Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 13.2s

Alternatives: 18

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot n\right)\\ \frac{\sqrt{\frac{1}{k}} \cdot \sqrt{t\_0}}{{t\_0}^{\left(k \cdot 0.5\right)}} \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* 2.0 (* PI n))))
   (/ (* (sqrt (/ 1.0 k)) (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 (sqrt((1.0 / k)) * 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.sqrt((1.0 / k)) * 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.sqrt((1.0 / k)) * 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(sqrt(Float64(1.0 / k)) * sqrt(t_0)) / (t_0 ^ Float64(k * 0.5)))
end

MATLAB

function tmp = code(k, n)
	t_0 = 2.0 * (pi * n);
	tmp = (sqrt((1.0 / k)) * 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[Sqrt[N[(1.0 / k), $MachinePrecision]], $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. Taylor expanded in k around 0 99.7%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(k, n)
	t_0 = 2.0 * (pi * n);
	tmp = (sqrt(t_0) * (k ^ -0.5)) / (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[Sqrt[t$95$0], $MachinePrecision] * N[Power[k, -0.5], $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%

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

Alternative 3: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Power[t$95$0, N[(k * -0.5), $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. Taylor expanded in k around inf 96.8%

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

   1. exp-to-pow 99.5%

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

   2. *-commutative 99.5%

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

   3. cancel-sign-sub-inv 99.5%

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

   4. unpow-prod-up 99.7%

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

   5. pow1/2 99.7%

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

   6. associate-*r* 99.7%

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

   7. *-commutative 99.7%

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

   8. associate-*r* 99.7%

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

   9. *-commutative 99.7%

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

   10. metadata-eval 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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-/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 99.5%

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

   5. fabs-sqr 99.5%

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

   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 31.3%

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

   19. pow-sqr 31.3%

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

   20. metadata-eval 31.3%

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

   21. unpow1/2 31.3%

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

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. Add Preprocessing

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-63}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{\pi}}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.7e-63)
   (* (/ 1.0 (sqrt (/ k PI))) (sqrt (* 2.0 n)))
   (pow (/ k (pow (* n (* 2.0 PI)) (- 1.0 k))) -0.5)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.7e-63) {
		tmp = (1.0 / sqrt((k / ((double) M_PI)))) * sqrt((2.0 * n));
	} else {
		tmp = pow((k / pow((n * (2.0 * ((double) M_PI))), (1.0 - k))), -0.5);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.7e-63) {
		tmp = (1.0 / Math.sqrt((k / Math.PI))) * Math.sqrt((2.0 * n));
	} else {
		tmp = Math.pow((k / Math.pow((n * (2.0 * Math.PI)), (1.0 - k))), -0.5);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.7e-63:
		tmp = (1.0 / math.sqrt((k / math.pi))) * math.sqrt((2.0 * n))
	else:
		tmp = math.pow((k / math.pow((n * (2.0 * math.pi)), (1.0 - k))), -0.5)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.7e-63)
		tmp = Float64(Float64(1.0 / sqrt(Float64(k / pi))) * sqrt(Float64(2.0 * n)));
	else
		tmp = Float64(k / (Float64(n * Float64(2.0 * pi)) ^ Float64(1.0 - k))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.7e-63)
		tmp = (1.0 / sqrt((k / pi))) * sqrt((2.0 * n));
	else
		tmp = (k / ((n * (2.0 * pi)) ^ (1.0 - k))) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.7e-63], N[(N[(1.0 / N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(k / N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.7000000000000002e-63

   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 66.5%

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

      1. *-commutative 66.5%

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

      2. associate-/l* 66.6%

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

   5. Simplified 66.6%

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

      1. sqrt-unprod 65.7%

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

   7. Applied egg-rr 65.7%

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

      1. pow1/2 65.7%

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

      2. associate-*r* 65.7%

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

      3. unpow-prod-down 98.4%

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

      4. pow1/2 98.4%

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

   9. Applied egg-rr 98.4%

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

       1. *-commutative 98.4%

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

       2. unpow1/2 98.4%

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

       3. *-commutative 98.4%

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

   11. Simplified 98.4%

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

       1. clear-num 98.4%

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

       2. sqrt-div 99.3%

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

       3. metadata-eval 99.3%

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

   13. Applied egg-rr 99.3%

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

   if 2.7000000000000002e-63 < 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 99.6%

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

      1. distribute-rgt-in 99.6%

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

      2. metadata-eval 99.6%

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

      3. associate-*l* 99.6%

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

      4. metadata-eval 99.6%

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

      5. *-commutative 99.6%

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

      6. neg-mul-1 99.6%

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

      7. sub-neg 99.6%

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

      8. *-commutative 99.6%

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

   6. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. sqrt-div 99.6%

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

      3. metadata-eval 99.6%

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

      4. associate-*r* 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. pow1/2 99.6%

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

      2. pow-flip 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*r* 99.6%

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

      5. metadata-eval 99.6%

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

   10. Applied egg-rr 99.6%

       \[\leadsto \color{blue}{{\left(\frac{k}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-63}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{\pi}}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{\pi}}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.3e-36)
   (* (/ 1.0 (sqrt (/ k PI))) (sqrt (* 2.0 n)))
   (sqrt (/ (pow (* 2.0 (* PI n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.3e-36) {
		tmp = (1.0 / sqrt((k / ((double) M_PI)))) * sqrt((2.0 * n));
	} 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 <= 2.3e-36) {
		tmp = (1.0 / Math.sqrt((k / Math.PI))) * Math.sqrt((2.0 * n));
	} 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 <= 2.3e-36:
		tmp = (1.0 / math.sqrt((k / math.pi))) * math.sqrt((2.0 * n))
	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 <= 2.3e-36)
		tmp = Float64(Float64(1.0 / sqrt(Float64(k / pi))) * sqrt(Float64(2.0 * n)));
	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 <= 2.3e-36)
		tmp = (1.0 / sqrt((k / pi))) * sqrt((2.0 * n));
	else
		tmp = sqrt((((2.0 * (pi * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.3e-36], N[(N[(1.0 / N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * n), $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 < 2.29999999999999996e-36

   1. Initial program 99.2%

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

   3. Taylor expanded in k around 0 70.0%

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

      1. *-commutative 70.0%

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

      2. associate-/l* 70.0%

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

   5. Simplified 70.0%

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

      1. sqrt-unprod 69.2%

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

   7. Applied egg-rr 69.2%

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

      1. pow1/2 69.2%

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

      2. associate-*r* 69.2%

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

      3. unpow-prod-down 98.5%

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

      4. pow1/2 98.5%

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

   9. Applied egg-rr 98.5%

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

       1. *-commutative 98.5%

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

       2. unpow1/2 98.5%

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

       3. *-commutative 98.5%

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

   11. Simplified 98.5%

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

       1. clear-num 98.6%

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

       2. sqrt-div 99.3%

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

       3. metadata-eval 99.3%

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

   13. Applied egg-rr 99.3%

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

   if 2.29999999999999996e-36 < k

   1. Initial program 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. distribute-rgt-in 99.6%

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

      2. metadata-eval 99.6%

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

      3. associate-*l* 99.6%

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

      4. metadata-eval 99.6%

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

      5. *-commutative 99.6%

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

      6. neg-mul-1 99.6%

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

      7. sub-neg 99.6%

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

      8. *-commutative 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{\pi}}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.6 \cdot 10^{+32}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{\pi}}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \mathsf{fma}\left(2, n \cdot \frac{\pi}{k}, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5.6e+32)
   (* (/ 1.0 (sqrt (/ k PI))) (sqrt (* 2.0 n)))
   (sqrt (+ 1.0 (fma 2.0 (* n (/ PI k)) -1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5.6e+32) {
		tmp = (1.0 / sqrt((k / ((double) M_PI)))) * sqrt((2.0 * n));
	} else {
		tmp = sqrt((1.0 + fma(2.0, (n * (((double) M_PI) / k)), -1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5.6e+32)
		tmp = Float64(Float64(1.0 / sqrt(Float64(k / pi))) * sqrt(Float64(2.0 * n)));
	else
		tmp = sqrt(Float64(1.0 + fma(2.0, Float64(n * Float64(pi / k)), -1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5.6e+32], N[(N[(1.0 / N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(1.0 + N[(2.0 * N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.6e32

   1. Initial program 98.9%

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

   3. Taylor expanded in k around 0 61.6%

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

      1. *-commutative 61.6%

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

      2. associate-/l* 61.6%

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

   5. Simplified 61.6%

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

      1. sqrt-unprod 61.0%

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

   7. Applied egg-rr 61.0%

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

      1. pow1/2 61.0%

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

      2. associate-*r* 61.0%

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

      3. unpow-prod-down 84.3%

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

      4. pow1/2 84.3%

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

   9. Applied egg-rr 84.3%

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

       1. *-commutative 84.3%

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

       2. unpow1/2 84.3%

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

       3. *-commutative 84.3%

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

   11. Simplified 84.3%

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

       1. clear-num 84.3%

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

       2. sqrt-div 84.9%

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

       3. metadata-eval 84.9%

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

   13. Applied egg-rr 84.9%

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

   if 5.6e32 < 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 2.7%

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

      1. *-commutative 2.7%

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

      2. associate-/l* 2.7%

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

   5. Simplified 2.7%

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

      1. sqrt-unprod 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. associate-*r/ 2.7%

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

      2. *-commutative 2.7%

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

   9. Applied egg-rr 2.7%

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

       1. associate-*r/ 2.7%

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

       2. *-commutative 2.7%

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

       3. associate-*l* 2.7%

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

       4. associate-*r/ 2.7%

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

       5. expm1-log1p-u 2.7%

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

       6. expm1-undefine 26.6%

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

       7. *-commutative 26.6%

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

       8. associate-*l* 26.6%

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

   11. Applied egg-rr 26.6%

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

       1. log1p-undefine 26.6%

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

       2. rem-exp-log 26.6%

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

       3. associate-+r- 26.6%

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

       4. associate-*r* 26.6%

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

       5. associate-*r/ 26.6%

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

       6. *-commutative 26.6%

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

       7. associate-*r* 26.6%

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

       8. associate-*r/ 26.6%

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

       9. associate-*r/ 26.6%

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

       10. fmm-def 26.6%

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

       11. metadata-eval 26.6%

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

   13. Simplified 26.6%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.6 \cdot 10^{+32}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{\pi}}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \mathsf{fma}\left(2, n \cdot \frac{\pi}{k}, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \mathsf{fma}\left(2, n \cdot \frac{\pi}{k}, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.5e+30)
   (/ (sqrt (* n (* 2.0 PI))) (sqrt k))
   (sqrt (+ 1.0 (fma 2.0 (* n (/ PI k)) -1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.5e+30) {
		tmp = sqrt((n * (2.0 * ((double) M_PI)))) / sqrt(k);
	} else {
		tmp = sqrt((1.0 + fma(2.0, (n * (((double) M_PI) / k)), -1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.5e+30)
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * pi))) / sqrt(k));
	else
		tmp = sqrt(Float64(1.0 + fma(2.0, Float64(n * Float64(pi / k)), -1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.5e+30], N[(N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(1.0 + N[(2.0 * N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4999999999999999e30

   1. Initial program 98.9%

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

   3. Taylor expanded in k around 0 62.0%

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

      1. *-commutative 62.0%

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

      2. associate-/l* 62.1%

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

   5. Simplified 62.1%

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

      1. sqrt-unprod 61.5%

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

   7. Applied egg-rr 61.5%

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

      1. associate-*r/ 61.5%

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

      2. *-commutative 61.5%

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

   9. Applied egg-rr 61.5%

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

       1. associate-*r/ 61.5%

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

       2. *-commutative 61.5%

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

       3. associate-*l* 61.5%

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

       4. sqrt-div 85.5%

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

       5. *-commutative 85.5%

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

       6. associate-*r* 85.5%

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

   11. Applied egg-rr 85.5%

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

   if 2.4999999999999999e30 < 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 2.7%

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

      1. *-commutative 2.7%

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

      2. associate-/l* 2.7%

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

   5. Simplified 2.7%

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

      1. sqrt-unprod 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. associate-*r/ 2.7%

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

      2. *-commutative 2.7%

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

   9. Applied egg-rr 2.7%

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

       1. associate-*r/ 2.7%

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

       2. *-commutative 2.7%

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

       3. associate-*l* 2.7%

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

       4. associate-*r/ 2.7%

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

       5. expm1-log1p-u 2.7%

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

       6. expm1-undefine 26.4%

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

       7. *-commutative 26.4%

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

       8. associate-*l* 26.4%

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

   11. Applied egg-rr 26.4%

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

       1. log1p-undefine 26.4%

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

       2. rem-exp-log 26.4%

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

       3. associate-+r- 26.4%

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

       4. associate-*r* 26.4%

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

       5. associate-*r/ 26.4%

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

       6. *-commutative 26.4%

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

       7. associate-*r* 26.4%

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

       8. associate-*r/ 26.4%

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

       9. associate-*r/ 26.4%

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

       10. fmm-def 26.4%

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

       11. metadata-eval 26.4%

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

   13. Simplified 26.4%

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

Alternative 9: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 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 99.5%

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

4. Final simplification 99.5%

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

Alternative 10: 99.5% 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. metadata-eval 99.5%

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

   2. div-sub 99.5%

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

   3. associate-*r* 99.5%

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

   4. div-inv 99.4%

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

   5. associate-*r* 99.4%

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

   6. div-sub 99.4%

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

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. div-inv 99.4%

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

   10. metadata-eval 99.4%

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

   11. distribute-rgt-neg-in 99.4%

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

   12. metadata-eval 99.4%

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

   13. pow1/2 99.4%

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

   14. pow-flip 99.5%

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

   15. 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 11: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

Alternative 12: 49.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(n * N[(2.0 * Pi), $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 33.0%

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

   1. *-commutative 33.0%

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

   2. associate-/l* 33.1%

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

5. Simplified 33.1%

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

   1. sqrt-unprod 32.7%

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

7. Applied egg-rr 32.7%

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

   1. associate-*r/ 32.8%

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

   2. *-commutative 32.8%

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

9. Applied egg-rr 32.8%

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

    1. associate-*r/ 32.8%

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

    2. *-commutative 32.8%

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

    3. associate-*l* 32.8%

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

    4. sqrt-div 45.1%

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

    5. *-commutative 45.1%

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

    6. associate-*r* 45.1%

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

11. Applied egg-rr 45.1%

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

Alternative 13: 49.4% 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 33.0%

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

   1. *-commutative 33.0%

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

   2. associate-/l* 33.1%

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

5. Simplified 33.1%

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

   1. sqrt-unprod 32.7%

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

7. Applied egg-rr 32.7%

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

   1. pow1/2 32.7%

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

   2. associate-*r* 32.7%

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

   3. unpow-prod-down 44.8%

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

   4. pow1/2 44.8%

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

9. Applied egg-rr 44.8%

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

    1. *-commutative 44.8%

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

    2. unpow1/2 44.8%

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

    3. *-commutative 44.8%

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

11. Simplified 44.8%

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

12. Final simplification 44.8%

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

Alternative 14: 49.4% 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 33.0%

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

   1. *-commutative 33.0%

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

   2. associate-/l* 33.1%

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

5. Simplified 33.1%

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

   1. sqrt-unprod 32.7%

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

7. Applied egg-rr 32.7%

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

   1. sqrt-prod 33.1%

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

   2. *-commutative 33.1%

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

   3. sqrt-unprod 44.8%

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

   4. associate-*r* 44.7%

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

   5. sqrt-unprod 44.8%

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

9. Applied egg-rr 44.8%

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

Alternative 15: 38.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{k \cdot \frac{\frac{0.5}{n}}{\pi}}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (/ 1.0 (sqrt (* k (/ (/ 0.5 n) PI)))))

C

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

Java

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

Python

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

Julia

function code(k, n)
	return Float64(1.0 / sqrt(Float64(k * Float64(Float64(0.5 / n) / pi))))
end

MATLAB

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

Wolfram

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

4. Applied egg-rr 87.2%

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

   1. distribute-rgt-in 87.2%

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

   2. metadata-eval 87.2%

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

   3. associate-*l* 87.2%

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

   4. metadata-eval 87.2%

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

   5. *-commutative 87.2%

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

   6. neg-mul-1 87.2%

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

   7. sub-neg 87.2%

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

   8. *-commutative 87.2%

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

6. Simplified 87.2%

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

   1. clear-num 87.2%

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

   2. sqrt-div 88.5%

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

   3. metadata-eval 88.5%

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

   4. associate-*r* 88.5%

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

8. Applied egg-rr 88.5%

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

9. Taylor expanded in k around 0 34.1%

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

    1. *-commutative 34.1%

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

    2. associate-*l/ 34.1%

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

    3. associate-*r/ 34.1%

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

    4. associate-/r* 34.1%

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

11. Simplified 34.1%

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

Alternative 16: 38.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(2.0 / N[(k / N[(Pi * 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 33.0%

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

   1. *-commutative 33.0%

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

   2. associate-/l* 33.1%

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

5. Simplified 33.1%

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

   1. sqrt-unprod 32.7%

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

7. Applied egg-rr 32.7%

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

   1. associate-*r/ 32.8%

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

   2. *-commutative 32.8%

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

9. Applied egg-rr 32.8%

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

    1. clear-num 32.8%

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

    2. un-div-inv 32.8%

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

    3. *-commutative 32.8%

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

11. Applied egg-rr 32.8%

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

12. Final simplification 32.8%

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

Alternative 17: 38.0% 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 33.0%

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

   1. *-commutative 33.0%

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

   2. associate-/l* 33.1%

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

5. Simplified 33.1%

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

   1. sqrt-unprod 32.7%

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

7. Applied egg-rr 32.7%

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

   1. associate-*r/ 32.8%

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

   2. *-commutative 32.8%

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

9. Applied egg-rr 32.8%

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

10. Taylor expanded in n around 0 32.8%

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

    1. *-commutative 32.8%

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

    2. associate-*r/ 32.8%

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

12. Simplified 32.8%

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

Alternative 18: 38.0% 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 33.0%

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

   1. *-commutative 33.0%

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

   2. associate-/l* 33.1%

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

5. Simplified 33.1%

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

   1. sqrt-unprod 32.7%

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

7. Applied egg-rr 32.7%

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

Reproduce

herbie shell --seed 2024146 
(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))))