Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 14.9s

Alternatives: 11

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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot n\right)\\ \frac{{t_0}^{\left(k \cdot -0.5\right)} \cdot \sqrt{t_0}}{\sqrt{k}} \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* 2.0 (* PI n))))
   (/ (* (pow t_0 (* k -0.5)) (sqrt t_0)) (sqrt k))))

C

double code(double k, double n) {
	double t_0 = 2.0 * (((double) M_PI) * n);
	return (pow(t_0, (k * -0.5)) * sqrt(t_0)) / sqrt(k);
}

Java

public static double code(double k, double n) {
	double t_0 = 2.0 * (Math.PI * n);
	return (Math.pow(t_0, (k * -0.5)) * Math.sqrt(t_0)) / Math.sqrt(k);
}

Python

def code(k, n):
	t_0 = 2.0 * (math.pi * n)
	return (math.pow(t_0, (k * -0.5)) * math.sqrt(t_0)) / math.sqrt(k)

Julia

function code(k, n)
	t_0 = Float64(2.0 * Float64(pi * n))
	return Float64(Float64((t_0 ^ Float64(k * -0.5)) * sqrt(t_0)) / sqrt(k))
end

MATLAB

function tmp = code(k, n)
	t_0 = 2.0 * (pi * n);
	tmp = ((t_0 ^ (k * -0.5)) * sqrt(t_0)) / sqrt(k);
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$0, N[(k * -0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $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/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

   5. div-sub 99.5%

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

   6. sub-neg 99.5%

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

   7. distribute-frac-neg 99.5%

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

   8. metadata-eval 99.5%

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

   9. neg-mul-1 99.5%

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

   10. associate-/l* 99.5%

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

   11. associate-/r/ 99.5%

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

   12. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. +-commutative 99.5%

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

   2. unpow-prod-up 99.7%

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

   3. associate-*r* 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*l* 99.7%

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

   6. *-commutative 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-*r* 99.7%

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

   9. *-commutative 99.7%

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

   10. associate-*l* 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot n\right)\\ \mathbf{if}\;k \leq 2 \cdot 10^{-27}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{t_0}^{\left(1 - k\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* 2.0 n))))
   (if (<= k 2e-27)
     (* (pow k -0.5) (sqrt t_0))
     (/ 1.0 (sqrt (/ k (pow t_0 (- 1.0 k))))))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (2.0 * n);
	double tmp;
	if (k <= 2e-27) {
		tmp = pow(k, -0.5) * sqrt(t_0);
	} 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 = Math.PI * (2.0 * n);
	double tmp;
	if (k <= 2e-27) {
		tmp = Math.pow(k, -0.5) * Math.sqrt(t_0);
	} else {
		tmp = 1.0 / Math.sqrt((k / Math.pow(t_0, (1.0 - k))));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.pi * (2.0 * n)
	tmp = 0
	if k <= 2e-27:
		tmp = math.pow(k, -0.5) * math.sqrt(t_0)
	else:
		tmp = 1.0 / math.sqrt((k / math.pow(t_0, (1.0 - k))))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(2.0 * n))
	tmp = 0.0
	if (k <= 2e-27)
		tmp = Float64((k ^ -0.5) * sqrt(t_0));
	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 = pi * (2.0 * n);
	tmp = 0.0;
	if (k <= 2e-27)
		tmp = (k ^ -0.5) * sqrt(t_0);
	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[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2e-27], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[t$95$0], $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 < 2.0000000000000001e-27

   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. Step-by-step derivation

      1. add-sqr-sqrt 98.9%

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

      2. sqrt-unprod 74.5%

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

      3. associate-*l/ 74.6%

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

      4. *-un-lft-identity 74.6%

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

      5. associate-*l/ 74.6%

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

      6. *-un-lft-identity 74.6%

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

      7. frac-times 74.6%

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

   3. Applied egg-rr 74.7%

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

   5. Simplified 74.8%

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

   6. Taylor expanded in k around 0 74.8%

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

      1. associate-*r* 74.8%

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

      2. *-commutative 74.8%

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

      3. *-commutative 74.8%

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

   8. Simplified 74.8%

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

      1. sqrt-div 99.4%

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

      2. associate-*r* 99.4%

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

      3. sqrt-unprod 99.2%

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

      4. div-inv 99.1%

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

      5. sqrt-unprod 99.3%

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

      6. associate-*r* 99.3%

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

      7. pow1/2 99.3%

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

      8. pow-flip 99.4%

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

      9. metadata-eval 99.4%

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

   10. Applied egg-rr 99.4%

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

   if 2.0000000000000001e-27 < 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. 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. associate-*l/ 99.6%

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

      4. *-un-lft-identity 99.6%

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

      5. associate-*l/ 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. frac-times 99.6%

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

   3. Applied egg-rr 99.6%

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

   5. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. sqrt-div 99.6%

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

      3. metadata-eval 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

   9. Simplified 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-27}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {k}^{-0.5} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (pow k -0.5) (pow (* n (* 2.0 PI)) (/ (- 1.0 k) 2.0))))

C

double code(double k, double n) {
	return pow(k, -0.5) * pow((n * (2.0 * ((double) M_PI))), ((1.0 - k) / 2.0));
}

Java

public static double code(double k, double n) {
	return Math.pow(k, -0.5) * Math.pow((n * (2.0 * Math.PI)), ((1.0 - k) / 2.0));
}

Python

def code(k, n):
	return math.pow(k, -0.5) * math.pow((n * (2.0 * math.pi)), ((1.0 - k) / 2.0))

Julia

function code(k, n)
	return Float64((k ^ -0.5) * (Float64(n * Float64(2.0 * pi)) ^ Float64(Float64(1.0 - k) / 2.0)))
end

MATLAB

function tmp = code(k, n)
	tmp = (k ^ -0.5) * ((n * (2.0 * pi)) ^ ((1.0 - k) / 2.0));
end

Wolfram

code[k_, n_] := N[(N[Power[k, -0.5], $MachinePrecision] * N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $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. Step-by-step derivation

   1. expm1-log1p-u 96.3%

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

   2. expm1-udef 75.6%

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

   3. inv-pow 75.6%

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

   4. sqrt-pow2 75.6%

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

   5. metadata-eval 75.6%

      \[\leadsto \left(e^{\mathsf{log1p}\left({k}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

3. Applied egg-rr 75.6%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-0.5}\right)} - 1\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
4. Step-by-step derivation

   1. expm1-def 96.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. expm1-log1p 99.5%

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot n\right)\\ \mathbf{if}\;k \leq 2 \cdot 10^{-27}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* 2.0 n))))
   (if (<= k 2e-27)
     (* (pow k -0.5) (sqrt t_0))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (2.0 * n);
	double tmp;
	if (k <= 2e-27) {
		tmp = pow(k, -0.5) * sqrt(t_0);
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (2.0 * n);
	double tmp;
	if (k <= 2e-27) {
		tmp = Math.pow(k, -0.5) * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.pi * (2.0 * n)
	tmp = 0
	if k <= 2e-27:
		tmp = math.pow(k, -0.5) * math.sqrt(t_0)
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(2.0 * n))
	tmp = 0.0
	if (k <= 2e-27)
		tmp = Float64((k ^ -0.5) * sqrt(t_0));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = pi * (2.0 * n);
	tmp = 0.0;
	if (k <= 2e-27)
		tmp = (k ^ -0.5) * sqrt(t_0);
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2e-27], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.0000000000000001e-27

   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. Step-by-step derivation

      1. add-sqr-sqrt 98.9%

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

      2. sqrt-unprod 74.5%

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

      3. associate-*l/ 74.6%

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

      4. *-un-lft-identity 74.6%

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

      5. associate-*l/ 74.6%

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

      6. *-un-lft-identity 74.6%

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

      7. frac-times 74.6%

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

   3. Applied egg-rr 74.7%

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

   5. Simplified 74.8%

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

   6. Taylor expanded in k around 0 74.8%

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

      1. associate-*r* 74.8%

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

      2. *-commutative 74.8%

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

      3. *-commutative 74.8%

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

   8. Simplified 74.8%

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

      1. sqrt-div 99.4%

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

      2. associate-*r* 99.4%

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

      3. sqrt-unprod 99.2%

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

      4. div-inv 99.1%

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

      5. sqrt-unprod 99.3%

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

      6. associate-*r* 99.3%

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

      7. pow1/2 99.3%

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

      8. pow-flip 99.4%

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

      9. metadata-eval 99.4%

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

   10. Applied egg-rr 99.4%

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

   if 2.0000000000000001e-27 < 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. 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. associate-*l/ 99.6%

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

      4. *-un-lft-identity 99.6%

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

      5. associate-*l/ 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. frac-times 99.6%

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

   3. Applied egg-rr 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-27}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(k * -0.5), $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/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

   5. div-sub 99.5%

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

   6. sub-neg 99.5%

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

   7. distribute-frac-neg 99.5%

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

   8. metadata-eval 99.5%

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

   9. neg-mul-1 99.5%

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

   10. associate-/l* 99.5%

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

   11. associate-/r/ 99.5%

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

   12. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 6: 50.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{+253}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{2}{\frac{k}{\pi \cdot n}}\right)}^{2}\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6.8e+253)
   (* (pow k -0.5) (sqrt (* PI (* 2.0 n))))
   (pow (pow (/ 2.0 (/ k (* PI n))) 2.0) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6.8e+253) {
		tmp = pow(k, -0.5) * sqrt((((double) M_PI) * (2.0 * n)));
	} else {
		tmp = pow(pow((2.0 / (k / (((double) M_PI) * n))), 2.0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 6.8e+253) {
		tmp = Math.pow(k, -0.5) * Math.sqrt((Math.PI * (2.0 * n)));
	} else {
		tmp = Math.pow(Math.pow((2.0 / (k / (Math.PI * n))), 2.0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 6.8e+253:
		tmp = math.pow(k, -0.5) * math.sqrt((math.pi * (2.0 * n)))
	else:
		tmp = math.pow(math.pow((2.0 / (k / (math.pi * n))), 2.0), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 6.8e+253)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(pi * Float64(2.0 * n))));
	else
		tmp = (Float64(2.0 / Float64(k / Float64(pi * n))) ^ 2.0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 6.8e+253)
		tmp = (k ^ -0.5) * sqrt((pi * (2.0 * n)));
	else
		tmp = ((2.0 / (k / (pi * n))) ^ 2.0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6.8e+253], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 / N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.80000000000000035e253

   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. Step-by-step derivation

      1. add-sqr-sqrt 99.2%

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

      2. sqrt-unprod 86.7%

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

      3. associate-*l/ 86.7%

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

      4. *-un-lft-identity 86.7%

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

      5. associate-*l/ 86.7%

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

      6. *-un-lft-identity 86.7%

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

      7. frac-times 86.7%

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

   3. Applied egg-rr 86.8%

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

   5. Simplified 86.8%

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

   6. Taylor expanded in k around 0 44.6%

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

      1. associate-*r* 44.6%

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

      2. *-commutative 44.6%

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

      3. *-commutative 44.6%

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

   8. Simplified 44.6%

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

      1. sqrt-div 57.3%

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

      2. associate-*r* 57.3%

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

      3. sqrt-unprod 57.1%

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

      4. div-inv 57.1%

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

      5. sqrt-unprod 57.2%

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

      6. associate-*r* 57.2%

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

      7. pow1/2 57.2%

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

      8. pow-flip 57.3%

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

      9. metadata-eval 57.3%

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

   10. Applied egg-rr 57.3%

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

   if 6.80000000000000035e253 < 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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. associate-*l/ 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. frac-times 100.0%

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

   3. Applied egg-rr 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in k around 0 2.9%

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

      1. associate-*r* 2.9%

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

      2. *-commutative 2.9%

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

      3. *-commutative 2.9%

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

   8. Simplified 2.9%

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

      1. associate-*r* 2.9%

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

      2. *-commutative 2.9%

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

      3. *-un-lft-identity 2.9%

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

      4. times-frac 2.9%

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

      5. metadata-eval 2.9%

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

   10. Applied egg-rr 2.9%

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

       1. pow1/2 2.9%

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

       2. metadata-eval 2.9%

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

       3. pow-prod-up 2.9%

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

       4. pow-prod-down 20.0%

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

       5. pow2 20.0%

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

       6. clear-num 20.0%

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

       7. un-div-inv 20.0%

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

   12. Applied egg-rr 20.0%

       \[\leadsto \color{blue}{{\left({\left(\frac{2}{\frac{k}{\pi \cdot n}}\right)}^{2}\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{+253}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{2}{\frac{k}{\pi \cdot n}}\right)}^{2}\right)}^{0.25}\\ \end{array} \]

Alternative 7: 48.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(Pi * N[(2.0 * n), $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. Step-by-step derivation

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 88.1%

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

   3. associate-*l/ 88.2%

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

   4. *-un-lft-identity 88.2%

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

   5. associate-*l/ 88.2%

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

   6. *-un-lft-identity 88.2%

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

   7. frac-times 88.2%

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

3. Applied egg-rr 88.2%

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

5. Simplified 88.3%

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

6. Taylor expanded in k around 0 40.0%

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

   1. associate-*r* 40.0%

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

   2. *-commutative 40.0%

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

   3. *-commutative 40.0%

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

8. Simplified 40.0%

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

   1. sqrt-div 51.3%

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

   2. associate-*r* 51.3%

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

   3. sqrt-unprod 51.2%

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

   4. div-inv 51.2%

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

   5. sqrt-unprod 51.3%

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

   6. associate-*r* 51.3%

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

   7. pow1/2 51.3%

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

   8. pow-flip 51.3%

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

   9. metadata-eval 51.3%

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

10. Applied egg-rr 51.3%

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

11. Final simplification 51.3%

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

Alternative 8: 48.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(Pi * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / 2.0), $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. Step-by-step derivation

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 88.1%

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

   3. associate-*l/ 88.2%

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

   4. *-un-lft-identity 88.2%

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

   5. associate-*l/ 88.2%

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

   6. *-un-lft-identity 88.2%

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

   7. frac-times 88.2%

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

3. Applied egg-rr 88.2%

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

5. Simplified 88.3%

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

6. Taylor expanded in k around 0 40.0%

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

   1. associate-*r* 40.0%

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

   2. *-commutative 40.0%

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

   3. *-commutative 40.0%

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

8. Simplified 40.0%

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

   1. sqrt-div 51.3%

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

   2. associate-*r* 51.3%

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

   3. *-commutative 51.3%

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

   4. sqrt-unprod 51.2%

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

   5. associate-/l* 51.2%

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

   6. *-commutative 51.2%

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

   7. sqrt-undiv 51.3%

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

10. Applied egg-rr 51.3%

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

11. Final simplification 51.3%

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

Alternative 9: 38.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 88.1%

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

   3. associate-*l/ 88.2%

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

   4. *-un-lft-identity 88.2%

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

   5. associate-*l/ 88.2%

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

   6. *-un-lft-identity 88.2%

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

   7. frac-times 88.2%

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

3. Applied egg-rr 88.2%

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

5. Simplified 88.3%

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

6. Taylor expanded in k around 0 40.0%

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

   1. associate-*r* 40.0%

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

   2. *-commutative 40.0%

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

   3. *-commutative 40.0%

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

8. Simplified 40.0%

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

   1. clear-num 40.0%

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

   2. sqrt-div 40.9%

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

   3. metadata-eval 40.9%

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

   4. *-un-lft-identity 40.9%

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

   5. associate-*r* 40.9%

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

   6. *-commutative 40.9%

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

   7. times-frac 40.9%

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

   8. metadata-eval 40.9%

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

10. Applied egg-rr 40.9%

    \[\leadsto \color{blue}{\frac{1}{\sqrt{0.5 \cdot \frac{k}{\pi \cdot n}}}} \]
11. Step-by-step derivation

    1. expm1-log1p-u 39.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{0.5 \cdot \frac{k}{\pi \cdot n}}}\right)\right)} \]

    2. expm1-udef 41.6%

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

    3. inv-pow 41.6%

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

    4. sqrt-pow2 41.6%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(0.5 \cdot \frac{k}{\pi \cdot n}\right)}^{\left(\frac{-1}{2}\right)}}\right)} - 1 \]

    5. metadata-eval 41.6%

       \[\leadsto e^{\mathsf{log1p}\left({\left(0.5 \cdot \frac{k}{\pi \cdot n}\right)}^{\color{blue}{-0.5}}\right)} - 1 \]

12. Applied egg-rr 41.6%

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

    1. expm1-def 39.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(0.5 \cdot \frac{k}{\pi \cdot n}\right)}^{-0.5}\right)\right)} \]

    2. expm1-log1p 41.0%

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

    3. associate-*r/ 41.0%

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

    4. *-commutative 41.0%

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

    5. associate-/l* 41.0%

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

    6. metadata-eval 41.0%

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

    7. associate-/l* 41.0%

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

    8. *-commutative 41.0%

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

    9. /-rgt-identity 41.0%

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

    10. *-commutative 41.0%

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

    11. associate-*l* 41.0%

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

14. Simplified 41.0%

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

15. Final simplification 41.0%

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

Alternative 10: 37.6% accurate, 1.5× 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. Step-by-step derivation

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 88.1%

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

   3. associate-*l/ 88.2%

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

   4. *-un-lft-identity 88.2%

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

   5. associate-*l/ 88.2%

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

   6. *-un-lft-identity 88.2%

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

   7. frac-times 88.2%

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

3. Applied egg-rr 88.2%

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

5. Simplified 88.3%

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

6. Taylor expanded in k around 0 40.0%

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

   1. associate-*r* 40.0%

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

   2. *-commutative 40.0%

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

   3. *-commutative 40.0%

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

8. Simplified 40.0%

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

   1. associate-*r* 40.0%

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

   2. *-commutative 40.0%

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

   3. *-un-lft-identity 40.0%

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

   4. times-frac 40.0%

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

   5. metadata-eval 40.0%

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

10. Applied egg-rr 40.0%

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

    1. associate-/l* 40.0%

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

    2. associate-/r/ 40.1%

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

12. Applied egg-rr 40.1%

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

13. Final simplification 40.1%

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

Alternative 11: 37.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(Pi * N[(N[(2.0 * n), $MachinePrecision] / 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. Step-by-step derivation

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 88.1%

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

   3. associate-*l/ 88.2%

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

   4. *-un-lft-identity 88.2%

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

   5. associate-*l/ 88.2%

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

   6. *-un-lft-identity 88.2%

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

   7. frac-times 88.2%

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

3. Applied egg-rr 88.2%

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

5. Simplified 88.3%

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

6. Taylor expanded in k around 0 40.0%

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

   1. associate-*r* 40.0%

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

   2. *-commutative 40.0%

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

   3. *-commutative 40.0%

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

8. Simplified 40.0%

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

9. Taylor expanded in n around 0 40.0%

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

    1. associate-*r/ 40.0%

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

    2. associate-*r* 40.0%

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

    3. *-commutative 40.0%

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

    4. associate-*l/ 40.1%

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

    5. *-commutative 40.1%

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

    6. *-commutative 40.1%

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

11. Simplified 40.1%

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

12. Final simplification 40.1%

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

Reproduce

herbie shell --seed 2023318 
(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))))