Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 98.8%

Time: 20.0s

Alternatives: 14

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: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6.6e-66)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6.6e-66) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = sqrt((pow((((double) M_PI) * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 6.6e-66) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((2.0 * n));
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 6.6e-66:
		tmp = math.sqrt((math.pi / k)) * math.sqrt((2.0 * n))
	else:
		tmp = math.sqrt((math.pow((math.pi * (2.0 * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 6.6e-66)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(2.0 * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 6.6e-66)
		tmp = sqrt((pi / k)) * sqrt((2.0 * n));
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6.6e-66], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.5999999999999998e-66

   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. Taylor expanded in k around 0 99.1%

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

      1. associate-*l/ 99.2%

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

      2. *-un-lft-identity 99.2%

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

      3. *-commutative 99.2%

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

      4. sqrt-prod 99.3%

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

      5. associate-*r* 99.3%

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

      6. *-commutative 99.3%

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

      7. pow1 99.3%

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

      8. sqrt-undiv 68.7%

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

   4. Applied egg-rr 68.7%

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

      1. unpow1 68.7%

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

      2. associate-/l* 68.7%

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

      3. associate-/r/ 68.7%

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

   6. Simplified 68.7%

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

      1. sqrt-prod 99.5%

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

   8. Applied egg-rr 99.5%

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

   if 6.5999999999999998e-66 < 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. Step-by-step derivation

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

      3. sqr-pow 99.8%

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

      4. pow-sqr 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*l* 99.8%

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

      7. associate-*r/ 99.8%

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

      8. associate-/l* 99.7%

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

      9. associate-/r/ 99.8%

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

      10. metadata-eval 99.8%

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

      11. *-lft-identity 99.8%

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

      12. div-sub 99.8%

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

      13. sub-neg 99.8%

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

      14. distribute-frac-neg 99.8%

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

      15. metadata-eval 99.8%

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

      16. neg-mul-1 99.8%

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

      17. associate-/l* 99.8%

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

      18. associate-/r/ 99.8%

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

      19. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in k around inf 98.8%

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

      1. add-sqr-sqrt 98.8%

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

      2. sqrt-unprod 98.8%

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

      3. frac-times 98.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r* 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*l* 99.8%

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

      4. distribute-lft-in 99.8%

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

      5. metadata-eval 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*r* 99.8%

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

      8. metadata-eval 99.8%

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

      9. neg-mul-1 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. *-un-lft-identity 99.6%

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

   5. div-sub 99.6%

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

   6. metadata-eval 99.6%

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

   7. pow-sub 99.7%

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

   8. unpow1/2 99.7%

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

   9. associate-/l/ 99.7%

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

   10. associate-*r* 99.7%

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

   11. *-commutative 99.7%

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

   12. associate-*l* 99.7%

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

3. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*r* 99.7%

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

   3. *-commutative 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*r* 99.7%

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

   6. *-commutative 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. expm1-log1p-u 96.1%

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

   2. expm1-udef 75.0%

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

   3. pow1/2 75.0%

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

   4. pow-flip 75.0%

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

   5. metadata-eval 75.0%

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

3. Applied egg-rr 75.0%

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

   1. expm1-def 96.1%

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

   2. expm1-log1p 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. sqr-pow 99.4%

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

   4. pow-sqr 99.6%

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

   5. *-commutative 99.6%

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

   6. associate-*l* 99.6%

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

   7. associate-*r/ 99.6%

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

   8. associate-/l* 99.6%

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

   9. associate-/r/ 99.6%

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

   10. metadata-eval 99.6%

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

   11. *-lft-identity 99.6%

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

   12. div-sub 99.6%

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

   13. sub-neg 99.6%

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

   14. distribute-frac-neg 99.6%

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

   15. metadata-eval 99.6%

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

   16. neg-mul-1 99.6%

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

   17. associate-/l* 99.6%

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

   18. associate-/r/ 99.6%

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

   19. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 5: 51.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{+226}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\pi \cdot \left(2 \cdot \frac{n}{k}\right)\right)}^{2}\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 7.5e+226)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (pow (pow (* PI (* 2.0 (/ n k))) 2.0) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 7.5e+226) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = pow(pow((((double) M_PI) * (2.0 * (n / k))), 2.0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 7.5e+226) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((2.0 * n));
	} else {
		tmp = Math.pow(Math.pow((Math.PI * (2.0 * (n / k))), 2.0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 7.5e+226:
		tmp = math.sqrt((math.pi / k)) * math.sqrt((2.0 * n))
	else:
		tmp = math.pow(math.pow((math.pi * (2.0 * (n / k))), 2.0), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 7.5e+226)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = (Float64(pi * Float64(2.0 * Float64(n / k))) ^ 2.0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 7.5e+226)
		tmp = sqrt((pi / k)) * sqrt((2.0 * n));
	else
		tmp = ((pi * (2.0 * (n / k))) ^ 2.0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 7.5e+226], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(Pi * N[(2.0 * N[(n / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.49999999999999964e226

   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. Taylor expanded in k around 0 59.4%

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

      1. associate-*l/ 59.4%

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

      2. *-un-lft-identity 59.4%

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

      3. *-commutative 59.4%

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

      4. sqrt-prod 59.5%

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

      5. associate-*r* 59.5%

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

      6. *-commutative 59.5%

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

      7. pow1 59.5%

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

      8. sqrt-undiv 44.6%

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

   4. Applied egg-rr 44.6%

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

      1. unpow1 44.6%

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

      2. associate-/l* 44.6%

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

      3. associate-/r/ 44.6%

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

   6. Simplified 44.6%

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

      1. sqrt-prod 59.6%

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

   8. Applied egg-rr 59.6%

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

   if 7.49999999999999964e226 < 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. Taylor expanded in k around 0 3.0%

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

      1. associate-*l/ 3.0%

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

      2. *-un-lft-identity 3.0%

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

      3. *-commutative 3.0%

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

      4. sqrt-prod 3.0%

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

      5. associate-*r* 3.0%

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

      6. *-commutative 3.0%

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

      7. pow1 3.0%

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

      8. sqrt-undiv 3.0%

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

   4. Applied egg-rr 3.0%

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

      1. unpow1 3.0%

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

      2. associate-/l* 3.0%

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

      3. associate-/r/ 3.0%

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

   6. Simplified 3.0%

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

      1. pow1/2 3.0%

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

      2. *-commutative 3.0%

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

      3. associate-*l* 3.0%

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

      4. div-inv 3.0%

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

      5. associate-*l* 3.0%

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

      6. div-inv 3.0%

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

      7. metadata-eval 3.0%

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

      8. pow-sqr 3.0%

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

      9. pow-prod-down 20.3%

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

      10. *-commutative 20.3%

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

      11. *-commutative 20.3%

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

      12. swap-sqr 20.3%

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

      13. pow2 20.3%

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

      14. associate-/l* 20.3%

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

      15. associate-/r/ 20.3%

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

      16. metadata-eval 20.3%

          \[\leadsto {\left({\left(\frac{n}{k} \cdot \pi\right)}^{2} \cdot \color{blue}{4}\right)}^{0.25} \]

   8. Applied egg-rr 20.3%

      \[\leadsto \color{blue}{{\left({\left(\frac{n}{k} \cdot \pi\right)}^{2} \cdot 4\right)}^{0.25}} \]
   9. Step-by-step derivation

      1. *-commutative 20.3%

         \[\leadsto {\color{blue}{\left(4 \cdot {\left(\frac{n}{k} \cdot \pi\right)}^{2}\right)}}^{0.25} \]

      2. metadata-eval 20.3%

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

      3. unpow2 20.3%

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

      4. swap-sqr 20.3%

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

      5. unpow1 20.3%

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

      6. unpow1 20.3%

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

      7. pow-sqr 20.3%

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

      8. associate-*r* 20.3%

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

      9. *-commutative 20.3%

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

      10. metadata-eval 20.3%

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

   10. Simplified 20.3%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{+226}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\pi \cdot \left(2 \cdot \frac{n}{k}\right)\right)}^{2}\right)}^{0.25}\\ \end{array} \]

Alternative 6: 54.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\left(2 \cdot n\right) \cdot \frac{\pi}{k}\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.95e+147)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (pow (pow (* (* 2.0 n) (/ PI k)) 3.0) 0.16666666666666666)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.95e+147) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = pow(pow(((2.0 * n) * (((double) M_PI) / k)), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.95e+147) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((2.0 * n));
	} else {
		tmp = Math.pow(Math.pow(((2.0 * n) * (Math.PI / k)), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.95e+147:
		tmp = math.sqrt((math.pi / k)) * math.sqrt((2.0 * n))
	else:
		tmp = math.pow(math.pow(((2.0 * n) * (math.pi / k)), 3.0), 0.16666666666666666)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.95e+147)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = (Float64(Float64(2.0 * n) * Float64(pi / k)) ^ 3.0) ^ 0.16666666666666666;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.95e+147)
		tmp = sqrt((pi / k)) * sqrt((2.0 * n));
	else
		tmp = (((2.0 * n) * (pi / k)) ^ 3.0) ^ 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.95e+147], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(Pi / k), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.95000000000000008e147

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0 68.4%

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

      1. associate-*l/ 68.4%

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

      2. *-un-lft-identity 68.4%

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

      3. *-commutative 68.4%

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

      4. sqrt-prod 68.5%

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

      5. associate-*r* 68.5%

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

      6. *-commutative 68.5%

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

      7. pow1 68.5%

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

      8. sqrt-undiv 51.3%

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

   4. Applied egg-rr 51.3%

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

      1. unpow1 51.3%

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

      2. associate-/l* 51.3%

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

      3. associate-/r/ 51.3%

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

   6. Simplified 51.3%

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

      1. sqrt-prod 68.7%

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

   8. Applied egg-rr 68.7%

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

   if 1.95000000000000008e147 < k

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. pow2 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{-1 + k}{-4}\right)}}{{k}^{0.25}}\right)}^{2}} \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto {\left(\frac{{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot n\right)}}^{\left(\frac{-1 + k}{-4}\right)}}{{k}^{0.25}}\right)}^{2} \]

      2. +-commutative 100.0%

         \[\leadsto {\left(\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\color{blue}{k + -1}}{-4}\right)}}{{k}^{0.25}}\right)}^{2} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k + -1}{-4}\right)}}{{k}^{0.25}}\right)}^{2}} \]

   6. Taylor expanded in k around 0 2.8%

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

      1. sqrt-unprod 2.8%

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

      2. *-commutative 2.8%

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

      3. associate-/l* 2.8%

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

      4. associate-/r/ 2.8%

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

   8. Applied egg-rr 2.8%

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

      1. pow1/2 2.8%

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

      2. metadata-eval 2.8%

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

      3. pow-pow 9.2%

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

      4. sqr-pow 9.2%

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

      5. pow-prod-down 26.4%

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

      6. pow-sqr 26.4%

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

      7. associate-*l/ 26.4%

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

      8. associate-*r/ 26.4%

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

      9. metadata-eval 26.4%

         \[\leadsto {\left({\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{\color{blue}{3}}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      10. metadata-eval 26.4%

          \[\leadsto {\left({\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{3}\right)}^{\color{blue}{0.16666666666666666}} \]

   10. Applied egg-rr 26.4%

       \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{3}\right)}^{0.16666666666666666}} \]
   11. Step-by-step derivation

       1. associate-*r* 26.4%

          \[\leadsto {\left({\color{blue}{\left(\left(2 \cdot n\right) \cdot \frac{\pi}{k}\right)}}^{3}\right)}^{0.16666666666666666} \]

       2. *-commutative 26.4%

          \[\leadsto {\left({\color{blue}{\left(\frac{\pi}{k} \cdot \left(2 \cdot n\right)\right)}}^{3}\right)}^{0.16666666666666666} \]

       3. *-commutative 26.4%

          \[\leadsto {\left({\left(\frac{\pi}{k} \cdot \color{blue}{\left(n \cdot 2\right)}\right)}^{3}\right)}^{0.16666666666666666} \]

   12. Simplified 26.4%

       \[\leadsto \color{blue}{{\left({\left(\frac{\pi}{k} \cdot \left(n \cdot 2\right)\right)}^{3}\right)}^{0.16666666666666666}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\left(2 \cdot n\right) \cdot \frac{\pi}{k}\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \]

Alternative 7: 50.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{+226}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.7e+226)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (cbrt (pow (* 2.0 (* n (/ PI k))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.7e+226) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = cbrt(pow((2.0 * (n * (((double) M_PI) / k))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.7e+226) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((2.0 * n));
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * (n * (Math.PI / k))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.7e+226)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = cbrt((Float64(2.0 * Float64(n * Float64(pi / k))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.7e+226], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.69999999999999982e226

   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. Taylor expanded in k around 0 59.4%

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

      1. associate-*l/ 59.4%

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

      2. *-un-lft-identity 59.4%

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

      3. *-commutative 59.4%

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

      4. sqrt-prod 59.5%

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

      5. associate-*r* 59.5%

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

      6. *-commutative 59.5%

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

      7. pow1 59.5%

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

      8. sqrt-undiv 44.6%

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

   4. Applied egg-rr 44.6%

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

      1. unpow1 44.6%

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

      2. associate-/l* 44.6%

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

      3. associate-/r/ 44.6%

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

   6. Simplified 44.6%

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

      1. sqrt-prod 59.6%

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

   8. Applied egg-rr 59.6%

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

   if 3.69999999999999982e226 < 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. Taylor expanded in k around 0 3.0%

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

      1. associate-*l/ 3.0%

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

      2. *-un-lft-identity 3.0%

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

      3. *-commutative 3.0%

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

      4. sqrt-prod 3.0%

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

      5. associate-*r* 3.0%

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

      6. *-commutative 3.0%

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

      7. pow1 3.0%

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

      8. sqrt-undiv 3.0%

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

   4. Applied egg-rr 3.0%

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

      1. unpow1 3.0%

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

      2. associate-/l* 3.0%

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

      3. associate-/r/ 3.0%

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

   6. Simplified 3.0%

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

      1. add-log-exp 48.7%

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

      2. add-cbrt-cube 48.7%

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

      3. add-log-exp 48.7%

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

      4. add-log-exp 48.7%

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

      5. add-sqr-sqrt 48.7%

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

      6. *-commutative 48.7%

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

      7. add-log-exp 13.8%

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

      8. pow1/3 13.8%

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

   8. Applied egg-rr 13.8%

      \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   9. Step-by-step derivation

      1. unpow1/3 13.8%

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

      2. associate-*l/ 13.8%

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

      3. associate-*r/ 13.8%

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

   10. Simplified 13.8%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{+226}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1.5}}\\ \end{array} \]

Alternative 8: 49.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(Pi * N[(2.0 / 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. Taylor expanded in k around 0 49.7%

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

   1. associate-*l/ 49.7%

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

   2. *-un-lft-identity 49.7%

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

   3. *-commutative 49.7%

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

   4. sqrt-prod 49.8%

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

   5. associate-*r* 49.8%

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

   6. *-commutative 49.8%

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

   7. pow1 49.8%

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

   8. sqrt-undiv 37.5%

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

4. Applied egg-rr 37.5%

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

   1. unpow1 37.5%

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

   2. associate-/l* 37.5%

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

   3. associate-/r/ 37.5%

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

6. Simplified 37.5%

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

   1. associate-*r* 37.5%

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

   2. sqrt-prod 49.8%

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

   3. clear-num 49.8%

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

   4. metadata-eval 49.8%

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

   5. associate-*l/ 49.8%

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

   6. metadata-eval 49.8%

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

   7. metadata-eval 49.8%

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

8. Applied egg-rr 49.8%

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

   1. *-commutative 49.8%

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

   2. associate-/r/ 49.8%

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

10. Simplified 49.8%

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

11. Final simplification 49.8%

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

Alternative 9: 49.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $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. Taylor expanded in k around 0 49.7%

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

   1. associate-*l/ 49.7%

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

   2. *-un-lft-identity 49.7%

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

   3. *-commutative 49.7%

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

   4. sqrt-prod 49.8%

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

   5. associate-*r* 49.8%

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

   6. *-commutative 49.8%

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

   7. pow1 49.8%

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

   8. sqrt-undiv 37.5%

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

4. Applied egg-rr 37.5%

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

   1. unpow1 37.5%

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

   2. associate-/l* 37.5%

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

   3. associate-/r/ 37.5%

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

6. Simplified 37.5%

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

   1. sqrt-prod 49.9%

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

8. Applied egg-rr 49.9%

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

9. Final simplification 49.9%

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

Alternative 10: 38.2% accurate, 1.5× 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. Step-by-step derivation

   1. add-sqr-sqrt 99.5%

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

   2. pow2 99.5%

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

3. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{{\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{-1 + k}{-4}\right)}}{{k}^{0.25}}\right)}^{2}} \]
4. Step-by-step derivation

   1. associate-*r* 99.4%

      \[\leadsto {\left(\frac{{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot n\right)}}^{\left(\frac{-1 + k}{-4}\right)}}{{k}^{0.25}}\right)}^{2} \]

   2. +-commutative 99.4%

      \[\leadsto {\left(\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\color{blue}{k + -1}}{-4}\right)}}{{k}^{0.25}}\right)}^{2} \]

5. Simplified 99.4%

   \[\leadsto \color{blue}{{\left(\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k + -1}{-4}\right)}}{{k}^{0.25}}\right)}^{2}} \]

6. Taylor expanded in k around 0 37.4%

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

   1. sqrt-unprod 37.5%

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

   2. *-commutative 37.5%

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

   3. associate-/l* 37.4%

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

   4. associate-/r/ 37.4%

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

8. Applied egg-rr 37.4%

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

   1. metadata-eval 37.4%

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

   2. *-commutative 37.4%

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

   3. clear-num 37.4%

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

   4. div-inv 37.5%

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

   5. times-frac 37.5%

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

   6. associate-/l* 37.5%

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

   7. metadata-eval 37.5%

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

   8. add-sqr-sqrt 37.4%

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

   9. frac-times 37.4%

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

   10. sqrt-unprod 37.9%

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

   11. add-sqr-sqrt 38.0%

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

   12. pow1/2 38.0%

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

   13. pow-flip 38.1%

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

   14. *-un-lft-identity 38.1%

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

   15. times-frac 38.1%

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

   16. metadata-eval 38.1%

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

   17. metadata-eval 38.1%

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

10. Applied egg-rr 38.1%

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

    1. associate-/r* 38.1%

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

12. Simplified 38.1%

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

13. Final simplification 38.1%

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

Alternative 11: 37.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.5%

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

   2. pow2 99.5%

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

3. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{{\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{-1 + k}{-4}\right)}}{{k}^{0.25}}\right)}^{2}} \]
4. Step-by-step derivation

   1. associate-*r* 99.4%

      \[\leadsto {\left(\frac{{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot n\right)}}^{\left(\frac{-1 + k}{-4}\right)}}{{k}^{0.25}}\right)}^{2} \]

   2. +-commutative 99.4%

      \[\leadsto {\left(\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\color{blue}{k + -1}}{-4}\right)}}{{k}^{0.25}}\right)}^{2} \]

5. Simplified 99.4%

   \[\leadsto \color{blue}{{\left(\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k + -1}{-4}\right)}}{{k}^{0.25}}\right)}^{2}} \]

6. Taylor expanded in k around 0 37.4%

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

   1. sqrt-unprod 37.5%

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

   2. *-commutative 37.5%

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

   3. associate-/l* 37.4%

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

   4. associate-/r/ 37.4%

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

8. Applied egg-rr 37.4%

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

9. Final simplification 37.4%

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

Alternative 12: 37.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(k, n)
	return sqrt(Float64(Float64(2.0 * 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[(N[(2.0 * n), $MachinePrecision] * 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. Taylor expanded in k around 0 49.7%

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

   1. associate-*l/ 49.7%

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

   2. *-un-lft-identity 49.7%

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

   3. *-commutative 49.7%

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

   4. sqrt-prod 49.8%

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

   5. associate-*r* 49.8%

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

   6. *-commutative 49.8%

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

   7. pow1 49.8%

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

   8. sqrt-undiv 37.5%

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

4. Applied egg-rr 37.5%

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

   1. unpow1 37.5%

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

   2. associate-/l* 37.5%

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

   3. associate-/r/ 37.5%

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

6. Simplified 37.5%

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

7. Final simplification 37.5%

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

Alternative 13: 37.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(Pi / N[(k / N[(2.0 * n), $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. Taylor expanded in k around 0 49.7%

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

   1. associate-*l/ 49.7%

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

   2. *-un-lft-identity 49.7%

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

   3. *-commutative 49.7%

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

   4. sqrt-prod 49.8%

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

   5. associate-*r* 49.8%

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

   6. *-commutative 49.8%

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

   7. pow1 49.8%

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

   8. sqrt-undiv 37.5%

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

4. Applied egg-rr 37.5%

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

   1. unpow1 37.5%

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

   2. associate-/l* 37.5%

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

6. Simplified 37.5%

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

7. Final simplification 37.5%

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

Alternative 14: 37.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] / 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. Taylor expanded in k around 0 49.7%

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

   1. associate-*l/ 49.7%

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

   2. *-un-lft-identity 49.7%

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

   3. *-commutative 49.7%

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

   4. sqrt-prod 49.8%

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

   5. associate-*r* 49.8%

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

   6. *-commutative 49.8%

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

   7. sqrt-undiv 37.5%

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

4. Applied egg-rr 37.5%

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

5. Final simplification 37.5%

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

Reproduce

herbie shell --seed 2023301 
(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))))