Migdal et al, Equation (51)

Percentage Accurate: 98.5% → 99.6%

Time: 13.2s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. k = 4.94066e-324
  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: 98.5% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[k, -0.5], $MachinePrecision] * N[Power[N[Abs[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. associate-*r* 99.3%

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

   2. add-sqr-sqrt 79.4%

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

   3. sqrt-unprod 82.3%

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

   4. *-commutative 82.3%

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

   5. *-commutative 82.3%

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

   6. swap-sqr 82.3%

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

   7. pow2 82.3%

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

   8. metadata-eval 82.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{{\left(\pi \cdot n\right)}^{2} \cdot \color{blue}{4}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

4. Applied egg-rr 82.3%

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

   1. *-commutative 82.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{\color{blue}{4 \cdot {\left(\pi \cdot n\right)}^{2}}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. metadata-eval 82.3%

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

   3. unpow2 82.3%

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

   4. swap-sqr 82.3%

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

   5. rem-sqrt-square 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.7%

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

6. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. inv-pow 99.7%

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

   3. sqrt-pow2 99.7%

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

   4. metadata-eval 99.7%

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

8. Applied egg-rr 99.7%

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

   1. *-lft-identity 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;k \leq 4.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{t\_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t\_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* n 2.0))))
   (if (<= k 4.4e-17)
     (/ (sqrt t_0) (sqrt k))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (n * 2.0);
	double tmp;
	if (k <= 4.4e-17) {
		tmp = sqrt(t_0) / sqrt(k);
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (n * 2.0);
	double tmp;
	if (k <= 4.4e-17) {
		tmp = Math.sqrt(t_0) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.pi * (n * 2.0)
	tmp = 0
	if k <= 4.4e-17:
		tmp = math.sqrt(t_0) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(n * 2.0))
	tmp = 0.0
	if (k <= 4.4e-17)
		tmp = Float64(sqrt(t_0) / sqrt(k));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = pi * (n * 2.0);
	tmp = 0.0;
	if (k <= 4.4e-17)
		tmp = sqrt(t_0) / sqrt(k);
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 4.4e-17], N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.4e-17

   1. Initial program 99.2%

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

      1. associate-*r* 99.2%

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

      2. add-sqr-sqrt 99.2%

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

      3. sqrt-unprod 55.2%

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

      4. *-commutative 55.2%

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

      5. *-commutative 55.2%

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

      6. swap-sqr 55.2%

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

      7. pow2 55.2%

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

      8. metadata-eval 55.2%

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{{\left(\pi \cdot n\right)}^{2} \cdot \color{blue}{4}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

   4. Applied egg-rr 55.2%

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

      1. *-commutative 55.2%

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{\color{blue}{4 \cdot {\left(\pi \cdot n\right)}^{2}}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

      2. metadata-eval 55.2%

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

      3. unpow2 55.2%

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

      4. swap-sqr 55.2%

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

      5. rem-sqrt-square 99.2%

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

      6. *-commutative 99.2%

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

      7. associate-*l* 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in k around 0 79.2%

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

      1. associate-*r* 79.2%

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

      2. *-commutative 79.2%

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

      3. *-commutative 79.2%

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

      4. rem-square-sqrt 79.0%

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

      5. fabs-sqr 79.0%

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

      6. rem-square-sqrt 79.2%

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

      7. *-commutative 79.2%

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

      8. associate-/l* 79.2%

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

      9. *-commutative 79.2%

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

      10. associate-*r* 79.2%

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

   9. Simplified 79.2%

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

       1. sqrt-prod 78.9%

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

       2. associate-*r/ 79.0%

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

       3. *-commutative 79.0%

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

       4. add-sqr-sqrt 78.9%

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

       5. fabs-sqr 78.9%

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

       6. add-sqr-sqrt 79.0%

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

       7. *-commutative 79.0%

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

       8. sqrt-prod 79.2%

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

       9. associate-/l* 79.2%

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

       10. sqrt-div 99.4%

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

   11. Applied egg-rr 99.4%

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

   if 4.4e-17 < k

   1. Initial program 99.3%

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

      1. add-sqr-sqrt 99.3%

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

      2. sqrt-unprod 99.3%

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

      3. *-commutative 99.3%

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

      4. associate-*r* 99.3%

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

      5. div-sub 99.3%

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

      6. metadata-eval 99.3%

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

      7. div-inv 99.3%

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

      8. *-commutative 99.3%

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

   4. Applied egg-rr 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.65 \cdot 10^{+73}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n \cdot 2, \frac{\pi}{k}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.65e+73)
   (/ (sqrt (* PI (* n 2.0))) (sqrt k))
   (sqrt (+ -1.0 (fma (* n 2.0) (/ PI k) 1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.65e+73) {
		tmp = sqrt((((double) M_PI) * (n * 2.0))) / sqrt(k);
	} else {
		tmp = sqrt((-1.0 + fma((n * 2.0), (((double) M_PI) / k), 1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.65e+73)
		tmp = Float64(sqrt(Float64(pi * Float64(n * 2.0))) / sqrt(k));
	else
		tmp = sqrt(Float64(-1.0 + fma(Float64(n * 2.0), Float64(pi / k), 1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.65e+73], N[(N[Sqrt[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-1.0 + N[(N[(n * 2.0), $MachinePrecision] * N[(Pi / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.65000000000000015e73

   1. Initial program 98.7%

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

      1. associate-*r* 98.7%

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

      2. add-sqr-sqrt 91.7%

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

      3. sqrt-unprod 68.1%

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

      4. *-commutative 68.1%

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

      5. *-commutative 68.1%

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

      6. swap-sqr 68.1%

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

      7. pow2 68.1%

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

      8. metadata-eval 68.1%

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{{\left(\pi \cdot n\right)}^{2} \cdot \color{blue}{4}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

   4. Applied egg-rr 68.1%

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

      1. *-commutative 68.1%

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{\color{blue}{4 \cdot {\left(\pi \cdot n\right)}^{2}}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

      2. metadata-eval 68.1%

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

      3. unpow2 68.1%

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

      4. swap-sqr 68.1%

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

      5. rem-sqrt-square 99.4%

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

      6. *-commutative 99.4%

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

      7. associate-*l* 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in k around 0 57.5%

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

      1. associate-*r* 57.5%

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

      2. *-commutative 57.5%

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

      3. *-commutative 57.5%

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

      4. rem-square-sqrt 57.2%

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

      5. fabs-sqr 57.2%

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

      6. rem-square-sqrt 57.4%

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

      7. *-commutative 57.4%

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

      8. associate-/l* 57.4%

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

      9. *-commutative 57.4%

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

      10. associate-*r* 57.4%

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

   9. Simplified 57.4%

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

       1. sqrt-prod 57.2%

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

       2. associate-*r/ 57.2%

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

       3. *-commutative 57.2%

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

       4. add-sqr-sqrt 57.2%

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

       5. fabs-sqr 57.2%

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

       6. add-sqr-sqrt 57.4%

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

       7. *-commutative 57.4%

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

       8. sqrt-prod 57.5%

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

       9. associate-/l* 57.5%

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

       10. sqrt-div 71.9%

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

   11. Applied egg-rr 71.7%

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

   if 1.65000000000000015e73 < k

   1. Initial program 100.0%

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

      1. associate-*r* 100.0%

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

      2. add-sqr-sqrt 64.0%

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

      3. sqrt-unprod 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. swap-sqr 100.0%

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

      7. pow2 100.0%

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

      8. metadata-eval 100.0%

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{{\left(\pi \cdot n\right)}^{2} \cdot \color{blue}{4}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{\color{blue}{4 \cdot {\left(\pi \cdot n\right)}^{2}}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

      2. metadata-eval 100.0%

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

      3. unpow2 100.0%

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

      4. swap-sqr 100.0%

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

      5. rem-sqrt-square 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. inv-pow 100.0%

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

      3. sqrt-pow2 100.0%

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

      4. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in k around 0 2.9%

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

       1. fabs-mul 2.9%

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

       2. metadata-eval 2.9%

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

       3. associate-*r/ 2.9%

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

       4. *-commutative 2.9%

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

       5. rem-square-sqrt 1.7%

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

       6. *-commutative 1.7%

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

       7. *-commutative 1.7%

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

       8. fabs-sqr 1.7%

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

       9. *-commutative 1.7%

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

       10. *-commutative 1.7%

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

       11. rem-square-sqrt 1.7%

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

       12. associate-/l* 1.7%

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

   13. Simplified 1.7%

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

       1. expm1-log1p-u 1.7%

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

       2. expm1-undefine 39.4%

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

   15. Applied egg-rr 39.4%

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

       1. sub-neg 39.4%

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

       2. metadata-eval 39.4%

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

       3. +-commutative 39.4%

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

       4. log1p-undefine 39.4%

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

       5. rem-exp-log 39.4%

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

       6. +-commutative 39.4%

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

       7. associate-*r* 39.4%

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

       8. fma-define 39.4%

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

   17. Simplified 39.4%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.65 \cdot 10^{+73}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n \cdot 2, \frac{\pi}{k}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l/ 99.3%

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

   2. *-lft-identity 99.3%

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

   3. associate-*l* 99.3%

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

   4. div-sub 99.3%

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

   5. metadata-eval 99.3%

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 5: 40.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*r* 99.3%

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

   2. add-sqr-sqrt 79.4%

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

   3. sqrt-unprod 82.3%

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

   4. *-commutative 82.3%

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

   5. *-commutative 82.3%

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

   6. swap-sqr 82.3%

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

   7. pow2 82.3%

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

   8. metadata-eval 82.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{{\left(\pi \cdot n\right)}^{2} \cdot \color{blue}{4}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

4. Applied egg-rr 82.3%

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

   1. *-commutative 82.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{\color{blue}{4 \cdot {\left(\pi \cdot n\right)}^{2}}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. metadata-eval 82.3%

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

   3. unpow2 82.3%

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

   4. swap-sqr 82.3%

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

   5. rem-sqrt-square 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.7%

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

6. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. inv-pow 99.7%

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

   3. sqrt-pow2 99.7%

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

   4. metadata-eval 99.7%

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

8. Applied egg-rr 99.7%

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

   1. *-lft-identity 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in k around 0 33.2%

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

    1. fabs-mul 33.2%

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

    2. metadata-eval 33.2%

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

    3. associate-*r/ 33.2%

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

    4. *-commutative 33.2%

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

    5. rem-square-sqrt 32.5%

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

    6. *-commutative 32.5%

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

    7. *-commutative 32.5%

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

    8. fabs-sqr 32.5%

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

    9. *-commutative 32.5%

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

    10. *-commutative 32.5%

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

    11. rem-square-sqrt 32.6%

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

    12. associate-/l* 32.6%

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

13. Simplified 32.6%

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

    1. pow1/2 32.6%

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

    2. associate-*r/ 32.6%

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

    3. associate-*r/ 32.6%

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

    4. *-commutative 32.6%

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

    5. associate-*r* 32.6%

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

    6. associate-*l/ 32.6%

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

    7. *-commutative 32.6%

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

    8. associate-*l* 32.6%

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

    9. metadata-eval 32.6%

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

    10. unpow-prod-down 40.5%

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

    11. metadata-eval 40.5%

        \[\leadsto {n}^{\color{blue}{0.5}} \cdot {\left(2 \cdot \frac{\pi}{k}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

    12. pow1/2 40.5%

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

    13. metadata-eval 40.5%

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

15. Applied egg-rr 40.5%

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

    1. unpow1/2 40.5%

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

17. Simplified 40.5%

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

18. Final simplification 40.5%

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

Alternative 6: 40.3% 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 99.3%

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

   1. associate-*r* 99.3%

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

   2. add-sqr-sqrt 79.4%

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

   3. sqrt-unprod 82.3%

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

   4. *-commutative 82.3%

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

   5. *-commutative 82.3%

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

   6. swap-sqr 82.3%

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

   7. pow2 82.3%

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

   8. metadata-eval 82.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{{\left(\pi \cdot n\right)}^{2} \cdot \color{blue}{4}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

4. Applied egg-rr 82.3%

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

   1. *-commutative 82.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{\color{blue}{4 \cdot {\left(\pi \cdot n\right)}^{2}}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. metadata-eval 82.3%

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

   3. unpow2 82.3%

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

   4. swap-sqr 82.3%

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

   5. rem-sqrt-square 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in k around 0 33.2%

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

   1. associate-*r* 33.2%

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

   2. *-commutative 33.2%

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

   3. *-commutative 33.2%

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

   4. rem-square-sqrt 32.5%

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

   5. fabs-sqr 32.5%

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

   6. rem-square-sqrt 32.6%

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

   7. *-commutative 32.6%

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

   8. associate-/l* 32.6%

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

   9. *-commutative 32.6%

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

   10. associate-*r* 32.6%

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

9. Simplified 32.6%

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

    1. sqrt-prod 32.5%

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

    2. associate-*r/ 32.5%

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

    3. *-commutative 32.5%

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

    4. add-sqr-sqrt 32.5%

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

    5. fabs-sqr 32.5%

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

    6. add-sqr-sqrt 33.1%

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

    7. *-commutative 33.1%

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

    8. sqrt-prod 33.2%

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

    9. associate-/l* 33.2%

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

    10. sqrt-div 41.2%

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

11. Applied egg-rr 40.6%

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

12. Final simplification 40.6%

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

Alternative 7: 31.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*r* 99.3%

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

   2. add-sqr-sqrt 79.4%

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

   3. sqrt-unprod 82.3%

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

   4. *-commutative 82.3%

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

   5. *-commutative 82.3%

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

   6. swap-sqr 82.3%

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

   7. pow2 82.3%

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

   8. metadata-eval 82.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{{\left(\pi \cdot n\right)}^{2} \cdot \color{blue}{4}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

4. Applied egg-rr 82.3%

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

   1. *-commutative 82.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{\color{blue}{4 \cdot {\left(\pi \cdot n\right)}^{2}}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. metadata-eval 82.3%

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

   3. unpow2 82.3%

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

   4. swap-sqr 82.3%

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

   5. rem-sqrt-square 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in k around 0 33.2%

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

   1. associate-*r* 33.2%

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

   2. *-commutative 33.2%

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

   3. *-commutative 33.2%

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

   4. rem-square-sqrt 32.5%

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

   5. fabs-sqr 32.5%

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

   6. rem-square-sqrt 32.6%

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

   7. *-commutative 32.6%

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

   8. associate-/l* 32.6%

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

   9. *-commutative 32.6%

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

   10. associate-*r* 32.6%

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

9. Simplified 32.6%

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

    1. sqrt-prod 32.5%

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

    2. associate-*r/ 32.5%

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

    3. *-commutative 32.5%

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

    4. add-sqr-sqrt 32.5%

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

    5. fabs-sqr 32.5%

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

    6. add-sqr-sqrt 33.1%

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

    7. *-commutative 33.1%

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

    8. sqrt-prod 33.2%

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

    9. associate-/l* 33.2%

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

    10. clear-num 33.2%

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

    11. sqrt-div 34.1%

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

    12. metadata-eval 34.1%

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

    13. *-commutative 34.1%

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

    14. add-sqr-sqrt 33.4%

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

    15. fabs-sqr 33.4%

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

    16. add-sqr-sqrt 33.5%

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

    17. *-commutative 33.5%

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

11. Applied egg-rr 33.5%

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

12. Final simplification 33.5%

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

Alternative 8: 31.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*r* 99.3%

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

   2. add-sqr-sqrt 79.4%

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

   3. sqrt-unprod 82.3%

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

   4. *-commutative 82.3%

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

   5. *-commutative 82.3%

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

   6. swap-sqr 82.3%

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

   7. pow2 82.3%

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

   8. metadata-eval 82.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{{\left(\pi \cdot n\right)}^{2} \cdot \color{blue}{4}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

4. Applied egg-rr 82.3%

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

   1. *-commutative 82.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{\color{blue}{4 \cdot {\left(\pi \cdot n\right)}^{2}}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. metadata-eval 82.3%

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

   3. unpow2 82.3%

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

   4. swap-sqr 82.3%

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

   5. rem-sqrt-square 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in k around 0 33.2%

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

   1. associate-*r* 33.2%

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

   2. *-commutative 33.2%

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

   3. *-commutative 33.2%

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

   4. rem-square-sqrt 32.5%

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

   5. fabs-sqr 32.5%

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

   6. rem-square-sqrt 32.6%

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

   7. *-commutative 32.6%

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

   8. associate-/l* 32.6%

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

   9. *-commutative 32.6%

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

   10. associate-*r* 32.6%

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

9. Simplified 32.6%

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

10. Final simplification 32.6%

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

Alternative 9: 31.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*r* 99.3%

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

   2. add-sqr-sqrt 79.4%

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

   3. sqrt-unprod 82.3%

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

   4. *-commutative 82.3%

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

   5. *-commutative 82.3%

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

   6. swap-sqr 82.3%

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

   7. pow2 82.3%

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

   8. metadata-eval 82.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{{\left(\pi \cdot n\right)}^{2} \cdot \color{blue}{4}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

4. Applied egg-rr 82.3%

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

   1. *-commutative 82.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{\color{blue}{4 \cdot {\left(\pi \cdot n\right)}^{2}}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. metadata-eval 82.3%

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

   3. unpow2 82.3%

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

   4. swap-sqr 82.3%

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

   5. rem-sqrt-square 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in k around 0 33.2%

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

   1. fabs-mul 33.2%

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

   2. metadata-eval 33.2%

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

9. Simplified 33.2%

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

    1. associate-/l* 33.2%

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

    2. *-commutative 33.2%

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

    3. add-sqr-sqrt 32.5%

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

    4. fabs-sqr 32.5%

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

    5. add-sqr-sqrt 32.6%

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

    6. *-commutative 32.6%

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

    7. associate-*r/ 32.6%

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

    8. associate-*r* 32.6%

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

    9. *-commutative 32.6%

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

11. Applied egg-rr 32.6%

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

12. Final simplification 32.6%

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

Alternative 10: 31.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*r* 99.3%

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

   2. add-sqr-sqrt 79.4%

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

   3. sqrt-unprod 82.3%

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

   4. *-commutative 82.3%

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

   5. *-commutative 82.3%

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

   6. swap-sqr 82.3%

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

   7. pow2 82.3%

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

   8. metadata-eval 82.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{{\left(\pi \cdot n\right)}^{2} \cdot \color{blue}{4}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

4. Applied egg-rr 82.3%

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

   1. *-commutative 82.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{\color{blue}{4 \cdot {\left(\pi \cdot n\right)}^{2}}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. metadata-eval 82.3%

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

   3. unpow2 82.3%

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

   4. swap-sqr 82.3%

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

   5. rem-sqrt-square 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.7%

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

6. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. inv-pow 99.7%

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

   3. sqrt-pow2 99.7%

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

   4. metadata-eval 99.7%

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

8. Applied egg-rr 99.7%

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

   1. *-lft-identity 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in k around 0 33.2%

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

    1. fabs-mul 33.2%

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

    2. metadata-eval 33.2%

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

    3. associate-*r/ 33.2%

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

    4. *-commutative 33.2%

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

    5. rem-square-sqrt 32.5%

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

    6. *-commutative 32.5%

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

    7. *-commutative 32.5%

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

    8. fabs-sqr 32.5%

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

    9. *-commutative 32.5%

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

    10. *-commutative 32.5%

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

    11. rem-square-sqrt 32.6%

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

    12. associate-/l* 32.6%

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

13. Simplified 32.6%

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

    1. associate-*r/ 32.6%

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

    2. associate-*r/ 32.6%

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

    3. *-commutative 32.6%

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

    4. associate-*r* 32.6%

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

    5. associate-*l/ 32.6%

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

    6. clear-num 32.6%

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

    7. associate-*l/ 32.6%

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

    8. *-un-lft-identity 32.6%

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

15. Applied egg-rr 32.6%

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

16. Final simplification 32.6%

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

Reproduce

herbie shell --seed 2024059 
(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))))