Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 16.3s

Alternatives: 16

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 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. Add Preprocessing

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (2.0 * 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 = Math.PI * (2.0 * 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 = math.pi * (2.0 * 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(pi * Float64(2.0 * n))
	return Float64((k ^ -0.5) * Float64(sqrt(t_0) / (t_0 ^ Float64(k * 0.5))))
end

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[k, -0.5], $MachinePrecision] * N[(N[Sqrt[t$95$0], $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-*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. Step-by-step derivation

   1. associate-/l* 99.7%

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

   2. *-rgt-identity 99.7%

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

   3. associate-*l* 99.7%

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

   4. *-lft-identity 99.7%

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

   5. associate-*r* 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.7%

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

   8. associate-*r* 99.7%

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

   9. *-commutative 99.7%

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

   10. associate-*l* 99.7%

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

6. Simplified 99.7%

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

Alternative 3: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(k, n)
	t_0 = 2.0 * (pi * n);
	tmp = (sqrt(t_0) / (t_0 ^ (k * 0.5))) / sqrt(k);
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[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. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 99.4%

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

   2. *-un-lft-identity 99.4%

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

   3. associate-*r* 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

   6. pow-div 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. *-rgt-identity 99.7%

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

   2. *-commutative 99.7%

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

   3. times-frac 99.6%

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

   4. associate-*l/ 99.7%

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

   5. *-lft-identity 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

   1. associate-*l/ 99.4%

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

   2. *-un-lft-identity 99.4%

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

   3. associate-*r* 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

   6. pow-div 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-/l/ 99.7%

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

   2. unpow1/2 99.7%

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

   3. metadata-eval 99.7%

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

   4. pow-sqr 99.5%

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

   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 26.8%

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

   19. pow-sqr 26.8%

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

   20. metadata-eval 26.8%

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

   21. unpow1/2 26.8%

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

   22. neg-mul-1 26.8%

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

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. Final simplification 99.7%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot n\right)\\ \mathbf{if}\;k \leq 2.5 \cdot 10^{-46}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t\_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* 2.0 (* PI n))))
   (if (<= k 2.5e-46)
     (* (pow k -0.5) (sqrt t_0))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = 2.0 * (((double) M_PI) * n);
	double tmp;
	if (k <= 2.5e-46) {
		tmp = pow(k, -0.5) * sqrt(t_0);
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = 2.0 * (Math.PI * n);
	double tmp;
	if (k <= 2.5e-46) {
		tmp = Math.pow(k, -0.5) * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = 2.0 * (math.pi * n)
	tmp = 0
	if k <= 2.5e-46:
		tmp = math.pow(k, -0.5) * math.sqrt(t_0)
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(2.0 * Float64(pi * n))
	tmp = 0.0
	if (k <= 2.5e-46)
		tmp = Float64((k ^ -0.5) * sqrt(t_0));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = 2.0 * (pi * n);
	tmp = 0.0;
	if (k <= 2.5e-46)
		tmp = (k ^ -0.5) * sqrt(t_0);
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2.5e-46], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.49999999999999996e-46

   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. Step-by-step derivation

      1. associate-/r/ 99.3%

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

      2. associate-*r* 99.3%

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

      3. div-sub 99.3%

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

      4. metadata-eval 99.3%

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

      5. sub-neg 99.3%

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

      6. div-inv 99.3%

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

      7. metadata-eval 99.3%

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

      8. distribute-rgt-neg-in 99.3%

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

      9. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in k around 0 75.2%

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

      1. associate-*r/ 75.3%

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

      2. *-rgt-identity 75.3%

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

      3. *-commutative 75.3%

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

      4. associate-/r* 75.3%

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

   7. Simplified 75.3%

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

      1. associate-/l/ 75.3%

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

      2. un-div-inv 75.2%

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

      3. *-commutative 75.2%

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

      4. sqrt-div 99.0%

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

      5. frac-times 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. sqrt-prod 99.3%

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

      8. associate-*r* 99.3%

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

      9. *-commutative 99.3%

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

      10. clear-num 99.4%

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

      11. sqrt-div 71.4%

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

      12. *-commutative 71.4%

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

      13. associate-*r* 71.4%

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

      14. un-div-inv 71.4%

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

      15. *-commutative 71.4%

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

      16. associate-*r* 71.4%

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

      17. *-commutative 71.4%

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

   9. Applied egg-rr 99.5%

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

   if 2.49999999999999996e-46 < k

   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 99.4%

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

      1. *-commutative 99.4%

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

      2. distribute-lft-in 99.4%

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

      3. metadata-eval 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-*r* 99.4%

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

      6. metadata-eval 99.4%

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

      7. neg-mul-1 99.4%

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

      8. sub-neg 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-46}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{-43}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(-1 + \mathsf{fma}\left(\pi, \frac{2}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.05e-43)
   (* (pow k -0.5) (sqrt (* 2.0 (* PI n))))
   (sqrt (* n (+ -1.0 (fma PI (/ 2.0 k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.05e-43) {
		tmp = pow(k, -0.5) * sqrt((2.0 * (((double) M_PI) * n)));
	} else {
		tmp = sqrt((n * (-1.0 + fma(((double) M_PI), (2.0 / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.05e-43)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(2.0 * Float64(pi * n))));
	else
		tmp = sqrt(Float64(n * Float64(-1.0 + fma(pi, Float64(2.0 / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.05e-43], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(n * N[(-1.0 + N[(Pi * N[(2.0 / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e-43

   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. Step-by-step derivation

      1. associate-/r/ 99.3%

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

      2. associate-*r* 99.3%

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

      3. div-sub 99.3%

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

      4. metadata-eval 99.3%

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

      5. sub-neg 99.3%

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

      6. div-inv 99.3%

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

      7. metadata-eval 99.3%

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

      8. distribute-rgt-neg-in 99.3%

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

      9. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in k around 0 75.7%

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

      1. associate-*r/ 75.8%

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

      2. *-rgt-identity 75.8%

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

      3. *-commutative 75.8%

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

      4. associate-/r* 75.8%

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

   7. Simplified 75.8%

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

      1. associate-/l/ 75.8%

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

      2. un-div-inv 75.7%

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

      3. *-commutative 75.7%

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

      4. sqrt-div 99.1%

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

      5. frac-times 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. sqrt-prod 99.3%

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

      8. associate-*r* 99.3%

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

      9. *-commutative 99.3%

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

      10. clear-num 99.4%

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

      11. sqrt-div 71.9%

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

      12. *-commutative 71.9%

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

      13. associate-*r* 71.9%

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

      14. un-div-inv 71.9%

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

      15. *-commutative 71.9%

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

      16. associate-*r* 71.9%

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

      17. *-commutative 71.9%

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

   9. Applied egg-rr 99.5%

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

   if 1.05e-43 < k

   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 13.0%

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

      1. *-commutative 13.0%

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

      2. associate-/l* 13.0%

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

   5. Simplified 13.0%

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

      1. pow1 13.0%

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

      2. sqrt-unprod 13.1%

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

   7. Applied egg-rr 13.1%

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

      1. unpow1 13.1%

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

      2. *-commutative 13.1%

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

      3. associate-*l* 13.1%

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

   9. Simplified 13.1%

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

       1. expm1-log1p-u 12.9%

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

       2. expm1-undefine 52.2%

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

       3. *-commutative 52.2%

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

   11. Applied egg-rr 52.2%

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

       1. sub-neg 52.2%

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

       2. metadata-eval 52.2%

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

       3. +-commutative 52.2%

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

       4. log1p-undefine 52.2%

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

       5. rem-exp-log 52.4%

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

       6. +-commutative 52.4%

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

       7. associate-*r/ 52.4%

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

       8. *-commutative 52.4%

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

       9. associate-/l* 52.4%

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

       10. fma-define 52.4%

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

   13. Simplified 52.4%

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

Alternative 7: 50.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{+240}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 8e+240)
   (* (pow k -0.5) (sqrt (* 2.0 (* PI n))))
   (cbrt (pow (* 2.0 (* n (/ PI k))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 8e+240) {
		tmp = pow(k, -0.5) * sqrt((2.0 * (((double) M_PI) * n)));
	} else {
		tmp = cbrt(pow((2.0 * (n * (((double) M_PI) / k))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 8e+240) {
		tmp = Math.pow(k, -0.5) * Math.sqrt((2.0 * (Math.PI * n)));
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * (n * (Math.PI / k))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 8e+240)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(2.0 * Float64(pi * n))));
	else
		tmp = cbrt((Float64(2.0 * Float64(n * Float64(pi / k))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 8e+240], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.00000000000000011e240

   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. Step-by-step derivation

      1. associate-/r/ 99.3%

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

      2. associate-*r* 99.3%

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

      3. div-sub 99.3%

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

      4. metadata-eval 99.3%

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

      5. sub-neg 99.3%

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

      6. div-inv 99.3%

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

      7. metadata-eval 99.3%

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

      8. distribute-rgt-neg-in 99.3%

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

      9. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in k around 0 45.3%

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

      1. associate-*r/ 45.4%

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

      2. *-rgt-identity 45.4%

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

      3. *-commutative 45.4%

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

      4. associate-/r* 45.4%

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

   7. Simplified 45.4%

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

      1. associate-/l/ 45.4%

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

      2. un-div-inv 45.3%

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

      3. *-commutative 45.3%

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

      4. sqrt-div 56.8%

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

      5. frac-times 56.9%

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

      6. *-un-lft-identity 56.9%

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

      7. sqrt-prod 57.0%

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

      8. associate-*r* 57.0%

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

      9. *-commutative 57.0%

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

      10. clear-num 57.0%

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

      11. sqrt-div 43.5%

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

      12. *-commutative 43.5%

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

      13. associate-*r* 43.5%

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

      14. un-div-inv 43.5%

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

      15. *-commutative 43.5%

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

      16. associate-*r* 43.5%

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

      17. *-commutative 43.5%

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

   9. Applied egg-rr 57.1%

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

   if 8.00000000000000011e240 < 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. pow1 2.9%

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

      2. sqrt-unprod 2.9%

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

   7. Applied egg-rr 2.9%

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

      1. unpow1 2.9%

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

      2. *-commutative 2.9%

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

      3. associate-*l* 2.9%

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

   9. Simplified 2.9%

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

       1. add-cbrt-cube 16.9%

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

       2. pow1/3 16.9%

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

   11. Applied egg-rr 16.9%

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

       1. unpow1/3 16.9%

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

       2. *-commutative 16.9%

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

       3. *-commutative 16.9%

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

       4. associate-/l* 16.9%

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

   13. Simplified 16.9%

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

Alternative 8: 50.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+241}:\\ \;\;\;\;\sqrt{\frac{2}{k}} \cdot \sqrt{\pi \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5e+241)
   (* (sqrt (/ 2.0 k)) (sqrt (* PI n)))
   (cbrt (pow (* 2.0 (* n (/ PI k))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5e+241) {
		tmp = sqrt((2.0 / k)) * sqrt((((double) M_PI) * n));
	} else {
		tmp = cbrt(pow((2.0 * (n * (((double) M_PI) / k))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 5e+241) {
		tmp = Math.sqrt((2.0 / k)) * Math.sqrt((Math.PI * n));
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * (n * (Math.PI / k))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5e+241)
		tmp = Float64(sqrt(Float64(2.0 / k)) * sqrt(Float64(pi * n)));
	else
		tmp = cbrt((Float64(2.0 * Float64(n * Float64(pi / k))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5e+241], N[(N[Sqrt[N[(2.0 / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.00000000000000025e241

   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 43.4%

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

      1. *-commutative 43.4%

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

      2. associate-/l* 43.4%

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

   5. Simplified 43.4%

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

      1. pow1 43.4%

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

      2. sqrt-unprod 43.5%

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

   7. Applied egg-rr 43.5%

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

      1. unpow1 43.5%

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

      2. *-commutative 43.5%

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

      3. associate-*l* 43.5%

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

   9. Simplified 43.5%

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

   10. Taylor expanded in k around 0 43.5%

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

       1. associate-*r/ 43.5%

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

       2. *-commutative 43.5%

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

       3. associate-/l* 43.5%

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

   12. Simplified 43.5%

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

       1. associate-*r* 43.5%

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

       2. *-commutative 43.5%

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

       3. sqrt-prod 57.0%

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

   14. Applied egg-rr 57.0%

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

       1. *-commutative 57.0%

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

       2. *-commutative 57.0%

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

   16. Simplified 57.0%

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

   if 5.00000000000000025e241 < 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. pow1 2.9%

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

      2. sqrt-unprod 2.9%

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

   7. Applied egg-rr 2.9%

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

      1. unpow1 2.9%

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

      2. *-commutative 2.9%

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

      3. associate-*l* 2.9%

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

   9. Simplified 2.9%

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

       1. add-cbrt-cube 16.9%

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

       2. pow1/3 16.9%

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

   11. Applied egg-rr 16.9%

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

       1. unpow1/3 16.9%

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

       2. *-commutative 16.9%

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

       3. *-commutative 16.9%

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

       4. associate-/l* 16.9%

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

   13. Simplified 16.9%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+241}:\\ \;\;\;\;\sqrt{\frac{2}{k}} \cdot \sqrt{\pi \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. associate-*l* 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}} \]
4. Add Preprocessing
5. 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. sub-neg 99.4%

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

   3. div-inv 99.4%

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

   4. metadata-eval 99.4%

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

   5. distribute-rgt-neg-in 99.4%

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

   6. metadata-eval 99.4%

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

   7. pow1/2 99.4%

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

   8. pow-flip 99.5%

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

   9. 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 10: 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/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. associate-*l* 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

3. Simplified 99.4%

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

Alternative 11: 49.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 38.0%

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

   1. *-commutative 38.0%

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

   2. associate-/l* 38.0%

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

5. Simplified 38.0%

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

   1. pow1 38.0%

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

   2. sqrt-unprod 38.1%

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

7. Applied egg-rr 38.1%

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

   1. unpow1 38.1%

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

   2. *-commutative 38.1%

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

   3. associate-*l* 38.1%

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

9. Simplified 38.1%

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

10. Taylor expanded in k around 0 38.1%

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

    1. associate-*r/ 38.1%

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

    2. *-commutative 38.1%

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

    3. associate-/l* 38.1%

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

12. Simplified 38.1%

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

    1. associate-*r* 38.1%

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

    2. *-commutative 38.1%

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

    3. sqrt-prod 49.8%

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

14. Applied egg-rr 49.8%

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

    1. *-commutative 49.8%

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

    2. *-commutative 49.8%

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

16. Simplified 49.8%

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

17. Final simplification 49.8%

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

Alternative 12: 49.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 38.0%

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

   1. *-commutative 38.0%

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

   2. associate-/l* 38.0%

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

5. Simplified 38.0%

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

   1. pow1 38.0%

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

   2. sqrt-unprod 38.1%

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

7. Applied egg-rr 38.1%

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

   1. unpow1 38.1%

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

   2. *-commutative 38.1%

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

   3. associate-*l* 38.1%

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

9. Simplified 38.1%

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

    1. *-commutative 38.1%

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

    2. sqrt-prod 49.8%

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

    3. *-commutative 49.8%

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

11. Applied egg-rr 49.8%

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

Alternative 13: 49.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 38.0%

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

   1. *-commutative 38.0%

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

   2. associate-/l* 38.0%

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

5. Simplified 38.0%

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

   1. pow1 38.0%

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

   2. sqrt-unprod 38.1%

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

7. Applied egg-rr 38.1%

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

   1. unpow1 38.1%

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

   2. *-commutative 38.1%

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

   3. associate-*l* 38.1%

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

9. Simplified 38.1%

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

    1. sqrt-prod 49.8%

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

    2. *-commutative 49.8%

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

11. Applied egg-rr 49.8%

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

    1. associate-*r/ 49.8%

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

    2. *-commutative 49.8%

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

    3. associate-/l* 49.8%

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

13. Simplified 49.8%

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

Alternative 14: 38.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-/r/ 99.4%

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

   2. associate-*r* 99.4%

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

   3. div-sub 99.4%

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

   4. metadata-eval 99.4%

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

   5. sub-neg 99.4%

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

   6. div-inv 99.4%

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

   7. metadata-eval 99.4%

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

   8. distribute-rgt-neg-in 99.4%

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

   9. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in k around 0 39.7%

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

   1. associate-*r/ 39.7%

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

   2. *-rgt-identity 39.7%

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

   3. *-commutative 39.7%

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

   4. associate-/r* 39.7%

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

7. Simplified 39.7%

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

   1. inv-pow 39.7%

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

   2. add-sqr-sqrt 39.6%

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

   3. sqrt-unprod 39.7%

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

   4. sqrt-pow2 39.7%

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

9. Applied egg-rr 39.9%

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

Alternative 15: 37.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 38.0%

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

   1. *-commutative 38.0%

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

   2. associate-/l* 38.0%

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

5. Simplified 38.0%

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

   1. pow1 38.0%

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

   2. sqrt-unprod 38.1%

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

7. Applied egg-rr 38.1%

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

   1. unpow1 38.1%

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

   2. *-commutative 38.1%

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

   3. associate-*l* 38.1%

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

9. Simplified 38.1%

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

10. Final simplification 38.1%

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

Alternative 16: 37.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 38.0%

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

   1. *-commutative 38.0%

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

   2. associate-/l* 38.0%

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

5. Simplified 38.0%

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

   1. pow1 38.0%

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

   2. sqrt-unprod 38.1%

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

7. Applied egg-rr 38.1%

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

   1. unpow1 38.1%

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

   2. *-commutative 38.1%

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

   3. associate-*l* 38.1%

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

9. Simplified 38.1%

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

10. Taylor expanded in k around 0 38.1%

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

    1. associate-*r/ 38.1%

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

    2. *-commutative 38.1%

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

    3. associate-/l* 38.1%

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

12. Simplified 38.1%

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

Reproduce

herbie shell --seed 2024135 
(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))))