Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 98.5%

Time: 14.0s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{\pi \cdot \frac{2}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.5e-75)
   (* (sqrt (* PI (/ 2.0 k))) (sqrt n))
   (sqrt (/ (pow (* PI (* n 2.0)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.5e-75) {
		tmp = sqrt((((double) M_PI) * (2.0 / k))) * sqrt(n);
	} else {
		tmp = sqrt((pow((((double) M_PI) * (n * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.5e-75) {
		tmp = Math.sqrt((Math.PI * (2.0 / k))) * Math.sqrt(n);
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (n * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 4.5e-75:
		tmp = math.sqrt((math.pi * (2.0 / k))) * math.sqrt(n)
	else:
		tmp = math.sqrt((math.pow((math.pi * (n * 2.0)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.5e-75)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 / k))) * sqrt(n));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(n * 2.0)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 4.5e-75)
		tmp = sqrt((pi * (2.0 / k))) * sqrt(n);
	else
		tmp = sqrt((((pi * (n * 2.0)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.5e-75], N[(N[Sqrt[N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.5000000000000003e-75

   1. Initial program 98.3%

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

   3. Taylor expanded in k around 0 68.2%

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

      1. *-commutative 68.2%

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

      2. associate-/l* 68.2%

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

   5. Simplified 68.2%

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

      1. pow1 68.2%

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

      2. sqrt-unprod 68.5%

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

      3. clear-num 68.5%

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

      4. un-div-inv 68.4%

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

   7. Applied egg-rr 68.4%

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

      1. unpow1 68.4%

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

      2. associate-/r/ 68.4%

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

   9. Simplified 68.4%

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

       1. sqrt-prod 68.2%

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

       2. associate-*l/ 68.2%

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

       3. associate-/l* 68.2%

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

       4. sqrt-prod 99.1%

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

       5. *-commutative 99.1%

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

       6. associate-*r* 99.1%

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

       7. sqrt-unprod 99.4%

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

   11. Applied egg-rr 99.4%

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

       1. associate-*r/ 99.4%

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

       2. *-commutative 99.4%

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

       3. associate-/l* 99.5%

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

   13. Simplified 99.5%

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

   if 4.5000000000000003e-75 < k

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 99.6%

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

      2. sqrt-unprod 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r* 99.7%

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

      5. div-sub 99.7%

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

      6. metadata-eval 99.7%

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

      7. div-inv 99.7%

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

      8. *-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{\pi \cdot \frac{2}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l/ 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. associate-*r* 99.3%

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

   4. div-sub 99.3%

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

   5. metadata-eval 99.3%

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

   6. pow-div 99.4%

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

   7. pow1/2 99.4%

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

   8. associate-/l/ 99.4%

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

   9. div-inv 99.4%

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

   10. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. *-commutative 99.4%

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

   4. *-commutative 99.4%

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

   5. associate-*l* 99.4%

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

   6. *-commutative 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Alternative 3: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.1:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.1) (* (sqrt n) (sqrt (* 2.0 (/ PI k)))) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.1) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = sqrt(0.0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.1) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.1:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (math.pi / k)))
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.1)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = sqrt(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.1)
		tmp = sqrt(n) * sqrt((2.0 * (pi / k)));
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.1], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.10000000000000009

   1. Initial program 98.2%

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

   3. Taylor expanded in k around 0 74.3%

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

      1. *-commutative 74.3%

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

      2. associate-/l* 74.2%

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

   5. Simplified 74.2%

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

      1. pow1 74.2%

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

      2. sqrt-unprod 74.5%

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

      3. clear-num 74.5%

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

      4. un-div-inv 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. unpow1 74.5%

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

      2. associate-/r/ 74.5%

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

   9. Simplified 74.5%

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

       1. sqrt-prod 74.3%

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

       2. associate-*l/ 74.3%

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

       3. associate-/l* 74.2%

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

       4. sqrt-prod 96.4%

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

       5. *-commutative 96.4%

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

       6. associate-*r* 96.3%

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

       7. sqrt-unprod 96.7%

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

   11. Applied egg-rr 96.7%

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

   if 3.10000000000000009 < 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. pow1 2.7%

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

      2. sqrt-unprod 2.7%

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

      3. clear-num 2.7%

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

      4. un-div-inv 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. associate-/r/ 2.7%

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

   9. Simplified 2.7%

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

       1. associate-*l/ 2.7%

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

       2. expm1-log1p-u 2.7%

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

       3. expm1-undefine 22.7%

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

       4. associate-*l/ 22.7%

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

       5. *-commutative 22.7%

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

       6. clear-num 22.7%

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

       7. un-div-inv 22.7%

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

   11. Applied egg-rr 22.7%

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

       1. sub-neg 22.7%

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

       2. metadata-eval 22.7%

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

       3. +-commutative 22.7%

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

       4. log1p-undefine 22.7%

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

       5. rem-exp-log 22.7%

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

       6. +-commutative 22.7%

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

       7. associate-/r/ 22.7%

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

       8. *-commutative 22.7%

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

       9. fma-define 22.7%

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

   13. Simplified 22.7%

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

   14. Taylor expanded in n around 0 51.1%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.2:\\ \;\;\;\;\sqrt{\pi \cdot \frac{2}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.2) (* (sqrt (* PI (/ 2.0 k))) (sqrt n)) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.2) {
		tmp = sqrt((((double) M_PI) * (2.0 / k))) * sqrt(n);
	} else {
		tmp = sqrt(0.0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.2) {
		tmp = Math.sqrt((Math.PI * (2.0 / k))) * Math.sqrt(n);
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.2:
		tmp = math.sqrt((math.pi * (2.0 / k))) * math.sqrt(n)
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.2)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 / k))) * sqrt(n));
	else
		tmp = sqrt(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.2)
		tmp = sqrt((pi * (2.0 / k))) * sqrt(n);
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.2], N[(N[Sqrt[N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.2000000000000002

   1. Initial program 98.2%

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

   3. Taylor expanded in k around 0 74.3%

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

      1. *-commutative 74.3%

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

      2. associate-/l* 74.2%

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

   5. Simplified 74.2%

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

      1. pow1 74.2%

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

      2. sqrt-unprod 74.5%

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

      3. clear-num 74.5%

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

      4. un-div-inv 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. unpow1 74.5%

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

      2. associate-/r/ 74.5%

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

   9. Simplified 74.5%

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

       1. sqrt-prod 74.3%

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

       2. associate-*l/ 74.3%

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

       3. associate-/l* 74.2%

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

       4. sqrt-prod 96.4%

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

       5. *-commutative 96.4%

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

       6. associate-*r* 96.3%

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

       7. sqrt-unprod 96.7%

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

   11. Applied egg-rr 96.7%

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

       1. associate-*r/ 96.7%

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

       2. *-commutative 96.7%

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

       3. associate-/l* 96.7%

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

   13. Simplified 96.7%

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

   if 3.2000000000000002 < 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. pow1 2.7%

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

      2. sqrt-unprod 2.7%

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

      3. clear-num 2.7%

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

      4. un-div-inv 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. associate-/r/ 2.7%

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

   9. Simplified 2.7%

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

       1. associate-*l/ 2.7%

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

       2. expm1-log1p-u 2.7%

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

       3. expm1-undefine 22.7%

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

       4. associate-*l/ 22.7%

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

       5. *-commutative 22.7%

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

       6. clear-num 22.7%

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

       7. un-div-inv 22.7%

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

   11. Applied egg-rr 22.7%

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

       1. sub-neg 22.7%

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

       2. metadata-eval 22.7%

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

       3. +-commutative 22.7%

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

       4. log1p-undefine 22.7%

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

       5. rem-exp-log 22.7%

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

       6. +-commutative 22.7%

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

       7. associate-/r/ 22.7%

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

       8. *-commutative 22.7%

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

       9. fma-define 22.7%

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

   13. Simplified 22.7%

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

   14. Taylor expanded in n around 0 51.1%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2:\\ \;\;\;\;\sqrt{\pi \cdot \frac{2}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 99.2%

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

   1. associate-*l/ 99.3%

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

   2. *-lft-identity 99.3%

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

   3. associate-*l* 99.3%

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

   4. div-sub 99.3%

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

   5. metadata-eval 99.3%

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 6: 63.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.9:\\ \;\;\;\;\frac{1}{\sqrt{0.5 \cdot \frac{\frac{k}{\pi}}{n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.9) (/ 1.0 (sqrt (* 0.5 (/ (/ k PI) n)))) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.9) {
		tmp = 1.0 / sqrt((0.5 * ((k / ((double) M_PI)) / n)));
	} else {
		tmp = sqrt(0.0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.9) {
		tmp = 1.0 / Math.sqrt((0.5 * ((k / Math.PI) / n)));
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.9:
		tmp = 1.0 / math.sqrt((0.5 * ((k / math.pi) / n)))
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.9)
		tmp = Float64(1.0 / sqrt(Float64(0.5 * Float64(Float64(k / pi) / n))));
	else
		tmp = sqrt(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.9)
		tmp = 1.0 / sqrt((0.5 * ((k / pi) / n)));
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.9], N[(1.0 / N[Sqrt[N[(0.5 * N[(N[(k / Pi), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.89999999999999991

   1. Initial program 98.2%

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

   3. Taylor expanded in k around 0 74.3%

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

      1. *-commutative 74.3%

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

      2. associate-/l* 74.2%

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

   5. Simplified 74.2%

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

      1. pow1 74.2%

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

      2. sqrt-unprod 74.5%

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

      3. clear-num 74.5%

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

      4. un-div-inv 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. unpow1 74.5%

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

      2. associate-/r/ 74.5%

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

   9. Simplified 74.5%

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

   10. Taylor expanded in n around 0 74.5%

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

       1. associate-*r/ 74.5%

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

   12. Simplified 74.5%

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

       1. sqrt-prod 74.2%

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

       2. associate-*r/ 74.3%

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

       3. *-commutative 74.3%

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

       4. clear-num 74.3%

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

       5. sqrt-prod 74.5%

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

       6. div-inv 74.5%

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

       7. clear-num 74.5%

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

       8. sqrt-div 75.5%

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

       9. metadata-eval 75.5%

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

       10. div-inv 75.5%

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

       11. metadata-eval 75.5%

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

   14. Applied egg-rr 75.5%

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

       1. *-commutative 75.5%

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

       2. associate-/r* 75.5%

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

   16. Simplified 75.5%

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

   if 2.89999999999999991 < 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. pow1 2.7%

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

      2. sqrt-unprod 2.7%

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

      3. clear-num 2.7%

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

      4. un-div-inv 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. associate-/r/ 2.7%

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

   9. Simplified 2.7%

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

       1. associate-*l/ 2.7%

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

       2. expm1-log1p-u 2.7%

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

       3. expm1-undefine 22.7%

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

       4. associate-*l/ 22.7%

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

       5. *-commutative 22.7%

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

       6. clear-num 22.7%

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

       7. un-div-inv 22.7%

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

   11. Applied egg-rr 22.7%

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

       1. sub-neg 22.7%

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

       2. metadata-eval 22.7%

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

       3. +-commutative 22.7%

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

       4. log1p-undefine 22.7%

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

       5. rem-exp-log 22.7%

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

       6. +-commutative 22.7%

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

       7. associate-/r/ 22.7%

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

       8. *-commutative 22.7%

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

       9. fma-define 22.7%

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

   13. Simplified 22.7%

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

   14. Taylor expanded in n around 0 51.1%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.9:\\ \;\;\;\;\frac{1}{\sqrt{0.5 \cdot \frac{\frac{k}{\pi}}{n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.1:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.1) (sqrt (* 2.0 (* n (/ PI k)))) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.1) {
		tmp = sqrt((2.0 * (n * (((double) M_PI) / k))));
	} else {
		tmp = sqrt(0.0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.1) {
		tmp = Math.sqrt((2.0 * (n * (Math.PI / k))));
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.1:
		tmp = math.sqrt((2.0 * (n * (math.pi / k))))
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.1)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(pi / k))));
	else
		tmp = sqrt(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.1)
		tmp = sqrt((2.0 * (n * (pi / k))));
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.1], N[Sqrt[N[(2.0 * N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.10000000000000009

   1. Initial program 98.2%

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

   3. Taylor expanded in k around 0 74.3%

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

      1. *-commutative 74.3%

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

      2. associate-/l* 74.2%

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

   5. Simplified 74.2%

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

      1. pow1 74.2%

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

      2. sqrt-unprod 74.5%

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

      3. clear-num 74.5%

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

      4. un-div-inv 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. unpow1 74.5%

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

      2. associate-/r/ 74.5%

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

   9. Simplified 74.5%

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

   10. Taylor expanded in n around 0 74.5%

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

       1. associate-*r/ 74.5%

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

   12. Simplified 74.5%

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

   if 3.10000000000000009 < 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. pow1 2.7%

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

      2. sqrt-unprod 2.7%

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

      3. clear-num 2.7%

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

      4. un-div-inv 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. associate-/r/ 2.7%

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

   9. Simplified 2.7%

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

       1. associate-*l/ 2.7%

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

       2. expm1-log1p-u 2.7%

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

       3. expm1-undefine 22.7%

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

       4. associate-*l/ 22.7%

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

       5. *-commutative 22.7%

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

       6. clear-num 22.7%

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

       7. un-div-inv 22.7%

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

   11. Applied egg-rr 22.7%

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

       1. sub-neg 22.7%

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

       2. metadata-eval 22.7%

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

       3. +-commutative 22.7%

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

       4. log1p-undefine 22.7%

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

       5. rem-exp-log 22.7%

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

       6. +-commutative 22.7%

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

       7. associate-/r/ 22.7%

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

       8. *-commutative 22.7%

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

       9. fma-define 22.7%

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

   13. Simplified 22.7%

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

   14. Taylor expanded in n around 0 51.1%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.1:\\ \;\;\;\;\sqrt{2 \cdot \frac{n}{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.1) (sqrt (* 2.0 (/ n (/ k PI)))) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.1) {
		tmp = sqrt((2.0 * (n / (k / ((double) M_PI)))));
	} else {
		tmp = sqrt(0.0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.1) {
		tmp = Math.sqrt((2.0 * (n / (k / Math.PI))));
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.1:
		tmp = math.sqrt((2.0 * (n / (k / math.pi))))
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.1)
		tmp = sqrt(Float64(2.0 * Float64(n / Float64(k / pi))));
	else
		tmp = sqrt(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.1)
		tmp = sqrt((2.0 * (n / (k / pi))));
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.1], N[Sqrt[N[(2.0 * N[(n / N[(k / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.10000000000000009

   1. Initial program 98.2%

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

   3. Taylor expanded in k around 0 74.3%

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

      1. *-commutative 74.3%

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

      2. associate-/l* 74.2%

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

   5. Simplified 74.2%

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

      1. sqrt-unprod 74.5%

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

      2. clear-num 74.5%

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

      3. un-div-inv 74.5%

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

   7. Applied egg-rr 74.5%

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

   if 3.10000000000000009 < 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. pow1 2.7%

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

      2. sqrt-unprod 2.7%

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

      3. clear-num 2.7%

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

      4. un-div-inv 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. associate-/r/ 2.7%

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

   9. Simplified 2.7%

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

       1. associate-*l/ 2.7%

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

       2. expm1-log1p-u 2.7%

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

       3. expm1-undefine 22.7%

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

       4. associate-*l/ 22.7%

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

       5. *-commutative 22.7%

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

       6. clear-num 22.7%

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

       7. un-div-inv 22.7%

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

   11. Applied egg-rr 22.7%

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

       1. sub-neg 22.7%

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

       2. metadata-eval 22.7%

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

       3. +-commutative 22.7%

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

       4. log1p-undefine 22.7%

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

       5. rem-exp-log 22.7%

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

       6. +-commutative 22.7%

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

       7. associate-/r/ 22.7%

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

       8. *-commutative 22.7%

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

       9. fma-define 22.7%

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

   13. Simplified 22.7%

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

   14. Taylor expanded in n around 0 51.1%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1:\\ \;\;\;\;\sqrt{2 \cdot \frac{n}{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 27.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{0} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt 0.0))

C

double code(double k, double n) {
	return sqrt(0.0);
}

Fortran

real(8) function code(k, n)
    real(8), intent (in) :: k
    real(8), intent (in) :: n
    code = sqrt(0.0d0)
end function

Java

public static double code(double k, double n) {
	return Math.sqrt(0.0);
}

Python

def code(k, n):
	return math.sqrt(0.0)

Julia

function code(k, n)
	return sqrt(0.0)
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt(0.0);
end

Wolfram

code[k_, n_] := N[Sqrt[0.0], $MachinePrecision]

Derivation

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 33.2%

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

   1. *-commutative 33.2%

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

   2. associate-/l* 33.2%

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

5. Simplified 33.2%

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

   1. pow1 33.2%

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

   2. sqrt-unprod 33.3%

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

   3. clear-num 33.3%

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

   4. un-div-inv 33.3%

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

7. Applied egg-rr 33.3%

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

   1. unpow1 33.3%

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

   2. associate-/r/ 33.3%

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

9. Simplified 33.3%

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

    1. associate-*l/ 33.3%

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

    2. expm1-log1p-u 31.8%

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

    3. expm1-undefine 34.1%

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

    4. associate-*l/ 34.1%

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

    5. *-commutative 34.1%

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

    6. clear-num 34.1%

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

    7. un-div-inv 34.1%

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

11. Applied egg-rr 34.1%

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

    1. sub-neg 34.1%

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

    2. metadata-eval 34.1%

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

    3. +-commutative 34.1%

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

    4. log1p-undefine 34.1%

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

    5. rem-exp-log 35.6%

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

    6. +-commutative 35.6%

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

    7. associate-/r/ 35.6%

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

    8. *-commutative 35.6%

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

    9. fma-define 35.6%

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

13. Simplified 35.6%

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

14. Taylor expanded in n around 0 30.5%

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

15. Final simplification 30.5%

    \[\leadsto \sqrt{0} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(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))))