Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 18.9s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. k = 4.94066e-324
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / N[Power[t$95$0, N[(k * 0.5), $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-*r* 99.4%

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

   2. div-sub 99.4%

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

   3. metadata-eval 99.4%

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

   4. pow-div 99.6%

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

   5. pow1/2 99.6%

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

   6. associate-*r/ 99.6%

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

   7. pow1/2 99.6%

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

   8. pow-flip 99.7%

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

   9. metadata-eval 99.7%

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

   10. div-inv 99.7%

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

   11. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(2.0 * Pi), $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 99.4%

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

   1. associate-*l/ 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. associate-*r* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

   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-*r* 99.7%

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

   2. associate-*r* 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.5e-35)
   (/ (sqrt (* PI (* 2.0 n))) (sqrt k))
   (sqrt (/ (pow (* n (* 2.0 PI)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.5e-35) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) / sqrt(k);
	} else {
		tmp = sqrt((pow((n * (2.0 * ((double) M_PI))), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.5e-35) {
		tmp = Math.sqrt((Math.PI * (2.0 * n))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow((n * (2.0 * Math.PI)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.5e-35:
		tmp = math.sqrt((math.pi * (2.0 * n))) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow((n * (2.0 * math.pi)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.5e-35)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) / sqrt(k));
	else
		tmp = sqrt(Float64((Float64(n * Float64(2.0 * pi)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.5e-35)
		tmp = sqrt((pi * (2.0 * n))) / sqrt(k);
	else
		tmp = sqrt((((n * (2.0 * pi)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.5e-35], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.49999999999999996e-35

   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 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-/l* 70.7%

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

   5. Simplified 70.7%

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

      1. sqrt-unprod 70.9%

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

      2. associate-*r/ 71.0%

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

      3. *-commutative 71.0%

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

      4. associate-/l* 70.9%

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

   7. Applied egg-rr 70.9%

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

   8. Taylor expanded in n around 0 71.0%

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

      1. associate-/l* 70.9%

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

   10. Simplified 70.9%

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

       1. associate-*r* 70.9%

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

       2. *-commutative 70.9%

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

       3. associate-*r/ 71.0%

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

       4. *-commutative 71.0%

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

       5. sqrt-div 99.4%

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

   12. Applied egg-rr 99.4%

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

   if 3.49999999999999996e-35 < k

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 99.5%

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

      2. sqrt-unprod 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*r* 99.5%

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

      5. div-sub 99.5%

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

      6. metadata-eval 99.5%

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

      7. div-inv 99.5%

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

      8. *-commutative 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.9 \cdot 10^{+189}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(n \cdot \left(2 \cdot \frac{\pi}{k}\right)\right)}^{2}\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.9e+189)
   (/ (sqrt (* PI (* 2.0 n))) (sqrt k))
   (pow (pow (* n (* 2.0 (/ PI k))) 2.0) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.9e+189) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) / sqrt(k);
	} else {
		tmp = pow(pow((n * (2.0 * (((double) M_PI) / k))), 2.0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.9e+189) {
		tmp = Math.sqrt((Math.PI * (2.0 * n))) / Math.sqrt(k);
	} else {
		tmp = Math.pow(Math.pow((n * (2.0 * (Math.PI / k))), 2.0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.9e+189:
		tmp = math.sqrt((math.pi * (2.0 * n))) / math.sqrt(k)
	else:
		tmp = math.pow(math.pow((n * (2.0 * (math.pi / k))), 2.0), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.9e+189)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) / sqrt(k));
	else
		tmp = (Float64(n * Float64(2.0 * Float64(pi / k))) ^ 2.0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.9e+189)
		tmp = sqrt((pi * (2.0 * n))) / sqrt(k);
	else
		tmp = ((n * (2.0 * (pi / k))) ^ 2.0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.9e+189], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(n * N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.9e189

   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 45.3%

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

      1. *-commutative 45.3%

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

      2. associate-/l* 45.3%

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

   5. Simplified 45.3%

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

      1. sqrt-unprod 45.4%

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

      2. associate-*r/ 45.5%

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

      3. *-commutative 45.5%

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

      4. associate-/l* 45.5%

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

   7. Applied egg-rr 45.5%

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

   8. Taylor expanded in n around 0 45.5%

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

      1. associate-/l* 45.4%

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

   10. Simplified 45.4%

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

       1. associate-*r* 45.4%

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

       2. *-commutative 45.4%

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

       3. associate-*r/ 45.5%

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

       4. *-commutative 45.5%

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

       5. sqrt-div 61.0%

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

   12. Applied egg-rr 61.0%

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

   if 3.9e189 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.9%

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

      1. *-commutative 2.9%

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

      2. associate-/l* 2.9%

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

   5. Simplified 2.9%

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

      1. sqrt-unprod 2.9%

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

      2. associate-*r/ 2.9%

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

      3. *-commutative 2.9%

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

      4. associate-/l* 2.9%

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in n around 0 2.9%

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

      1. associate-/l* 2.9%

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

   10. Simplified 2.9%

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

       1. sqrt-prod 2.9%

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

       2. *-commutative 2.9%

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

       3. associate-*l/ 2.9%

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

       4. associate-*r/ 2.9%

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

       5. sqrt-prod 2.9%

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

       6. pow1/2 2.9%

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

       7. metadata-eval 2.9%

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

       8. pow-prod-up 2.9%

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

       9. pow-prod-down 17.1%

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

       10. *-commutative 17.1%

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

       11. *-commutative 17.1%

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

       12. swap-sqr 17.1%

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

       13. pow2 17.1%

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

       14. metadata-eval 17.1%

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

   12. Applied egg-rr 17.1%

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

   14. Simplified 17.1%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.9 \cdot 10^{+189}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(n \cdot \left(2 \cdot \frac{\pi}{k}\right)\right)}^{2}\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. associate-*l* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.4%

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

   2. div-inv 99.4%

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

   3. metadata-eval 99.4%

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

   4. pow1/2 99.4%

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

   5. pow-flip 99.5%

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

   6. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 6: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[k, -0.5], $MachinePrecision] / N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(-0.5 + N[(k * 0.5), $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. Step-by-step derivation

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. associate-*l* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.4%

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

   2. div-inv 99.4%

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

   3. metadata-eval 99.4%

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

   4. pow1/2 99.4%

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

   5. pow-flip 99.5%

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

   6. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

   1. *-commutative 99.5%

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

   2. metadata-eval 99.5%

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

   3. pow-flip 99.4%

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

   4. pow1/2 99.4%

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

   5. associate-/r/ 99.4%

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

   6. *-commutative 99.4%

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

   7. *-commutative 99.4%

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

   8. exp-to-pow 96.3%

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

   9. exp-to-pow 99.4%

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

   10. associate-*r* 99.4%

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

   11. *-commutative 99.4%

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

   12. sub-neg 99.4%

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

   13. *-commutative 99.4%

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

   14. distribute-rgt-neg-in 99.4%

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

   15. metadata-eval 99.4%

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

8. Applied egg-rr 99.4%

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

   1. inv-pow 99.4%

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

   2. div-inv 99.4%

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

   3. unpow-prod-down 99.4%

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

   4. inv-pow 99.4%

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

   5. pow1/2 99.4%

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

   6. pow-flip 99.4%

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

   7. metadata-eval 99.4%

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

   8. pow-flip 99.4%

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

   9. +-commutative 99.4%

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

   10. fma-define 99.4%

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

10. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{{k}^{-0.5} \cdot {\left({\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(-\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}\right)}^{-1}} \]
11. Step-by-step derivation

    1. unpow-1 99.4%

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

    2. associate-*r/ 99.5%

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

    3. *-commutative 99.5%

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

    4. *-lft-identity 99.5%

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

    5. *-commutative 99.5%

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

    6. neg-sub0 99.5%

       \[\leadsto \frac{{k}^{-0.5}}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\color{blue}{\left(0 - \mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}} \]

    7. fma-undefine 99.5%

       \[\leadsto \frac{{k}^{-0.5}}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(0 - \color{blue}{\left(k \cdot -0.5 + 0.5\right)}\right)}} \]

    8. *-commutative 99.5%

       \[\leadsto \frac{{k}^{-0.5}}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(0 - \left(\color{blue}{-0.5 \cdot k} + 0.5\right)\right)}} \]

    9. +-commutative 99.5%

       \[\leadsto \frac{{k}^{-0.5}}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(0 - \color{blue}{\left(0.5 + -0.5 \cdot k\right)}\right)}} \]

    10. metadata-eval 99.5%

        \[\leadsto \frac{{k}^{-0.5}}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(0 - \left(0.5 + \color{blue}{\left(-0.5\right)} \cdot k\right)\right)}} \]

    11. cancel-sign-sub-inv 99.5%

        \[\leadsto \frac{{k}^{-0.5}}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(0 - \color{blue}{\left(0.5 - 0.5 \cdot k\right)}\right)}} \]

    12. associate--r- 99.5%

        \[\leadsto \frac{{k}^{-0.5}}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\color{blue}{\left(\left(0 - 0.5\right) + 0.5 \cdot k\right)}}} \]

    13. metadata-eval 99.5%

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

    14. *-commutative 99.5%

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

12. Simplified 99.5%

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

13. Final simplification 99.5%

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

Alternative 7: 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/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. associate-*l* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 8: 49.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 37.5%

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

   1. *-commutative 37.5%

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

   2. associate-/l* 37.5%

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

5. Simplified 37.5%

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

   1. sqrt-unprod 37.6%

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

   2. associate-*r/ 37.7%

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

   3. *-commutative 37.7%

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

   4. associate-/l* 37.6%

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

7. Applied egg-rr 37.6%

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

8. Taylor expanded in n around 0 37.7%

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

   1. associate-/l* 37.6%

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

10. Simplified 37.6%

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

    1. associate-*r* 37.6%

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

    2. *-commutative 37.6%

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

    3. sqrt-prod 50.3%

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

12. Applied egg-rr 50.3%

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

13. Final simplification 50.3%

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

Alternative 9: 49.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(Pi * N[(2.0 * n), $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. Add Preprocessing

3. Taylor expanded in k around 0 37.5%

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

   1. *-commutative 37.5%

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

   2. associate-/l* 37.5%

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

5. Simplified 37.5%

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

   1. sqrt-unprod 37.6%

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

   2. associate-*r/ 37.7%

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

   3. *-commutative 37.7%

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

   4. associate-/l* 37.6%

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

7. Applied egg-rr 37.6%

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

8. Taylor expanded in n around 0 37.7%

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

   1. associate-/l* 37.6%

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

10. Simplified 37.6%

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

    1. associate-*r* 37.6%

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

    2. *-commutative 37.6%

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

    3. associate-*r/ 37.7%

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

    4. *-commutative 37.7%

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

    5. sqrt-div 50.3%

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

12. Applied egg-rr 50.3%

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

13. Final simplification 50.3%

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

Alternative 10: 38.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 37.5%

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

   1. *-commutative 37.5%

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

   2. associate-/l* 37.5%

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

5. Simplified 37.5%

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

   1. sqrt-unprod 37.6%

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

   2. associate-*r/ 37.7%

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

   3. *-commutative 37.7%

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

   4. associate-/l* 37.6%

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

7. Applied egg-rr 37.6%

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

8. Taylor expanded in n around 0 37.7%

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

   1. associate-/l* 37.6%

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

10. Simplified 37.6%

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

11. Final simplification 37.6%

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

Alternative 11: 38.2% 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 37.5%

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

   1. *-commutative 37.5%

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

   2. associate-/l* 37.5%

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

5. Simplified 37.5%

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

   1. sqrt-unprod 37.6%

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

   2. associate-*r/ 37.7%

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

   3. *-commutative 37.7%

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

   4. associate-/l* 37.6%

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

7. Applied egg-rr 37.6%

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

8. Final simplification 37.6%

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

Alternative 12: 38.2% 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 37.5%

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

   1. *-commutative 37.5%

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

   2. associate-/l* 37.5%

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

5. Simplified 37.5%

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

   1. sqrt-unprod 37.6%

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

   2. associate-*r/ 37.7%

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

   3. *-commutative 37.7%

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

   4. associate-/l* 37.6%

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

7. Applied egg-rr 37.6%

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

8. Taylor expanded in n around 0 37.7%

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

   1. associate-/l* 37.6%

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

10. Simplified 37.6%

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

    1. *-commutative 37.6%

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

    2. associate-*l/ 37.7%

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

    3. associate-*r/ 37.6%

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

    4. clear-num 37.6%

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

    5. un-div-inv 37.7%

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

12. Applied egg-rr 37.7%

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

13. Final simplification 37.7%

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

Alternative 13: 38.2% accurate, 2.0× speedup

Math

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

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(Float64(pi * 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[(N[(Pi * n), $MachinePrecision] / k), $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 37.5%

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

   1. *-commutative 37.5%

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

   2. associate-/l* 37.5%

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

5. Simplified 37.5%

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

   1. sqrt-unprod 37.6%

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

   2. associate-*r/ 37.7%

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

   3. *-commutative 37.7%

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

   4. associate-/l* 37.6%

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

7. Applied egg-rr 37.6%

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

   1. associate-*r/ 37.7%

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

9. Applied egg-rr 37.7%

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

10. Final simplification 37.7%

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

Reproduce

herbie shell --seed 2024057 
(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))))