Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 18.8s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.7%

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

   2. *-lft-identity 99.7%

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

   3. sqr-pow 99.5%

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

   4. pow-sqr 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l* 99.7%

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

   7. associate-*r/ 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-/l* 99.7%

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

   10. metadata-eval 99.7%

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

   11. /-rgt-identity 99.7%

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

   12. div-sub 99.7%

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

   13. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;k \leq 3.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{t_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* n (* PI 2.0))))
   (if (<= k 3.5e-42)
     (/ (sqrt t_0) (sqrt k))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = n * (((double) M_PI) * 2.0);
	double tmp;
	if (k <= 3.5e-42) {
		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 = n * (Math.PI * 2.0);
	double tmp;
	if (k <= 3.5e-42) {
		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 = n * (math.pi * 2.0)
	tmp = 0
	if k <= 3.5e-42:
		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(n * Float64(pi * 2.0))
	tmp = 0.0
	if (k <= 3.5e-42)
		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 = n * (pi * 2.0);
	tmp = 0.0;
	if (k <= 3.5e-42)
		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[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 3.5e-42], 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 < 3.5000000000000002e-42

   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

   4. Applied egg-rr 46.7%

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

   5. Taylor expanded in k around 0 70.8%

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

      1. associate-*r* 70.8%

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

      2. *-commutative 70.8%

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

      3. associate-*l* 70.8%

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

   7. Simplified 70.8%

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

      1. sqrt-div 99.5%

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

      2. clear-num 99.3%

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

      3. *-commutative 99.3%

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

      4. associate-*r* 99.3%

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

   9. Applied egg-rr 99.3%

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

       1. associate-/r/ 99.3%

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

       2. associate-*l/ 99.5%

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

       3. *-commutative 99.5%

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

       4. *-rgt-identity 99.5%

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

       5. associate-*r* 99.5%

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

       6. *-commutative 99.5%

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

   11. Simplified 99.5%

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

   if 3.5000000000000002e-42 < k

   1. Initial program 99.8%

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

   4. Applied egg-rr 96.1%

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

   5. Taylor expanded in k around inf 95.8%

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

      1. log-pow 98.7%

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

      2. *-commutative 98.7%

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

      3. *-commutative 98.7%

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

      4. cancel-sign-sub-inv 98.7%

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

      5. metadata-eval 98.7%

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

      6. +-commutative 98.7%

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

      7. *-commutative 98.7%

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

      8. fma-udef 98.7%

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

      9. associate-*r* 98.7%

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

      10. exp-to-pow 99.2%

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

      11. *-commutative 99.2%

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

      12. *-commutative 99.2%

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

      13. associate-*l* 99.2%

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

      14. *-commutative 99.2%

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

      15. fma-udef 99.2%

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

      16. *-commutative 99.2%

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

      17. +-commutative 99.2%

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

      18. distribute-lft-in 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{+199}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{n}{0.5 \cdot \frac{k}{\pi}}\right)}^{2}\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5.2e+199)
   (/ (sqrt (* n (* PI 2.0))) (sqrt k))
   (pow (pow (/ n (* 0.5 (/ k PI))) 2.0) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5.2e+199) {
		tmp = sqrt((n * (((double) M_PI) * 2.0))) / sqrt(k);
	} else {
		tmp = pow(pow((n / (0.5 * (k / ((double) M_PI)))), 2.0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 5.2e+199) {
		tmp = Math.sqrt((n * (Math.PI * 2.0))) / Math.sqrt(k);
	} else {
		tmp = Math.pow(Math.pow((n / (0.5 * (k / Math.PI))), 2.0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 5.2e+199:
		tmp = math.sqrt((n * (math.pi * 2.0))) / math.sqrt(k)
	else:
		tmp = math.pow(math.pow((n / (0.5 * (k / math.pi))), 2.0), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5.2e+199)
		tmp = Float64(sqrt(Float64(n * Float64(pi * 2.0))) / sqrt(k));
	else
		tmp = (Float64(n / Float64(0.5 * Float64(k / pi))) ^ 2.0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 5.2e+199)
		tmp = sqrt((n * (pi * 2.0))) / sqrt(k);
	else
		tmp = ((n / (0.5 * (k / pi))) ^ 2.0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5.2e+199], N[(N[Sqrt[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(n / N[(0.5 * N[(k / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.2000000000000003e199

   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

   4. Applied egg-rr 68.2%

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

   5. Taylor expanded in k around 0 46.4%

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

      1. associate-*r* 46.4%

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

      2. *-commutative 46.4%

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

      3. associate-*l* 46.4%

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

   7. Simplified 46.4%

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

      1. sqrt-div 61.9%

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

      2. clear-num 61.8%

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

      3. *-commutative 61.8%

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

      4. associate-*r* 61.8%

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

   9. Applied egg-rr 61.8%

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

       1. associate-/r/ 61.8%

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

       2. associate-*l/ 61.9%

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

       3. *-commutative 61.9%

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

       4. *-rgt-identity 61.9%

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

       5. associate-*r* 61.9%

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

       6. *-commutative 61.9%

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

   11. Simplified 61.9%

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

   if 5.2000000000000003e199 < 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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in k around 0 2.8%

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

      1. associate-*r* 2.8%

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

      2. *-commutative 2.8%

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

      3. associate-*l* 2.8%

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

   7. Simplified 2.8%

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

      1. *-commutative 2.8%

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

      2. associate-*r* 2.8%

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

      3. *-commutative 2.8%

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

      4. associate-*r/ 2.8%

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

      5. pow1/2 2.8%

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

      6. metadata-eval 2.8%

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

      7. pow-prod-up 2.8%

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

      8. pow-prod-down 22.5%

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

      9. pow2 22.5%

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

      10. associate-*r/ 22.5%

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

      11. *-commutative 22.5%

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

      12. associate-*r* 22.5%

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

      13. *-commutative 22.5%

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

      14. associate-/l* 22.5%

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

      15. *-un-lft-identity 22.5%

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

      16. times-frac 22.5%

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

      17. metadata-eval 22.5%

          \[\leadsto {\left({\left(\frac{n}{\color{blue}{0.5} \cdot \frac{k}{\pi}}\right)}^{2}\right)}^{0.25} \]

   9. Applied egg-rr 22.5%

      \[\leadsto \color{blue}{{\left({\left(\frac{n}{0.5 \cdot \frac{k}{\pi}}\right)}^{2}\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{+199}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{n}{0.5 \cdot \frac{k}{\pi}}\right)}^{2}\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{+239}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.9e+239)
   (/ (sqrt (* n (* PI 2.0))) (sqrt k))
   (cbrt (pow (* 2.0 (* PI (/ n k))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.9e+239) {
		tmp = sqrt((n * (((double) M_PI) * 2.0))) / sqrt(k);
	} else {
		tmp = cbrt(pow((2.0 * (((double) M_PI) * (n / k))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.9e+239) {
		tmp = Math.sqrt((n * (Math.PI * 2.0))) / Math.sqrt(k);
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * (Math.PI * (n / k))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.9e+239)
		tmp = Float64(sqrt(Float64(n * Float64(pi * 2.0))) / sqrt(k));
	else
		tmp = cbrt((Float64(2.0 * Float64(pi * Float64(n / k))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.9e+239], N[(N[Sqrt[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(Pi * N[(n / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9000000000000001e239

   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

   4. Applied egg-rr 71.1%

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

   5. Taylor expanded in k around 0 42.5%

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

      1. associate-*r* 42.5%

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

      2. *-commutative 42.5%

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

      3. associate-*l* 42.5%

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

   7. Simplified 42.5%

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

      1. sqrt-div 56.6%

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

      2. clear-num 56.5%

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

      3. *-commutative 56.5%

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

      4. associate-*r* 56.5%

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

   9. Applied egg-rr 56.5%

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

       1. associate-/r/ 56.5%

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

       2. associate-*l/ 56.6%

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

       3. *-commutative 56.6%

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

       4. *-rgt-identity 56.6%

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

       5. associate-*r* 56.6%

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

       6. *-commutative 56.6%

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

   11. Simplified 56.6%

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

   if 1.9000000000000001e239 < 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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in k around 0 2.8%

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

      1. associate-*r* 2.8%

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

      2. *-commutative 2.8%

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

      3. associate-*l* 2.8%

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

   7. Simplified 2.8%

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

      1. add-cbrt-cube 13.6%

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

      2. pow1/3 13.6%

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

   9. Applied egg-rr 13.6%

      \[\leadsto \color{blue}{{\left({\left(\frac{n}{0.5 \cdot \frac{k}{\pi}}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   10. Step-by-step derivation

       1. unpow1/3 13.6%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{n}{0.5 \cdot \frac{k}{\pi}}\right)}^{1.5}}} \]

       2. *-lft-identity 13.6%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{1 \cdot n}}{0.5 \cdot \frac{k}{\pi}}\right)}^{1.5}} \]

       3. times-frac 13.6%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{1}{0.5} \cdot \frac{n}{\frac{k}{\pi}}\right)}}^{1.5}} \]

       4. metadata-eval 13.6%

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

       5. associate-/l* 13.6%

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

   11. Simplified 13.6%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(2 \cdot \frac{n \cdot \pi}{k}\right)}^{1.5}}} \]
   12. Step-by-step derivation

       1. associate-/l* 13.6%

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

       2. associate-/r/ 13.6%

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

   13. Applied egg-rr 13.6%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{+239}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.6% 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 99.6%

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

4. Applied egg-rr 76.0%

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

5. Taylor expanded in k around 0 35.7%

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

   1. associate-*r* 35.7%

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

   2. *-commutative 35.7%

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

   3. associate-*l* 35.7%

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

7. Simplified 35.7%

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

8. Taylor expanded in n around 0 35.7%

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

   1. associate-*r/ 35.7%

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

   2. associate-*r* 35.7%

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

   3. *-commutative 35.7%

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

   4. associate-*r/ 35.7%

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

   5. associate-*r* 35.7%

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

10. Simplified 35.7%

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

    1. sqrt-prod 47.3%

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

    2. clear-num 47.3%

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

    3. un-div-inv 47.3%

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

12. Applied egg-rr 47.3%

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

    1. associate-/l* 47.3%

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

    2. associate-*r/ 47.3%

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

14. Simplified 47.3%

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

15. Final simplification 47.3%

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

Alternative 6: 49.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

4. Applied egg-rr 76.0%

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

5. Taylor expanded in k around 0 35.7%

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

   1. associate-*r* 35.7%

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

   2. *-commutative 35.7%

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

   3. associate-*l* 35.7%

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

7. Simplified 35.7%

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

8. Taylor expanded in n around 0 35.7%

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

   1. associate-*r/ 35.7%

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

   2. associate-*r* 35.7%

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

   3. *-commutative 35.7%

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

   4. associate-*r/ 35.7%

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

   5. associate-*r* 35.7%

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

10. Simplified 35.7%

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

    1. pow1/2 35.7%

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

    2. associate-*r* 35.7%

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

    3. unpow-prod-down 47.4%

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

    4. pow1/2 47.4%

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

12. Applied egg-rr 47.4%

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

    1. unpow1/2 47.4%

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

14. Simplified 47.4%

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

15. Final simplification 47.4%

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

Alternative 7: 49.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

4. Applied egg-rr 76.0%

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

5. Taylor expanded in k around 0 35.7%

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

   1. associate-*r* 35.7%

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

   2. *-commutative 35.7%

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

   3. associate-*l* 35.7%

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

7. Simplified 35.7%

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

   1. sqrt-div 47.4%

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

   2. clear-num 47.3%

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

   3. *-commutative 47.3%

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

   4. associate-*r* 47.3%

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

9. Applied egg-rr 47.3%

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

    1. associate-/r/ 47.3%

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

    2. associate-*l/ 47.4%

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

    3. *-commutative 47.4%

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

    4. *-rgt-identity 47.4%

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

    5. associate-*r* 47.4%

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

    6. *-commutative 47.4%

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

11. Simplified 47.4%

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

12. Final simplification 47.4%

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

Alternative 8: 37.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

4. Applied egg-rr 76.0%

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

5. Taylor expanded in k around 0 35.7%

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

   1. associate-*r* 35.7%

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

   2. *-commutative 35.7%

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

   3. associate-*l* 35.7%

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

7. Simplified 35.7%

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

8. Taylor expanded in n around 0 35.7%

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

   1. associate-*r/ 35.7%

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

   2. associate-*r* 35.7%

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

   3. *-commutative 35.7%

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

   4. associate-*r/ 35.7%

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

   5. associate-*r* 35.7%

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

10. Simplified 35.7%

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

    1. *-commutative 35.7%

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

    2. associate-*r* 35.7%

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

    3. clear-num 35.7%

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

    4. div-inv 35.7%

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

    5. associate-/l* 35.7%

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

    6. associate-/l* 35.7%

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

    7. associate-*l/ 35.7%

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

    8. associate-/l* 35.7%

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

    9. associate-*r* 35.7%

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

    10. clear-num 35.7%

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

    11. metadata-eval 35.7%

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

    12. add-sqr-sqrt 35.6%

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

    13. frac-times 35.6%

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

    14. sqrt-unprod 36.3%

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

    15. add-sqr-sqrt 36.4%

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

    16. inv-pow 36.4%

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

    17. sqrt-pow2 36.5%

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

12. Applied egg-rr 36.5%

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

13. Final simplification 36.5%

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

Alternative 9: 37.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

4. Applied egg-rr 76.0%

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

5. Taylor expanded in k around 0 35.7%

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

   1. associate-*r* 35.7%

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

   2. *-commutative 35.7%

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

   3. associate-*l* 35.7%

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

7. Simplified 35.7%

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

8. Taylor expanded in n around 0 35.7%

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

   1. associate-*r/ 35.7%

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

   2. associate-*r* 35.7%

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

   3. *-commutative 35.7%

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

   4. associate-*r/ 35.7%

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

   5. associate-*r* 35.7%

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

10. Simplified 35.7%

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

    1. *-commutative 35.7%

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

    2. associate-*r* 35.7%

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

    3. clear-num 35.7%

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

    4. div-inv 35.7%

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

    5. associate-/l* 35.7%

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

    6. associate-/l* 35.7%

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

    7. associate-*l/ 35.7%

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

    8. associate-/l* 35.7%

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

    9. associate-*r* 35.7%

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

    10. clear-num 35.7%

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

    11. metadata-eval 35.7%

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

    12. add-sqr-sqrt 35.6%

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

    13. frac-times 35.6%

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

    14. sqrt-unprod 36.3%

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

    15. add-sqr-sqrt 36.4%

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

    16. inv-pow 36.4%

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

    17. sqrt-pow2 36.5%

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

12. Applied egg-rr 36.5%

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

    1. associate-*r/ 36.5%

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

    2. associate-*l/ 36.5%

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

    3. *-commutative 36.5%

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

14. Simplified 36.5%

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

15. Final simplification 36.5%

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

Alternative 10: 37.3% accurate, 2.0× speedup

Math

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

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(n * Float64(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 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.6%

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

4. Applied egg-rr 76.0%

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

5. Taylor expanded in k around 0 35.7%

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

   1. associate-*r* 35.7%

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

   2. *-commutative 35.7%

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

   3. associate-*l* 35.7%

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

7. Simplified 35.7%

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

8. Taylor expanded in n around 0 35.7%

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

   1. associate-*r/ 35.7%

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

   2. associate-*r* 35.7%

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

   3. *-commutative 35.7%

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

   4. associate-*r/ 35.7%

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

   5. associate-*r* 35.7%

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

10. Simplified 35.7%

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

11. Final simplification 35.7%

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

Alternative 11: 37.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

4. Applied egg-rr 76.0%

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

5. Taylor expanded in k around 0 35.7%

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

   1. associate-*r* 35.7%

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

   2. *-commutative 35.7%

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

   3. associate-*l* 35.7%

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

7. Simplified 35.7%

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

   1. *-commutative 35.7%

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

   2. associate-*r* 35.7%

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

   3. *-commutative 35.7%

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

   4. associate-*r/ 35.7%

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

   5. *-commutative 35.7%

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

   6. associate-/l* 35.7%

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

9. Applied egg-rr 35.7%

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

10. Final simplification 35.7%

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

Reproduce

herbie shell --seed 2024018 
(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))))