2%

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

       2. pow1/3 20.2%

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

       3. add-sqr-sqrt 20.2%

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

       4. pow1 20.2%

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

       5. pow1/2 20.2%

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

       6. pow-prod-up 20.2%

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

       7. associate-*r* 20.2%

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

       8. *-commutative 20.2%

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

       9. associate-*l* 20.2%

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

       10. metadata-eval 20.2%

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

   11. Applied egg-rr 20.2%

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

       1. unpow1/3 20.2%

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

       2. associate-*r/ 20.2%

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

       3. *-commutative 20.2%

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

       4. associate-/l* 20.2%

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

   13. Simplified 20.2%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.55 \cdot 10^{+241}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(2.0 * N[(n * 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. associate-*l* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 6: 49.7% accurate, 1.0× speedup

Math

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

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(Float64(2.0 * 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[(N[(2.0 * Pi), $MachinePrecision] / k), $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 35.8%

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

   1. associate-/l* 35.8%

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

5. Simplified 35.8%

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

   1. pow1 35.8%

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

   2. *-commutative 35.8%

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

   3. sqrt-unprod 36.0%

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

7. Applied egg-rr 36.0%

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

   1. unpow1 36.0%

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

9. Simplified 36.0%

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

    1. clear-num 35.9%

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

    2. un-div-inv 35.9%

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

11. Applied egg-rr 35.9%

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

    1. associate-*r/ 35.9%

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

    2. *-commutative 35.9%

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

    3. div-inv 35.9%

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

    4. clear-num 36.0%

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

    5. div-inv 35.9%

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

    6. associate-*r* 35.9%

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

    7. *-commutative 35.9%

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

    8. associate-*r* 35.9%

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

    9. *-commutative 35.9%

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

    10. *-commutative 35.9%

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

    11. associate-*r* 35.9%

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

    12. *-commutative 35.9%

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

    13. div-inv 35.9%

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

    14. associate-*r* 35.9%

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

    15. *-commutative 35.9%

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

    16. sqrt-prod 44.7%

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

    17. associate-*r/ 44.7%

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

13. Applied egg-rr 44.7%

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

14. Final simplification 44.7%

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

Alternative 7: 49.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 35.8%

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

   1. associate-/l* 35.8%

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

5. Simplified 35.8%

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

   1. pow1 35.8%

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

   2. *-commutative 35.8%

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

   3. sqrt-unprod 36.0%

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

7. Applied egg-rr 36.0%

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

   1. unpow1 36.0%

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

9. Simplified 36.0%

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

    1. associate-*r* 36.0%

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

    2. *-commutative 36.0%

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

    3. sqrt-prod 44.7%

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

11. Applied egg-rr 44.7%

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

Alternative 8: 38.3% 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 99.5%

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

3. Taylor expanded in k around 0 35.8%

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

   1. associate-/l* 35.8%

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

5. Simplified 35.8%

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

   1. pow1 35.8%

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

   2. *-commutative 35.8%

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

   3. sqrt-unprod 36.0%

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

7. Applied egg-rr 36.0%

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

   1. unpow1 36.0%

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

9. Simplified 36.0%

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

Reproduce

herbie shell --seed 2024086 
(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))))