Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 11.1s

Alternatives: 10

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 7.651187167493154e+121
  2. n = -7.688943455564925e+52

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, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-sqrt.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. associate-*l/ N/A

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

   11. neg-mul-1 N/A

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

   12. frac-2neg N/A

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

4. Applied rewrites 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. Add Preprocessing

Alternative 2: 68.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{{\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}^{-0.25}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* (/ 1.0 (sqrt k)) (pow (* n (* 2.0 PI)) (/ (- 1.0 k) 2.0))) 0.0)
   (sqrt (pow (* k (* k (* k k))) -0.25))
   (/ (sqrt (* 2.0 (* PI n))) (sqrt k))))

C

double code(double k, double n) {
	double tmp;
	if (((1.0 / sqrt(k)) * pow((n * (2.0 * ((double) M_PI))), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = sqrt(pow((k * (k * (k * k))), -0.25));
	} else {
		tmp = sqrt((2.0 * (((double) M_PI) * n))) / sqrt(k);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (((1.0 / Math.sqrt(k)) * Math.pow((n * (2.0 * Math.PI)), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = Math.sqrt(Math.pow((k * (k * (k * k))), -0.25));
	} else {
		tmp = Math.sqrt((2.0 * (Math.PI * n))) / Math.sqrt(k);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if ((1.0 / math.sqrt(k)) * math.pow((n * (2.0 * math.pi)), ((1.0 - k) / 2.0))) <= 0.0:
		tmp = math.sqrt(math.pow((k * (k * (k * k))), -0.25))
	else:
		tmp = math.sqrt((2.0 * (math.pi * n))) / math.sqrt(k)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(k)) * (Float64(n * Float64(2.0 * pi)) ^ Float64(Float64(1.0 - k) / 2.0))) <= 0.0)
		tmp = sqrt((Float64(k * Float64(k * Float64(k * k))) ^ -0.25));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi * n))) / sqrt(k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (((1.0 / sqrt(k)) * ((n * (2.0 * pi)) ^ ((1.0 - k) / 2.0))) <= 0.0)
		tmp = sqrt(((k * (k * (k * k))) ^ -0.25));
	else
		tmp = sqrt((2.0 * (pi * n))) / sqrt(k);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[Sqrt[N[Power[N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.25], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 0.0

   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 inf

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

      1. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 3.8

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

   8. Applied rewrites 3.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 3.8

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

      2. rem-square-sqrt N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 3.8

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

   10. Applied rewrites 3.8%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt{k} \cdot \sqrt{k}}}} \]
   11. Step-by-step derivation

       1. sqrt-unprod N/A

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

       2. pow1/2 N/A

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

       3. pow-flip N/A

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

       4. rem-square-sqrt N/A

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

       5. sqrt-unprod N/A

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

       6. metadata-eval N/A

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

       7. sqrt-pow2 N/A

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

       8. metadata-eval N/A

          \[\leadsto \sqrt{{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}^{\color{blue}{\frac{-1}{4}}}} \]

       9. metadata-eval N/A

          \[\leadsto \sqrt{{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}^{\color{blue}{\left(-1 \cdot \frac{1}{4}\right)}}} \]

       10. lower-pow.f64 N/A

           \[\leadsto \sqrt{\color{blue}{{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}^{\left(-1 \cdot \frac{1}{4}\right)}}} \]

       11. associate-*l* N/A

           \[\leadsto \sqrt{{\color{blue}{\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}}^{\left(-1 \cdot \frac{1}{4}\right)}} \]

       12. cube-mult N/A

           \[\leadsto \sqrt{{\left(k \cdot \color{blue}{{k}^{3}}\right)}^{\left(-1 \cdot \frac{1}{4}\right)}} \]

       13. lower-*.f64 N/A

           \[\leadsto \sqrt{{\color{blue}{\left(k \cdot {k}^{3}\right)}}^{\left(-1 \cdot \frac{1}{4}\right)}} \]

       14. cube-mult N/A

           \[\leadsto \sqrt{{\left(k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}\right)}^{\left(-1 \cdot \frac{1}{4}\right)}} \]

       15. lower-*.f64 N/A

           \[\leadsto \sqrt{{\left(k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}\right)}^{\left(-1 \cdot \frac{1}{4}\right)}} \]

       16. lower-*.f64 N/A

           \[\leadsto \sqrt{{\left(k \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)}^{\left(-1 \cdot \frac{1}{4}\right)}} \]

       17. metadata-eval 75.0

           \[\leadsto \sqrt{{\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}^{\color{blue}{-0.25}}} \]

   12. Applied rewrites 75.0%

       \[\leadsto \sqrt{\color{blue}{{\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}^{-0.25}}} \]

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   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

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 52.9

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

   5. Applied rewrites 52.9%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lower-/.f64 66.7

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

   7. Applied rewrites 66.7%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{{\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}^{-0.25}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* (/ 1.0 (sqrt k)) (pow (* n (* 2.0 PI)) (/ (- 1.0 k) 2.0))) 0.0)
   (sqrt (sqrt (/ 1.0 (* k k))))
   (/ (sqrt (* 2.0 (* PI n))) (sqrt k))))

C

double code(double k, double n) {
	double tmp;
	if (((1.0 / sqrt(k)) * pow((n * (2.0 * ((double) M_PI))), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = sqrt(sqrt((1.0 / (k * k))));
	} else {
		tmp = sqrt((2.0 * (((double) M_PI) * n))) / sqrt(k);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (((1.0 / Math.sqrt(k)) * Math.pow((n * (2.0 * Math.PI)), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = Math.sqrt(Math.sqrt((1.0 / (k * k))));
	} else {
		tmp = Math.sqrt((2.0 * (Math.PI * n))) / Math.sqrt(k);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if ((1.0 / math.sqrt(k)) * math.pow((n * (2.0 * math.pi)), ((1.0 - k) / 2.0))) <= 0.0:
		tmp = math.sqrt(math.sqrt((1.0 / (k * k))))
	else:
		tmp = math.sqrt((2.0 * (math.pi * n))) / math.sqrt(k)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[Sqrt[N[Sqrt[N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 0.0

   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 inf

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

      1. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 3.8

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

   8. Applied rewrites 3.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 3.8

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

      2. rem-square-sqrt N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 3.8

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

   10. Applied rewrites 3.8%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt{k} \cdot \sqrt{k}}}} \]
   11. Step-by-step derivation

       1. rem-square-sqrt N/A

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

       2. lift-/.f64 3.8

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

       3. rem-square-sqrt N/A

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

       4. sqrt-unprod N/A

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

       5. lift-*.f64 N/A

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

       6. lower-sqrt.f64 45.1

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. frac-times N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f64 N/A

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

       13. lower-*.f64 50.1

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

   12. Applied rewrites 50.1%

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

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   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

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 52.9

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

   5. Applied rewrites 52.9%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lower-/.f64 66.7

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

   7. Applied rewrites 66.7%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{k}} \cdot \sqrt{\pi \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* (/ 1.0 (sqrt k)) (pow (* n (* 2.0 PI)) (/ (- 1.0 k) 2.0))) 0.0)
   (sqrt (sqrt (/ 1.0 (* k k))))
   (* (sqrt (/ 2.0 k)) (sqrt (* PI n)))))

C

double code(double k, double n) {
	double tmp;
	if (((1.0 / sqrt(k)) * pow((n * (2.0 * ((double) M_PI))), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = sqrt(sqrt((1.0 / (k * k))));
	} else {
		tmp = sqrt((2.0 / k)) * sqrt((((double) M_PI) * n));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (((1.0 / Math.sqrt(k)) * Math.pow((n * (2.0 * Math.PI)), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = Math.sqrt(Math.sqrt((1.0 / (k * k))));
	} else {
		tmp = Math.sqrt((2.0 / k)) * Math.sqrt((Math.PI * n));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if ((1.0 / math.sqrt(k)) * math.pow((n * (2.0 * math.pi)), ((1.0 - k) / 2.0))) <= 0.0:
		tmp = math.sqrt(math.sqrt((1.0 / (k * k))))
	else:
		tmp = math.sqrt((2.0 / k)) * math.sqrt((math.pi * n))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[Sqrt[N[Sqrt[N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 0.0

   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 inf

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

      1. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 3.8

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

   8. Applied rewrites 3.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 3.8

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

      2. rem-square-sqrt N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 3.8

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

   10. Applied rewrites 3.8%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt{k} \cdot \sqrt{k}}}} \]
   11. Step-by-step derivation

       1. rem-square-sqrt N/A

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

       2. lift-/.f64 3.8

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

       3. rem-square-sqrt N/A

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

       4. sqrt-unprod N/A

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

       5. lift-*.f64 N/A

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

       6. lower-sqrt.f64 45.1

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. frac-times N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f64 N/A

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

       13. lower-*.f64 50.1

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

   12. Applied rewrites 50.1%

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

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   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

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 52.9

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

   5. Applied rewrites 52.9%

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

      1. lift-PI.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 52.9

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

   7. Applied rewrites 52.9%

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

   9. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{k}} \cdot \sqrt{\pi \cdot n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{k}} \cdot \sqrt{\pi \cdot n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* (/ 1.0 (sqrt k)) (pow (* n (* 2.0 PI)) (/ (- 1.0 k) 2.0))) 0.0)
   (sqrt (sqrt (/ 1.0 (* k k))))
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))))

C

double code(double k, double n) {
	double tmp;
	if (((1.0 / sqrt(k)) * pow((n * (2.0 * ((double) M_PI))), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = sqrt(sqrt((1.0 / (k * k))));
	} else {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (((1.0 / Math.sqrt(k)) * Math.pow((n * (2.0 * Math.PI)), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = Math.sqrt(Math.sqrt((1.0 / (k * k))));
	} else {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if ((1.0 / math.sqrt(k)) * math.pow((n * (2.0 * math.pi)), ((1.0 - k) / 2.0))) <= 0.0:
		tmp = math.sqrt(math.sqrt((1.0 / (k * k))))
	else:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (math.pi / k)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[Sqrt[N[Sqrt[N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 0.0

   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 inf

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

      1. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 3.8

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

   8. Applied rewrites 3.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 3.8

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

      2. rem-square-sqrt N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 3.8

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

   10. Applied rewrites 3.8%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt{k} \cdot \sqrt{k}}}} \]
   11. Step-by-step derivation

       1. rem-square-sqrt N/A

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

       2. lift-/.f64 3.8

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

       3. rem-square-sqrt N/A

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

       4. sqrt-unprod N/A

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

       5. lift-*.f64 N/A

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

       6. lower-sqrt.f64 45.1

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. frac-times N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f64 N/A

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

       13. lower-*.f64 50.1

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

   12. Applied rewrites 50.1%

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

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   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

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 52.9

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

   5. Applied rewrites 52.9%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 66.6

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

   7. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\sqrt{n} \cdot \sqrt{\frac{\pi}{k} \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(k, n)
	t_0 = 2.0 * (pi * n);
	tmp = 0.0;
	if (k <= 1.0)
		tmp = sqrt(t_0) / sqrt(k);
	else
		tmp = (t_0 ^ (k * -0.5)) / sqrt(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, 1.0], N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[(N[Power[t$95$0, N[(k * -0.5), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1

   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

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 76.6

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

   5. Applied rewrites 76.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lower-/.f64 96.9

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

   7. Applied rewrites 96.9%

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

   if 1 < 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 inf

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

      1. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. associate-*l/ N/A

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

      16. *-lft-identity N/A

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

   7. Applied rewrites 100.0%

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

Alternative 7: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-sqrt.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. associate-*l/ N/A

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

   11. neg-mul-1 N/A

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

   12. frac-2neg N/A

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.6%

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

Alternative 8: 50.1% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 0.5) (sqrt (/ (* 2.0 (* PI n)) k)) (sqrt (sqrt (/ 1.0 (* k k))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 0.5

   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

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 76.6

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

   5. Applied rewrites 76.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-/.f64 76.8

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

   7. Applied rewrites 76.8%

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

   if 0.5 < 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 inf

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

      1. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 3.2

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

   8. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 3.2

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

      2. rem-square-sqrt N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 3.2

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

   10. Applied rewrites 3.2%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\sqrt{k} \cdot \sqrt{k}}}} \]
   11. Step-by-step derivation

       1. rem-square-sqrt N/A

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

       2. lift-/.f64 3.2

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

       3. rem-square-sqrt N/A

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

       4. sqrt-unprod N/A

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

       5. lift-*.f64 N/A

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

       6. lower-sqrt.f64 22.7

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. frac-times N/A

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

       11. metadata-eval N/A

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

       12. lower-/.f64 N/A

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

       13. lower-*.f64 25.1

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

   12. Applied rewrites 25.1%

       \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k \cdot k}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 38.3% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 41.6

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

5. Applied rewrites 41.6%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sqrt-unprod N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-/.f64 41.7

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

7. Applied rewrites 41.7%

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

Alternative 10: 5.1% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{k}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (/ 1.0 k)))

C

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

Fortran

real(8) function code(k, n)
    real(8), intent (in) :: k
    real(8), intent (in) :: n
    code = sqrt((1.0d0 / k))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around inf

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

   1. lower-*.f64 50.8

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

5. Applied rewrites 50.8%

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

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
7. Step-by-step derivation

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.0

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

8. Applied rewrites 5.0%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))))