Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.6%

Time: 22.0s

Alternatives: 13

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: 99.5% 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: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(2 \cdot \pi\right)\\ \frac{\frac{\sqrt{t\_0}}{{t\_0}^{\left(0.5 \cdot k\right)}}}{\sqrt{k}} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Power[t$95$0, N[(0.5 * k), $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. Add Preprocessing
3. 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. *-un-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-*r* 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. metadata-eval 99.5%

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

   7. pow-sub 99.7%

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

   8. pow1/2 99.7%

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

   9. associate-/l/ 99.7%

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

   10. associate-*r* 99.7%

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

   11. *-commutative 99.7%

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

   12. associate-*l* 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-/l/ 99.7%

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

   2. associate-*r* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*r* 99.7%

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

   5. *-commutative 99.7%

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

   6. *-commutative 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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. sqr-pow 99.3%

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

   4. pow-sqr 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-*l* 99.5%

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

   7. associate-*r/ 99.5%

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

   8. *-commutative 99.5%

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

   9. associate-/l* 99.5%

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

   10. metadata-eval 99.5%

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

   11. /-rgt-identity 99.5%

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

   12. div-sub 99.5%

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

   13. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.5%

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

   2. associate-*r* 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

   5. div-inv 99.5%

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

   6. metadata-eval 99.5%

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

   7. inv-pow 99.5%

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

   8. sqrt-pow2 99.5%

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

   9. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

   1. metadata-eval 99.5%

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

   2. pow-flip 99.5%

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

   3. pow1/2 99.5%

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

   4. div-inv 99.5%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;k \leq 5.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{t\_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{t\_0}^{\left(1 - k\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* n 2.0))))
   (if (<= k 5.5e-66)
     (/ (sqrt t_0) (sqrt k))
     (/ 1.0 (sqrt (/ k (pow t_0 (- 1.0 k))))))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (n * 2.0);
	double tmp;
	if (k <= 5.5e-66) {
		tmp = sqrt(t_0) / sqrt(k);
	} 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 * (n * 2.0);
	double tmp;
	if (k <= 5.5e-66) {
		tmp = Math.sqrt(t_0) / Math.sqrt(k);
	} 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 * (n * 2.0)
	tmp = 0
	if k <= 5.5e-66:
		tmp = math.sqrt(t_0) / math.sqrt(k)
	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(n * 2.0))
	tmp = 0.0
	if (k <= 5.5e-66)
		tmp = Float64(sqrt(t_0) / sqrt(k));
	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 * (n * 2.0);
	tmp = 0.0;
	if (k <= 5.5e-66)
		tmp = sqrt(t_0) / sqrt(k);
	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[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 5.5e-66], N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $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 < 5.50000000000000053e-66

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing
   3. 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 65.4%

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

      3. *-commutative 65.4%

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

      4. *-commutative 65.4%

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

      5. associate-*r* 65.4%

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

      6. div-sub 65.4%

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

      7. metadata-eval 65.4%

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

      8. div-inv 65.5%

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

      9. *-commutative 65.5%

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

   4. Applied egg-rr 65.5%

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

   6. Simplified 65.6%

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

      1. *-commutative 65.6%

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

      2. associate-*r* 65.6%

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

      3. sub-neg 65.6%

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

      4. unpow-prod-up 65.6%

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

      5. pow1 65.6%

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

      6. associate-*r* 65.6%

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

      7. *-commutative 65.6%

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

      8. *-commutative 65.6%

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

      9. associate-*r* 65.6%

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

      10. *-commutative 65.6%

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

      11. *-commutative 65.6%

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

   8. Applied egg-rr 65.6%

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

   9. Taylor expanded in k around 0 65.6%

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

       1. associate-/l* 65.6%

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

       2. associate-/r/ 65.6%

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

   11. Simplified 65.6%

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

       1. metadata-eval 65.6%

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

       2. associate-*l/ 65.6%

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

       3. times-frac 65.6%

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

       4. *-un-lft-identity 65.6%

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

       5. *-commutative 65.6%

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

       6. clear-num 65.5%

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

       7. metadata-eval 65.5%

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

       8. metadata-eval 65.5%

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

       9. div-inv 65.5%

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

       10. add-sqr-sqrt 65.3%

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

       11. frac-times 65.5%

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

       12. sqrt-unprod 66.7%

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

       13. add-sqr-sqrt 67.0%

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

       14. associate-/l/ 67.0%

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

       15. *-commutative 67.0%

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

       16. associate-*r* 67.0%

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

       17. sqrt-undiv 99.3%

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

   13. Applied egg-rr 99.5%

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

   if 5.50000000000000053e-66 < k

   1. Initial program 99.5%

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

      1. associate-*l/ 99.5%

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

      2. *-un-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-*r* 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. metadata-eval 99.5%

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

      7. pow-sub 99.8%

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

      8. pow1/2 99.8%

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

      9. associate-/l/ 99.8%

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

      10. associate-*r* 99.8%

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

      11. *-commutative 99.8%

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

      12. associate-*l* 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-/l/ 99.8%

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

      2. associate-*r* 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r* 99.8%

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

      5. *-commutative 99.8%

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

      6. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   8. Applied egg-rr 99.3%

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

      1. unpow-1 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*l* 99.3%

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

   10. Simplified 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5.5e-66)
   (/ (sqrt (* PI (* n 2.0))) (sqrt k))
   (sqrt (/ (pow (* n (* 2.0 PI)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5.5e-66) {
		tmp = sqrt((((double) M_PI) * (n * 2.0))) / sqrt(k);
	} else {
		tmp = sqrt((pow((n * (2.0 * ((double) M_PI))), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 5.5e-66) {
		tmp = Math.sqrt((Math.PI * (n * 2.0))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow((n * (2.0 * Math.PI)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 5.5e-66:
		tmp = math.sqrt((math.pi * (n * 2.0))) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow((n * (2.0 * math.pi)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5.5e-66)
		tmp = Float64(sqrt(Float64(pi * Float64(n * 2.0))) / sqrt(k));
	else
		tmp = sqrt(Float64((Float64(n * Float64(2.0 * pi)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 5.5e-66)
		tmp = sqrt((pi * (n * 2.0))) / sqrt(k);
	else
		tmp = sqrt((((n * (2.0 * pi)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5.5e-66], N[(N[Sqrt[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.50000000000000053e-66

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing
   3. 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 65.4%

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

      3. *-commutative 65.4%

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

      4. *-commutative 65.4%

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

      5. associate-*r* 65.4%

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

      6. div-sub 65.4%

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

      7. metadata-eval 65.4%

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

      8. div-inv 65.5%

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

      9. *-commutative 65.5%

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

   4. Applied egg-rr 65.5%

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

   6. Simplified 65.6%

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

      1. *-commutative 65.6%

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

      2. associate-*r* 65.6%

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

      3. sub-neg 65.6%

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

      4. unpow-prod-up 65.6%

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

      5. pow1 65.6%

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

      6. associate-*r* 65.6%

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

      7. *-commutative 65.6%

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

      8. *-commutative 65.6%

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

      9. associate-*r* 65.6%

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

      10. *-commutative 65.6%

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

      11. *-commutative 65.6%

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

   8. Applied egg-rr 65.6%

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

   9. Taylor expanded in k around 0 65.6%

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

       1. associate-/l* 65.6%

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

       2. associate-/r/ 65.6%

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

   11. Simplified 65.6%

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

       1. metadata-eval 65.6%

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

       2. associate-*l/ 65.6%

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

       3. times-frac 65.6%

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

       4. *-un-lft-identity 65.6%

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

       5. *-commutative 65.6%

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

       6. clear-num 65.5%

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

       7. metadata-eval 65.5%

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

       8. metadata-eval 65.5%

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

       9. div-inv 65.5%

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

       10. add-sqr-sqrt 65.3%

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

       11. frac-times 65.5%

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

       12. sqrt-unprod 66.7%

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

       13. add-sqr-sqrt 67.0%

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

       14. associate-/l/ 67.0%

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

       15. *-commutative 67.0%

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

       16. associate-*r* 67.0%

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

       17. sqrt-undiv 99.3%

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

   13. Applied egg-rr 99.5%

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

   if 5.50000000000000053e-66 < k

   1. Initial program 99.5%

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

      1. 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 99.0%

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

      3. *-commutative 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-*r* 99.0%

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

      6. div-sub 99.0%

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

      7. metadata-eval 99.0%

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

      8. div-inv 99.0%

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

      9. *-commutative 99.0%

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

   4. Applied egg-rr 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.5%

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

   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. sqr-pow 99.3%

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

   4. pow-sqr 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-*l* 99.5%

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

   7. associate-*r/ 99.5%

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

   8. *-commutative 99.5%

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

   9. associate-/l* 99.5%

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

   10. metadata-eval 99.5%

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

   11. /-rgt-identity 99.5%

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

   12. div-sub 99.5%

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

   13. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.5%

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

   2. associate-*r* 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

   5. div-inv 99.5%

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

   6. metadata-eval 99.5%

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

   7. inv-pow 99.5%

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

   8. sqrt-pow2 99.5%

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

   9. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

   \[\leadsto {\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(0.5 - 0.5 \cdot k\right)} \cdot {k}^{-0.5} \]
8. 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 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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. sqr-pow 99.3%

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

   4. pow-sqr 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-*l* 99.5%

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

   7. associate-*r/ 99.5%

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

   8. *-commutative 99.5%

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

   9. associate-/l* 99.5%

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

   10. metadata-eval 99.5%

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

   11. /-rgt-identity 99.5%

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

   12. div-sub 99.5%

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

   13. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 7: 49.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-/r/ 99.4%

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

   2. *-commutative 99.4%

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

   3. associate-*r* 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

   6. associate-*r* 99.4%

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

   7. *-commutative 99.4%

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

   8. associate-*l* 99.4%

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

   9. div-inv 99.4%

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

   10. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in k around 0 47.9%

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

   1. associate-/r/ 47.9%

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

   2. pow1/2 47.9%

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

   3. pow-flip 47.9%

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

   4. metadata-eval 47.9%

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

   5. *-commutative 47.9%

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

   6. sqrt-prod 48.0%

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

7. Applied egg-rr 48.0%

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

8. Final simplification 48.0%

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

Alternative 8: 49.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. 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 87.0%

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

   3. *-commutative 87.0%

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

   4. *-commutative 87.0%

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

   5. associate-*r* 87.0%

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

   6. div-sub 87.0%

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

   7. metadata-eval 87.0%

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

   8. div-inv 87.1%

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

   9. *-commutative 87.1%

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

4. Applied egg-rr 87.1%

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

6. Simplified 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-*r* 87.2%

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

   3. sub-neg 87.2%

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

   4. unpow-prod-up 87.4%

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

   5. pow1 87.4%

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

   6. associate-*r* 87.4%

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

   7. *-commutative 87.4%

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

   8. *-commutative 87.4%

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

   9. associate-*r* 87.4%

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

   10. *-commutative 87.4%

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

   11. *-commutative 87.4%

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

8. Applied egg-rr 87.4%

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

9. Taylor expanded in k around 0 35.7%

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

    1. associate-/l* 35.7%

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

    2. associate-/r/ 35.7%

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

11. Simplified 35.7%

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

    1. metadata-eval 35.7%

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

    2. associate-*l/ 35.7%

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

    3. times-frac 35.7%

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

    4. *-un-lft-identity 35.7%

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

    5. *-commutative 35.7%

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

    6. clear-num 35.7%

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

    7. metadata-eval 35.7%

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

    8. metadata-eval 35.7%

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

    9. div-inv 35.7%

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

    10. add-sqr-sqrt 35.6%

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

    11. frac-times 35.6%

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

    12. sqrt-unprod 36.2%

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

    13. add-sqr-sqrt 36.4%

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

    14. associate-/l/ 36.4%

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

    15. *-commutative 36.4%

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

    16. associate-*r* 36.4%

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

    17. sqrt-undiv 47.9%

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

13. Applied egg-rr 48.0%

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

14. Final simplification 48.0%

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

Alternative 9: 39.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. 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 87.0%

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

   3. *-commutative 87.0%

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

   4. *-commutative 87.0%

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

   5. associate-*r* 87.0%

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

   6. div-sub 87.0%

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

   7. metadata-eval 87.0%

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

   8. div-inv 87.1%

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

   9. *-commutative 87.1%

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

4. Applied egg-rr 87.1%

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

6. Simplified 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-*r* 87.2%

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

   3. sub-neg 87.2%

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

   4. unpow-prod-up 87.4%

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

   5. pow1 87.4%

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

   6. associate-*r* 87.4%

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

   7. *-commutative 87.4%

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

   8. *-commutative 87.4%

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

   9. associate-*r* 87.4%

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

   10. *-commutative 87.4%

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

   11. *-commutative 87.4%

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

8. Applied egg-rr 87.4%

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

9. Taylor expanded in k around 0 35.7%

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

    1. associate-/l* 35.7%

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

    2. associate-/r/ 35.7%

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

11. Simplified 35.7%

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

    1. metadata-eval 35.7%

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

    2. associate-*l/ 35.7%

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

    3. times-frac 35.7%

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

    4. *-un-lft-identity 35.7%

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

    5. *-commutative 35.7%

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

    6. clear-num 35.7%

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

    7. metadata-eval 35.7%

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

    8. metadata-eval 35.7%

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

    9. div-inv 35.7%

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

    10. add-sqr-sqrt 35.6%

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

    11. frac-times 35.6%

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

    12. sqrt-unprod 36.2%

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

    13. add-sqr-sqrt 36.4%

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

    14. pow1/2 36.4%

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

    15. pow-flip 36.5%

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

13. Applied egg-rr 36.5%

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

    1. associate-*l/ 36.5%

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

    2. associate-*r/ 36.4%

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

    3. associate-/r* 36.4%

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

    4. *-commutative 36.4%

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

15. Simplified 36.4%

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

16. Final simplification 36.4%

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

Alternative 10: 39.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. 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 87.0%

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

   3. *-commutative 87.0%

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

   4. *-commutative 87.0%

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

   5. associate-*r* 87.0%

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

   6. div-sub 87.0%

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

   7. metadata-eval 87.0%

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

   8. div-inv 87.1%

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

   9. *-commutative 87.1%

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

4. Applied egg-rr 87.1%

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

6. Simplified 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-*r* 87.2%

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

   3. sub-neg 87.2%

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

   4. unpow-prod-up 87.4%

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

   5. pow1 87.4%

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

   6. associate-*r* 87.4%

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

   7. *-commutative 87.4%

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

   8. *-commutative 87.4%

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

   9. associate-*r* 87.4%

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

   10. *-commutative 87.4%

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

   11. *-commutative 87.4%

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

8. Applied egg-rr 87.4%

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

9. Taylor expanded in k around 0 35.7%

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

    1. associate-/l* 35.7%

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

    2. associate-/r/ 35.7%

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

11. Simplified 35.7%

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

    1. metadata-eval 35.7%

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

    2. associate-*l/ 35.7%

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

    3. times-frac 35.7%

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

    4. *-un-lft-identity 35.7%

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

    5. *-commutative 35.7%

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

    6. clear-num 35.7%

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

    7. metadata-eval 35.7%

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

    8. metadata-eval 35.7%

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

    9. div-inv 35.7%

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

    10. add-sqr-sqrt 35.6%

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

    11. frac-times 35.6%

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

    12. sqrt-unprod 36.2%

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

    13. add-sqr-sqrt 36.4%

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

    14. pow1/2 36.4%

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

    15. pow-flip 36.5%

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

13. Applied egg-rr 36.5%

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

14. Final simplification 36.5%

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

Alternative 11: 38.6% 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 99.5%

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

   1. 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 87.0%

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

   3. *-commutative 87.0%

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

   4. *-commutative 87.0%

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

   5. associate-*r* 87.0%

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

   6. div-sub 87.0%

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

   7. metadata-eval 87.0%

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

   8. div-inv 87.1%

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

   9. *-commutative 87.1%

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

4. Applied egg-rr 87.1%

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

6. Simplified 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-*r* 87.2%

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

   3. sub-neg 87.2%

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

   4. unpow-prod-up 87.4%

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

   5. pow1 87.4%

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

   6. associate-*r* 87.4%

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

   7. *-commutative 87.4%

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

   8. *-commutative 87.4%

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

   9. associate-*r* 87.4%

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

   10. *-commutative 87.4%

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

   11. *-commutative 87.4%

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

8. Applied egg-rr 87.4%

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

9. Taylor expanded in k around 0 35.7%

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

    1. associate-/l* 35.7%

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

    2. associate-/r/ 35.7%

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

11. Simplified 35.7%

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

12. Final simplification 35.7%

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

Alternative 12: 38.6% accurate, 2.0× speedup

Math

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

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(Float64(n * 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), $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. Add Preprocessing
3. 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 87.0%

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

   3. *-commutative 87.0%

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

   4. *-commutative 87.0%

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

   5. associate-*r* 87.0%

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

   6. div-sub 87.0%

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

   7. metadata-eval 87.0%

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

   8. div-inv 87.1%

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

   9. *-commutative 87.1%

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

4. Applied egg-rr 87.1%

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

6. Simplified 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-*r* 87.2%

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

   3. sub-neg 87.2%

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

   4. unpow-prod-up 87.4%

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

   5. pow1 87.4%

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

   6. associate-*r* 87.4%

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

   7. *-commutative 87.4%

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

   8. *-commutative 87.4%

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

   9. associate-*r* 87.4%

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

   10. *-commutative 87.4%

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

   11. *-commutative 87.4%

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

8. Applied egg-rr 87.4%

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

9. Taylor expanded in k around 0 35.7%

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

10. Final simplification 35.7%

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

Alternative 13: 38.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{\frac{\pi}{\frac{k}{n}}}{0.5}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (/ (/ PI (/ k n)) 0.5)))

C

double code(double k, double n) {
	return sqrt(((((double) M_PI) / (k / n)) / 0.5));
}

Java

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

Python

def code(k, n):
	return math.sqrt(((math.pi / (k / n)) / 0.5))

Julia

function code(k, n)
	return sqrt(Float64(Float64(pi / Float64(k / n)) / 0.5))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt(((pi / (k / n)) / 0.5));
end

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. 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 87.0%

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

   3. *-commutative 87.0%

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

   4. *-commutative 87.0%

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

   5. associate-*r* 87.0%

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

   6. div-sub 87.0%

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

   7. metadata-eval 87.0%

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

   8. div-inv 87.1%

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

   9. *-commutative 87.1%

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

4. Applied egg-rr 87.1%

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

6. Simplified 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-*r* 87.2%

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

   3. sub-neg 87.2%

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

   4. unpow-prod-up 87.4%

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

   5. pow1 87.4%

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

   6. associate-*r* 87.4%

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

   7. *-commutative 87.4%

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

   8. *-commutative 87.4%

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

   9. associate-*r* 87.4%

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

   10. *-commutative 87.4%

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

   11. *-commutative 87.4%

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

8. Applied egg-rr 87.4%

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

9. Taylor expanded in k around 0 35.7%

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

    1. associate-/l* 35.7%

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

    2. associate-/r/ 35.7%

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

11. Simplified 35.7%

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

    1. metadata-eval 35.7%

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

    2. associate-*l/ 35.7%

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

    3. times-frac 35.7%

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

    4. *-un-lft-identity 35.7%

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

    5. *-commutative 35.7%

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

    6. *-un-lft-identity 35.7%

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

    7. times-frac 35.7%

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

13. Applied egg-rr 35.7%

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

    1. *-lft-identity 35.7%

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

    2. associate-*r/ 35.7%

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

    3. *-commutative 35.7%

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

    4. associate-*r/ 35.7%

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

    5. associate-/l* 35.7%

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

15. Simplified 35.7%

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

16. Final simplification 35.7%

    \[\leadsto \sqrt{\frac{\frac{\pi}{\frac{k}{n}}}{0.5}} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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))))