Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 14.5s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-un-lft-identity 99.4%

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

   3. associate-*r* 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

   6. pow-div 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-/l/ 99.7%

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

   2. unpow1/2 99.7%

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

   3. metadata-eval 99.7%

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

   4. pow-sqr 99.6%

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

   5. fabs-sqr 99.6%

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

   6. pow-sqr 99.7%

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

   7. metadata-eval 99.7%

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

   8. unpow1/2 99.7%

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

   9. fabs-neg 99.7%

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

   10. neg-mul-1 99.7%

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

   11. rem-square-sqrt 0.0%

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

   12. unpow1/2 0.0%

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

   13. metadata-eval 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot {k}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right|} \]

   14. pow-sqr 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \color{blue}{\left({k}^{0.25} \cdot {k}^{0.25}\right)}\right|} \]

   15. unswap-sqr 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\color{blue}{\left(\sqrt{-1} \cdot {k}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {k}^{0.25}\right)}\right|} \]

   16. fabs-sqr 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{\left(\sqrt{-1} \cdot {k}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {k}^{0.25}\right)}} \]

   17. unswap-sqr 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \left({k}^{0.25} \cdot {k}^{0.25}\right)}} \]

   18. rem-square-sqrt 24.4%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{-1} \cdot \left({k}^{0.25} \cdot {k}^{0.25}\right)} \]

   19. pow-sqr 24.4%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{-1 \cdot \color{blue}{{k}^{\left(2 \cdot 0.25\right)}}} \]

   20. metadata-eval 24.4%

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

   21. unpow1/2 24.4%

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

6. Simplified 99.7%

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

   1. div-inv 99.7%

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

   2. pow1/2 99.7%

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

   3. pow-unpow 99.7%

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

   4. pow-prod-down 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

   3. *-commutative 99.7%

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

   4. unpow1/2 99.7%

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

   5. *-commutative 99.7%

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

10. Simplified 99.7%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.08 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{n \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.08e-30)
   (* (sqrt (/ PI k)) (sqrt (* n 2.0)))
   (sqrt (/ (pow (* 2.0 (* PI n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.08e-30) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((n * 2.0));
	} else {
		tmp = sqrt((pow((2.0 * (((double) M_PI) * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.08e-30) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((n * 2.0));
	} else {
		tmp = Math.sqrt((Math.pow((2.0 * (Math.PI * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.08e-30:
		tmp = math.sqrt((math.pi / k)) * math.sqrt((n * 2.0))
	else:
		tmp = math.sqrt((math.pow((2.0 * (math.pi * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.08e-30)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(n * 2.0)));
	else
		tmp = sqrt(Float64((Float64(2.0 * Float64(pi * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.08e-30)
		tmp = sqrt((pi / k)) * sqrt((n * 2.0));
	else
		tmp = sqrt((((2.0 * (pi * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.08e-30], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.08000000000000005e-30

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-/l* 76.3%

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

   5. Simplified 76.3%

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

      1. pow1 76.3%

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

      2. *-commutative 76.3%

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

      3. sqrt-unprod 76.7%

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

   7. Applied egg-rr 76.7%

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

      1. unpow1 76.7%

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

      2. associate-*l* 76.7%

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

   9. Simplified 76.7%

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

       1. associate-*r* 76.7%

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

       2. div-inv 76.6%

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

       3. sqrt-prod 76.3%

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

       4. *-commutative 76.3%

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

       5. sqrt-prod 99.1%

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

       6. associate-*r* 99.1%

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

       7. sqrt-prod 99.3%

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

       8. *-commutative 99.3%

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

       9. div-inv 99.5%

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

   11. Applied egg-rr 99.5%

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

       1. *-commutative 99.5%

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

   13. Simplified 99.5%

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

   if 1.08000000000000005e-30 < k

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. distribute-rgt-in 99.5%

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

      2. metadata-eval 99.5%

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

      3. associate-*l* 99.5%

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

      4. metadata-eval 99.5%

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

      5. *-commutative 99.5%

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

      6. neg-mul-1 99.5%

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

      7. sub-neg 99.5%

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

      8. *-commutative 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.08 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{n \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{n \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(-1 + \mathsf{fma}\left(2, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.5e-31)
   (* (sqrt (/ PI k)) (sqrt (* n 2.0)))
   (sqrt (* n (+ -1.0 (fma 2.0 (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.5e-31) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((n * 2.0));
	} else {
		tmp = sqrt((n * (-1.0 + fma(2.0, (((double) M_PI) / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.5e-31)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(n * 2.0)));
	else
		tmp = sqrt(Float64(n * Float64(-1.0 + fma(2.0, Float64(pi / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.5e-31], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(n * N[(-1.0 + N[(2.0 * N[(Pi / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.5000000000000004e-31

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-/l* 76.3%

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

   5. Simplified 76.3%

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

      1. pow1 76.3%

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

      2. *-commutative 76.3%

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

      3. sqrt-unprod 76.7%

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

   7. Applied egg-rr 76.7%

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

      1. unpow1 76.7%

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

      2. associate-*l* 76.7%

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

   9. Simplified 76.7%

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

       1. associate-*r* 76.7%

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

       2. div-inv 76.6%

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

       3. sqrt-prod 76.3%

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

       4. *-commutative 76.3%

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

       5. sqrt-prod 99.1%

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

       6. associate-*r* 99.1%

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

       7. sqrt-prod 99.3%

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

       8. *-commutative 99.3%

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

       9. div-inv 99.5%

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

   11. Applied egg-rr 99.5%

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

       1. *-commutative 99.5%

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

   13. Simplified 99.5%

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

   if 4.5000000000000004e-31 < k

   1. Initial program 99.5%

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

   3. Taylor expanded in k around 0 11.7%

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

      1. *-commutative 11.7%

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

      2. associate-/l* 11.7%

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

   5. Simplified 11.7%

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

      1. pow1 11.7%

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

      2. *-commutative 11.7%

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

      3. sqrt-unprod 11.7%

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

   7. Applied egg-rr 11.7%

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

      1. unpow1 11.7%

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

      2. associate-*l* 11.7%

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

   9. Simplified 11.7%

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

       1. expm1-log1p-u 11.6%

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

       2. expm1-undefine 49.7%

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

       3. *-commutative 49.7%

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

   11. Applied egg-rr 49.7%

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

       1. sub-neg 49.7%

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

       2. metadata-eval 49.7%

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

       3. +-commutative 49.7%

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

       4. log1p-undefine 49.7%

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

       5. rem-exp-log 49.8%

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

       6. +-commutative 49.8%

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

       7. fma-define 49.8%

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

   13. Simplified 49.8%

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

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. associate-*l* 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

3. Simplified 99.4%

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

Alternative 5: 49.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 38.7%

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

   1. *-commutative 38.7%

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

   2. associate-/l* 38.7%

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

5. Simplified 38.7%

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

   1. pow1 38.7%

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

   2. *-commutative 38.7%

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

   3. sqrt-unprod 38.9%

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

7. Applied egg-rr 38.9%

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

   1. unpow1 38.9%

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

   2. associate-*l* 38.9%

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

9. Simplified 38.9%

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

    1. associate-*r* 38.9%

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

    2. div-inv 38.9%

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

    3. sqrt-prod 38.7%

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

    4. *-commutative 38.7%

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

    5. sqrt-prod 48.3%

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

    6. associate-*r* 48.3%

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

    7. sqrt-prod 48.4%

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

    8. *-commutative 48.4%

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

    9. div-inv 48.4%

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

11. Applied egg-rr 48.4%

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

    1. *-commutative 48.4%

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

13. Simplified 48.4%

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

Alternative 6: 49.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 38.7%

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

   1. *-commutative 38.7%

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

   2. associate-/l* 38.7%

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

5. Simplified 38.7%

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

   1. pow1 38.7%

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

   2. *-commutative 38.7%

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

   3. sqrt-unprod 38.9%

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

7. Applied egg-rr 38.9%

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

   1. unpow1 38.9%

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

   2. associate-*l* 38.9%

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

9. Simplified 38.9%

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

    1. *-commutative 38.9%

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

    2. sqrt-prod 48.4%

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

    3. *-commutative 48.4%

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

11. Applied egg-rr 48.4%

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

Alternative 7: 39.1% accurate, 2.0× speedup

Math

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

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(Float64(0.5 / 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[(N[(0.5 / Pi), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 38.7%

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

   1. *-commutative 38.7%

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

   2. associate-/l* 38.7%

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

5. Simplified 38.7%

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

   1. pow1 38.7%

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

   2. *-commutative 38.7%

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

   3. sqrt-unprod 38.9%

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

7. Applied egg-rr 38.9%

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

   1. unpow1 38.9%

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

   2. associate-*l* 38.9%

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

9. Simplified 38.9%

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

    1. associate-*r* 38.9%

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

    2. div-inv 38.9%

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

    3. sqrt-prod 38.7%

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

    4. *-commutative 38.7%

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

    5. sqrt-unprod 38.9%

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

    6. associate-*r* 38.8%

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

    7. associate-*r* 38.8%

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

    8. associate-*r* 38.8%

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

    9. *-commutative 38.8%

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

    10. div-inv 38.9%

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

    11. clear-num 38.9%

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

    12. sqrt-div 39.8%

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

    13. metadata-eval 39.8%

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

    14. *-un-lft-identity 39.8%

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

    15. *-commutative 39.8%

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

    16. associate-*r* 39.8%

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

    17. times-frac 39.7%

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

    18. metadata-eval 39.7%

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

    19. *-commutative 39.7%

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

11. Applied egg-rr 39.7%

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

    1. *-un-lft-identity 39.7%

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

    2. pow1/2 39.7%

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

    3. pow-flip 39.8%

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

    4. associate-*r/ 39.8%

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

    5. times-frac 39.8%

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

    6. metadata-eval 39.8%

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

13. Applied egg-rr 39.8%

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

    1. *-lft-identity 39.8%

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

    2. *-commutative 39.8%

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

    3. times-frac 39.8%

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

    4. associate-*r/ 39.8%

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

    5. *-commutative 39.8%

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

    6. associate-/r* 39.8%

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

15. Simplified 39.8%

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

Alternative 8: 39.1% 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 99.4%

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

3. Taylor expanded in k around 0 38.7%

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

   1. *-commutative 38.7%

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

   2. associate-/l* 38.7%

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

5. Simplified 38.7%

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

   1. pow1 38.7%

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

   2. *-commutative 38.7%

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

   3. sqrt-unprod 38.9%

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

7. Applied egg-rr 38.9%

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

   1. unpow1 38.9%

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

   2. associate-*l* 38.9%

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

9. Simplified 38.9%

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

    1. associate-*r* 38.9%

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

    2. div-inv 38.9%

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

    3. sqrt-prod 38.7%

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

    4. *-commutative 38.7%

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

    5. sqrt-unprod 38.9%

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

    6. associate-*r* 38.8%

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

    7. associate-*r* 38.8%

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

    8. associate-*r* 38.8%

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

    9. *-commutative 38.8%

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

    10. div-inv 38.9%

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

    11. clear-num 38.9%

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

    12. sqrt-div 39.8%

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

    13. metadata-eval 39.8%

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

    14. *-un-lft-identity 39.8%

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

    15. *-commutative 39.8%

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

    16. associate-*r* 39.8%

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

    17. times-frac 39.7%

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

    18. metadata-eval 39.7%

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

    19. *-commutative 39.7%

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

11. Applied egg-rr 39.7%

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

    1. *-un-lft-identity 39.7%

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

    2. pow1/2 39.7%

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

    3. pow-flip 39.8%

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

    4. associate-*r/ 39.8%

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

    5. times-frac 39.8%

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

    6. metadata-eval 39.8%

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

13. Applied egg-rr 39.8%

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

    1. *-lft-identity 39.8%

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

    2. times-frac 39.8%

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

    3. *-commutative 39.8%

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

    4. associate-*r/ 39.8%

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

15. Simplified 39.8%

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

16. Final simplification 39.8%

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

Alternative 9: 38.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 38.7%

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

   1. *-commutative 38.7%

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

   2. associate-/l* 38.7%

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

5. Simplified 38.7%

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

   1. pow1 38.7%

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

   2. *-commutative 38.7%

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

   3. sqrt-unprod 38.9%

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

7. Applied egg-rr 38.9%

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

   1. unpow1 38.9%

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

   2. associate-*l* 38.9%

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

9. Simplified 38.9%

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

    1. associate-*r* 38.9%

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

    2. div-inv 38.9%

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

    3. sqrt-prod 38.7%

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

    4. *-commutative 38.7%

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

    5. *-un-lft-identity 38.7%

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

    6. sqrt-unprod 38.9%

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

    7. div-inv 38.9%

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

    8. clear-num 38.8%

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

    9. un-div-inv 38.9%

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

11. Applied egg-rr 38.9%

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

    1. *-lft-identity 38.9%

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

    2. associate-/r/ 38.9%

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

13. Simplified 38.9%

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

    1. *-commutative 38.9%

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

    2. clear-num 38.9%

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

    3. un-div-inv 38.9%

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

15. Applied egg-rr 38.9%

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

Alternative 10: 38.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 38.7%

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

   1. *-commutative 38.7%

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

   2. associate-/l* 38.7%

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

5. Simplified 38.7%

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

   1. pow1 38.7%

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

   2. *-commutative 38.7%

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

   3. sqrt-unprod 38.9%

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

7. Applied egg-rr 38.9%

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

   1. unpow1 38.9%

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

   2. associate-*l* 38.9%

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

9. Simplified 38.9%

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

    1. associate-*r* 38.9%

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

    2. div-inv 38.9%

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

    3. sqrt-prod 38.7%

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

    4. *-commutative 38.7%

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

    5. *-un-lft-identity 38.7%

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

    6. sqrt-unprod 38.9%

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

    7. div-inv 38.9%

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

    8. clear-num 38.8%

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

    9. un-div-inv 38.9%

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

11. Applied egg-rr 38.9%

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

    1. *-lft-identity 38.9%

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

    2. associate-/r/ 38.9%

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

13. Simplified 38.9%

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

14. Final simplification 38.9%

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

Reproduce

herbie shell --seed 2024160 
(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))))