Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 18.2s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;k \leq 1.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{{k}^{-0.5}}{{t\_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{t\_0}^{\left(1 - k\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* n (* 2.0 PI))))
   (if (<= k 1.2e-39)
     (/ (pow k -0.5) (pow t_0 -0.5))
     (/ 1.0 (sqrt (/ k (pow t_0 (- 1.0 k))))))))

C

double code(double k, double n) {
	double t_0 = n * (2.0 * ((double) M_PI));
	double tmp;
	if (k <= 1.2e-39) {
		tmp = pow(k, -0.5) / pow(t_0, -0.5);
	} else {
		tmp = 1.0 / sqrt((k / pow(t_0, (1.0 - k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = n * (2.0 * Math.PI);
	double tmp;
	if (k <= 1.2e-39) {
		tmp = Math.pow(k, -0.5) / Math.pow(t_0, -0.5);
	} else {
		tmp = 1.0 / Math.sqrt((k / Math.pow(t_0, (1.0 - k))));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = n * (2.0 * math.pi)
	tmp = 0
	if k <= 1.2e-39:
		tmp = math.pow(k, -0.5) / math.pow(t_0, -0.5)
	else:
		tmp = 1.0 / math.sqrt((k / math.pow(t_0, (1.0 - k))))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(n * Float64(2.0 * pi))
	tmp = 0.0
	if (k <= 1.2e-39)
		tmp = Float64((k ^ -0.5) / (t_0 ^ -0.5));
	else
		tmp = Float64(1.0 / sqrt(Float64(k / (t_0 ^ Float64(1.0 - k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = n * (2.0 * pi);
	tmp = 0.0;
	if (k <= 1.2e-39)
		tmp = (k ^ -0.5) / (t_0 ^ -0.5);
	else
		tmp = 1.0 / sqrt((k / (t_0 ^ (1.0 - k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.2e-39], N[(N[Power[k, -0.5], $MachinePrecision] / N[Power[t$95$0, -0.5], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(k / N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.20000000000000008e-39

   1. Initial program 99.3%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing
   3. 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 69.2%

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

      3. *-commutative 69.2%

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

      4. *-commutative 69.2%

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

      5. associate-*r* 69.2%

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

      6. div-sub 69.2%

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

      7. metadata-eval 69.2%

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

      8. div-inv 69.2%

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

      9. *-commutative 69.2%

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

   4. Applied egg-rr 69.3%

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

   6. Simplified 69.5%

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

   7. Taylor expanded in k around 0 69.5%

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

      1. associate-/l* 69.5%

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

   9. Simplified 69.5%

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

       1. associate-/r/ 69.5%

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

   11. Applied egg-rr 69.5%

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

       1. *-commutative 69.5%

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

       2. associate-*l/ 69.5%

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

       3. associate-*l/ 69.5%

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

       4. associate-*r* 69.5%

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

       5. sqrt-undiv 99.3%

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

       6. sqrt-prod 98.9%

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

       7. *-commutative 98.9%

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

       8. sqrt-prod 99.3%

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

       9. clear-num 99.3%

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

       10. div-inv 99.2%

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

       11. associate-/r* 99.2%

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

       12. pow1/2 99.2%

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

       13. pow-flip 99.2%

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

       14. metadata-eval 99.2%

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

       15. pow1/2 99.2%

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

       16. pow-flip 99.4%

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

       17. metadata-eval 99.4%

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

   13. Applied egg-rr 99.4%

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

   if 1.20000000000000008e-39 < k

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. div-inv 99.9%

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

      9. *-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

   6. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. sqrt-div 99.9%

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

      3. metadata-eval 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*r* 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{{k}^{-0.5}}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-40}:\\ \;\;\;\;\frac{{k}^{-0.5}}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3e-40)
   (/ (pow k -0.5) (pow (* n (* 2.0 PI)) -0.5))
   (sqrt (/ (pow (* PI (* n 2.0)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3e-40) {
		tmp = pow(k, -0.5) / pow((n * (2.0 * ((double) M_PI))), -0.5);
	} else {
		tmp = sqrt((pow((((double) M_PI) * (n * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3e-40) {
		tmp = Math.pow(k, -0.5) / Math.pow((n * (2.0 * Math.PI)), -0.5);
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (n * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3e-40:
		tmp = math.pow(k, -0.5) / math.pow((n * (2.0 * math.pi)), -0.5)
	else:
		tmp = math.sqrt((math.pow((math.pi * (n * 2.0)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3e-40)
		tmp = Float64((k ^ -0.5) / (Float64(n * Float64(2.0 * pi)) ^ -0.5));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(n * 2.0)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3e-40)
		tmp = (k ^ -0.5) / ((n * (2.0 * pi)) ^ -0.5);
	else
		tmp = sqrt((((pi * (n * 2.0)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3e-40], N[(N[Power[k, -0.5], $MachinePrecision] / N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.0000000000000002e-40

   1. Initial program 99.3%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing
   3. 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 69.2%

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

      3. *-commutative 69.2%

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

      4. *-commutative 69.2%

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

      5. associate-*r* 69.2%

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

      6. div-sub 69.2%

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

      7. metadata-eval 69.2%

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

      8. div-inv 69.2%

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

      9. *-commutative 69.2%

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

   4. Applied egg-rr 69.3%

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

   6. Simplified 69.5%

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

   7. Taylor expanded in k around 0 69.5%

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

      1. associate-/l* 69.5%

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

   9. Simplified 69.5%

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

       1. associate-/r/ 69.5%

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

   11. Applied egg-rr 69.5%

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

       1. *-commutative 69.5%

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

       2. associate-*l/ 69.5%

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

       3. associate-*l/ 69.5%

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

       4. associate-*r* 69.5%

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

       5. sqrt-undiv 99.3%

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

       6. sqrt-prod 98.9%

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

       7. *-commutative 98.9%

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

       8. sqrt-prod 99.3%

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

       9. clear-num 99.3%

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

       10. div-inv 99.2%

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

       11. associate-/r* 99.2%

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

       12. pow1/2 99.2%

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

       13. pow-flip 99.2%

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

       14. metadata-eval 99.2%

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

       15. pow1/2 99.2%

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

       16. pow-flip 99.4%

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

       17. metadata-eval 99.4%

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

   13. Applied egg-rr 99.4%

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

   if 3.0000000000000002e-40 < k

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. div-inv 99.9%

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

      9. *-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-40}:\\ \;\;\;\;\frac{{k}^{-0.5}}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{\frac{2}{k}} \cdot \sqrt{n \cdot \pi}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, \frac{\pi}{\frac{k}{2}}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.75e+39)
   (* (sqrt (/ 2.0 k)) (sqrt (* n PI)))
   (sqrt (+ -1.0 (fma n (/ PI (/ k 2.0)) 1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.75e+39) {
		tmp = sqrt((2.0 / k)) * sqrt((n * ((double) M_PI)));
	} else {
		tmp = sqrt((-1.0 + fma(n, (((double) M_PI) / (k / 2.0)), 1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.75e+39)
		tmp = Float64(sqrt(Float64(2.0 / k)) * sqrt(Float64(n * pi)));
	else
		tmp = sqrt(Float64(-1.0 + fma(n, Float64(pi / Float64(k / 2.0)), 1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.75e+39], N[(N[Sqrt[N[(2.0 / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-1.0 + N[(n * N[(Pi / N[(k / 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.7500000000000001e39

   1. Initial program 99.3%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing
   3. 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 76.7%

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

      3. *-commutative 76.7%

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

      4. *-commutative 76.7%

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

      5. associate-*r* 76.7%

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

      6. div-sub 76.7%

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

      7. metadata-eval 76.7%

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

      8. div-inv 76.8%

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

      9. *-commutative 76.8%

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

   4. Applied egg-rr 76.8%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in k around 0 64.4%

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

      1. associate-/l* 64.3%

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

   9. Simplified 64.3%

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

       1. clear-num 64.2%

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

       2. un-div-inv 64.2%

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

       3. associate-/l/ 64.3%

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

       4. *-commutative 64.3%

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

       5. associate-/l* 64.4%

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

       6. *-un-lft-identity 64.4%

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

       7. associate-*l/ 64.3%

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

       8. associate-*r* 64.3%

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

       9. sqrt-prod 86.7%

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

       10. associate-*l/ 86.7%

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

       11. metadata-eval 86.7%

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

   11. Applied egg-rr 86.7%

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

   if 1.7500000000000001e39 < 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 2.7%

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

      1. metadata-eval 2.7%

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

      2. sqrt-div 2.7%

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

      3. *-commutative 2.7%

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

      4. *-commutative 2.7%

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

      5. sqrt-prod 2.7%

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

      6. sqrt-unprod 2.6%

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

   5. Applied egg-rr 2.6%

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

      1. expm1-log1p-u 2.6%

         \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{k} \cdot \left(2 \cdot \left(\pi \cdot n\right)\right)\right)\right)}} \]

      2. expm1-undefine 29.8%

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

      3. associate-*l/ 29.8%

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

      4. *-un-lft-identity 29.8%

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

      5. *-un-lft-identity 29.8%

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

      6. times-frac 29.8%

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

      7. metadata-eval 29.8%

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

      8. *-commutative 29.8%

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

      9. associate-*l/ 29.8%

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

      10. *-commutative 29.8%

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

   7. Applied egg-rr 29.8%

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

      1. sub-neg 29.8%

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

      2. metadata-eval 29.8%

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

      3. +-commutative 29.8%

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

      4. log1p-undefine 29.8%

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

      5. rem-exp-log 29.8%

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

      6. +-commutative 29.8%

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

      7. associate-*r* 29.8%

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

      8. associate-*r/ 29.8%

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

      9. associate-*l/ 29.8%

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

      10. *-commutative 29.8%

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

      11. fma-define 29.8%

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

      12. *-commutative 29.8%

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

      13. associate-/l* 29.8%

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

   9. Simplified 29.8%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{\frac{2}{k}} \cdot \sqrt{n \cdot \pi}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, \frac{\pi}{\frac{k}{2}}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\frac{{k}^{-0.5}}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, \frac{\pi}{\frac{k}{2}}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2e+39)
   (/ (pow k -0.5) (pow (* n (* 2.0 PI)) -0.5))
   (sqrt (+ -1.0 (fma n (/ PI (/ k 2.0)) 1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2e+39) {
		tmp = pow(k, -0.5) / pow((n * (2.0 * ((double) M_PI))), -0.5);
	} else {
		tmp = sqrt((-1.0 + fma(n, (((double) M_PI) / (k / 2.0)), 1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2e+39)
		tmp = Float64((k ^ -0.5) / (Float64(n * Float64(2.0 * pi)) ^ -0.5));
	else
		tmp = sqrt(Float64(-1.0 + fma(n, Float64(pi / Float64(k / 2.0)), 1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2e+39], N[(N[Power[k, -0.5], $MachinePrecision] / N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-1.0 + N[(n * N[(Pi / N[(k / 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999988e39

   1. Initial program 99.3%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing
   3. 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 76.7%

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

      3. *-commutative 76.7%

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

      4. *-commutative 76.7%

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

      5. associate-*r* 76.7%

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

      6. div-sub 76.7%

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

      7. metadata-eval 76.7%

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

      8. div-inv 76.8%

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

      9. *-commutative 76.8%

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

   4. Applied egg-rr 76.8%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in k around 0 64.4%

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

      1. associate-/l* 64.3%

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

   9. Simplified 64.3%

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

       1. associate-/r/ 64.4%

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

   11. Applied egg-rr 64.4%

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

       1. *-commutative 64.4%

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

       2. associate-*l/ 64.4%

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

       3. associate-*l/ 64.4%

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

       4. associate-*r* 64.4%

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

       5. sqrt-undiv 86.7%

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

       6. sqrt-prod 86.3%

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

       7. *-commutative 86.3%

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

       8. sqrt-prod 86.7%

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

       9. clear-num 86.6%

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

       10. div-inv 86.5%

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

       11. associate-/r* 86.5%

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

       12. pow1/2 86.5%

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

       13. pow-flip 86.6%

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

       14. metadata-eval 86.6%

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

       15. pow1/2 86.6%

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

       16. pow-flip 86.8%

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

       17. metadata-eval 86.8%

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

   13. Applied egg-rr 86.8%

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

   if 1.99999999999999988e39 < 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 2.7%

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

      1. metadata-eval 2.7%

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

      2. sqrt-div 2.7%

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

      3. *-commutative 2.7%

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

      4. *-commutative 2.7%

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

      5. sqrt-prod 2.7%

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

      6. sqrt-unprod 2.6%

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

   5. Applied egg-rr 2.6%

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

      1. expm1-log1p-u 2.6%

         \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{k} \cdot \left(2 \cdot \left(\pi \cdot n\right)\right)\right)\right)}} \]

      2. expm1-undefine 29.8%

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

      3. associate-*l/ 29.8%

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

      4. *-un-lft-identity 29.8%

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

      5. *-un-lft-identity 29.8%

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

      6. times-frac 29.8%

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

      7. metadata-eval 29.8%

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

      8. *-commutative 29.8%

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

      9. associate-*l/ 29.8%

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

      10. *-commutative 29.8%

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

   7. Applied egg-rr 29.8%

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

      1. sub-neg 29.8%

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

      2. metadata-eval 29.8%

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

      3. +-commutative 29.8%

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

      4. log1p-undefine 29.8%

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

      5. rem-exp-log 29.8%

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

      6. +-commutative 29.8%

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

      7. associate-*r* 29.8%

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

      8. associate-*r/ 29.8%

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

      9. associate-*l/ 29.8%

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

      10. *-commutative 29.8%

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

      11. fma-define 29.8%

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

      12. *-commutative 29.8%

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

      13. associate-/l* 29.8%

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

   9. Simplified 29.8%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\frac{{k}^{-0.5}}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, \frac{\pi}{\frac{k}{2}}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.7%

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

   2. *-lft-identity 99.7%

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

   3. sqr-pow 99.5%

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

   4. pow-sqr 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l* 99.7%

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

   7. associate-*r/ 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-/l* 99.7%

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

   10. metadata-eval 99.7%

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

   11. /-rgt-identity 99.7%

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

   12. div-sub 99.7%

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

   13. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. div-inv 99.6%

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

   2. associate-*r* 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

   5. div-inv 99.6%

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

   6. metadata-eval 99.6%

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

   7. inv-pow 99.6%

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

   8. sqrt-pow2 99.6%

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

   9. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

   1. metadata-eval 99.6%

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

   2. pow-flip 99.6%

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

   3. pow1/2 99.6%

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

   4. div-inv 99.7%

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

   5. clear-num 99.6%

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

   6. div-inv 99.6%

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

   7. associate-/r* 99.6%

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

   8. inv-pow 99.6%

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

   9. sqrt-pow2 99.6%

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

   10. metadata-eval 99.6%

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

   11. pow-flip 99.7%

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

   12. associate-*r* 99.7%

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

   13. *-commutative 99.7%

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

   14. sub-neg 99.7%

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

   15. distribute-rgt-neg-in 99.7%

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

   16. metadata-eval 99.7%

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

8. Applied egg-rr 99.7%

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

   1. +-commutative 99.7%

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

   2. distribute-neg-in 99.7%

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

   3. distribute-rgt-neg-in 99.7%

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

   4. metadata-eval 99.7%

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

   5. metadata-eval 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.7%

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

   2. *-lft-identity 99.7%

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

   3. sqr-pow 99.5%

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

   4. pow-sqr 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l* 99.7%

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

   7. associate-*r/ 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-/l* 99.7%

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

   10. metadata-eval 99.7%

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

   11. /-rgt-identity 99.7%

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

   12. div-sub 99.7%

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

   13. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 7: 49.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 87.3%

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

   3. *-commutative 87.3%

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

   4. *-commutative 87.3%

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

   5. associate-*r* 87.3%

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

   6. div-sub 87.3%

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

   7. metadata-eval 87.3%

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

   8. div-inv 87.3%

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

   9. *-commutative 87.3%

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

4. Applied egg-rr 87.3%

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

6. Simplified 87.4%

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

7. Taylor expanded in k around 0 36.4%

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

   1. associate-/l* 36.3%

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

9. Simplified 36.3%

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

    1. clear-num 36.3%

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

    2. un-div-inv 36.3%

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

    3. associate-/l/ 36.3%

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

    4. *-commutative 36.3%

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

    5. associate-/l* 36.4%

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

    6. *-un-lft-identity 36.4%

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

    7. associate-*l/ 36.3%

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

    8. associate-*r* 36.3%

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

    9. sqrt-prod 48.7%

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

    10. associate-*l/ 48.7%

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

    11. metadata-eval 48.7%

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

11. Applied egg-rr 48.7%

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

12. Final simplification 48.7%

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

Alternative 8: 39.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 87.3%

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

   3. *-commutative 87.3%

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

   4. *-commutative 87.3%

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

   5. associate-*r* 87.3%

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

   6. div-sub 87.3%

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

   7. metadata-eval 87.3%

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

   8. div-inv 87.3%

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

   9. *-commutative 87.3%

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

4. Applied egg-rr 87.3%

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

6. Simplified 87.4%

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

7. Taylor expanded in k around 0 36.4%

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

   1. associate-/l* 36.3%

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

9. Simplified 36.3%

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

    1. associate-/r/ 36.4%

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

11. Applied egg-rr 36.4%

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

    1. *-commutative 36.4%

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

    2. associate-*l/ 36.4%

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

    3. associate-*l/ 36.4%

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

    4. associate-*r* 36.4%

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

    5. sqrt-undiv 48.6%

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

    6. sqrt-prod 48.4%

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

    7. *-commutative 48.4%

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

    8. sqrt-prod 48.6%

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

    9. clear-num 48.6%

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

    10. inv-pow 48.6%

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

    11. sqrt-undiv 36.9%

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

    12. sqrt-pow2 36.9%

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

    13. *-commutative 36.9%

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

    14. associate-/r* 36.9%

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

    15. metadata-eval 36.9%

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

13. Applied egg-rr 36.9%

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

    1. associate-/l/ 36.9%

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

    2. associate-*r* 36.9%

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

    3. associate-/r* 36.9%

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

    4. *-commutative 36.9%

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

15. Simplified 36.9%

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

16. Final simplification 36.9%

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

Alternative 9: 38.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 87.3%

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

   3. *-commutative 87.3%

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

   4. *-commutative 87.3%

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

   5. associate-*r* 87.3%

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

   6. div-sub 87.3%

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

   7. metadata-eval 87.3%

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

   8. div-inv 87.3%

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

   9. *-commutative 87.3%

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

4. Applied egg-rr 87.3%

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

6. Simplified 87.4%

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

7. Taylor expanded in k around 0 36.4%

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

   1. associate-/l* 36.3%

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

9. Simplified 36.3%

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

    1. associate-/r/ 36.4%

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

11. Applied egg-rr 36.4%

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

12. Final simplification 36.4%

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

Reproduce

herbie shell --seed 2024036 
(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))))