Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 7.0s

Alternatives: 16

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 2.0133193751708975e+120
  2. n = -3.3792283056618906e+23

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. div-sub N/A

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

   8. metadata-eval N/A

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

   9. pow-sub N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lift-+.f64 N/A

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

5. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity 99.5

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. unpow1/2 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

       \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

   11. lift-PI.f64 37.2

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

4. Applied rewrites 37.2%

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

5. Taylor expanded in n around 0

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

   1. sum-log N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-/.f64 N/A

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

7. Applied rewrites 99.4%

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

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. distribute-lft-in N/A

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

   10. count-2-rev N/A

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

   11. sub-flip N/A

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

   12. +-commutative N/A

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

   13. pow-plus-rev N/A

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

   14. lower-*.f64 N/A

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

9. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. unpow1/2 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

       \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

   11. lift-PI.f64 37.2

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

4. Applied rewrites 37.2%

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

5. Taylor expanded in n around 0

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

   1. sum-log N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-/.f64 N/A

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

7. Applied rewrites 99.4%

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

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-sub N/A

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

   4. lift-sqrt.f64 N/A

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

   5. sqrt-pow2 N/A

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

   6. metadata-eval N/A

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

   7. pow1/2 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-PI.f64 N/A

       \[\leadsto \frac{\frac{\sqrt{n \cdot \left(\mathsf{PI}\left(\right) + \pi\right)}}{{\left(\sqrt{\left(\pi + \pi\right) \cdot n}\right)}^{k}}}{\sqrt{k}} \]

   11. lift-PI.f64 N/A

       \[\leadsto \frac{\frac{\sqrt{n \cdot \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}}{{\left(\sqrt{\left(\pi + \pi\right) \cdot n}\right)}^{k}}}{\sqrt{k}} \]

   12. lift-+.f64 N/A

       \[\leadsto \frac{\frac{\sqrt{n \cdot \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}}{{\left(\sqrt{\left(\pi + \pi\right) \cdot n}\right)}^{k}}}{\sqrt{k}} \]

   13. distribute-lft-in N/A

       \[\leadsto \frac{\frac{\sqrt{n \cdot \mathsf{PI}\left(\right) + n \cdot \mathsf{PI}\left(\right)}}{{\left(\sqrt{\left(\pi + \pi\right) \cdot n}\right)}^{k}}}{\sqrt{k}} \]

   14. count-2-rev N/A

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)}}{{\left(\sqrt{\left(\pi + \pi\right) \cdot n}\right)}^{k}}}{\sqrt{k}} \]

   15. lower-/.f64 N/A

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

9. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(\left(\pi + \pi\right) \cdot n\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)} \cdot \sqrt{\frac{1}{k}} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (pow (* (+ PI PI) n) (fma -0.5 k 0.5)) (sqrt (/ 1.0 k))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

3. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-div N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. mult-flip-rev N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.5%

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

Alternative 6: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. unpow1/2 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

       \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

   11. lift-PI.f64 37.2

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

4. Applied rewrites 37.2%

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

5. Taylor expanded in n around 0

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

   1. sum-log N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-/.f64 N/A

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

7. Applied rewrites 99.4%

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

Alternative 7: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   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. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

      11. lift-PI.f64 37.2

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

   4. Applied rewrites 37.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\mathsf{PI}\left(\right) + \pi\right) \cdot n}{k}} \]

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-+.f64 N/A

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

      11. lift-PI.f64 N/A

          \[\leadsto \sqrt{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot \frac{n}{k}} \]

      12. lift-PI.f64 N/A

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

      13. lower-/.f64 37.2

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

   6. Applied rewrites 37.2%

      \[\leadsto \sqrt{\left(\pi + \pi\right) \cdot \frac{n}{k}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

         \[\leadsto \sqrt{\left(\mathsf{PI}\left(\right) + \pi\right) \cdot \frac{n}{k}} \]

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-PI.f64 N/A

         \[\leadsto \sqrt{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot \frac{n}{k}} \]

      9. lift-PI.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\mathsf{PI}\left(\right) + \pi\right) \cdot n}{k}} \]

      12. lift-PI.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. count-2-rev N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

   8. Applied rewrites 49.0%

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

   if 1 < k

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

      11. lift-PI.f64 37.2

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

   4. Applied rewrites 37.2%

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

   5. Taylor expanded in n around 0

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

      1. sum-log N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in k around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{{\left(\sqrt{\left(\pi + \pi\right) \cdot n}\right)}^{\left(\mathsf{neg}\left(k\right)\right)}}{\sqrt{k}} \]

      2. lower-neg.f64 53.9

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

   10. Applied rewrites 53.9%

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

Alternative 8: 76.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\frac{\pi}{k}, n, \frac{\pi \cdot n}{k}\right)}{n \cdot n}} \cdot n\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+285}:\\ \;\;\;\;\sqrt{n \cdot \left(\pi + \pi\right)} \cdot \frac{\sqrt{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{\pi \cdot k}{n} \cdot 2}}{k} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0)))))
   (if (<= t_0 0.0)
     (* (sqrt (/ (fma (/ PI k) n (/ (* PI n) k)) (* n n))) n)
     (if (<= t_0 5e+285)
       (* (sqrt (* n (+ PI PI))) (/ (sqrt k) k))
       (* (/ (sqrt (* (/ (* PI k) n) 2.0)) k) n)))))

C

double code(double k, double n) {
	double t_0 = (1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt((fma((((double) M_PI) / k), n, ((((double) M_PI) * n) / k)) / (n * n))) * n;
	} else if (t_0 <= 5e+285) {
		tmp = sqrt((n * (((double) M_PI) + ((double) M_PI)))) * (sqrt(k) / k);
	} else {
		tmp = (sqrt((((((double) M_PI) * k) / n) * 2.0)) / k) * n;
	}
	return tmp;
}

Julia

function code(k, n)
	t_0 = Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(fma(Float64(pi / k), n, Float64(Float64(pi * n) / k)) / Float64(n * n))) * n);
	elseif (t_0 <= 5e+285)
		tmp = Float64(sqrt(Float64(n * Float64(pi + pi))) * Float64(sqrt(k) / k));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(pi * k) / n) * 2.0)) / k) * n);
	end
	return tmp
end

Wolfram

code[k_, n_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(N[(N[(Pi / k), $MachinePrecision] * n + N[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$0, 5e+285], N[(N[Sqrt[N[(n * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[(Pi * k), $MachinePrecision] / n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 3 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 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

      11. lift-PI.f64 37.2

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

   4. Applied rewrites 37.2%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \mathsf{PI}\left(\right)}{k \cdot n}} \cdot n \]

      9. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{k \cdot n}} \cdot n \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      11. lower-*.f64 49.3

          \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

   7. Applied rewrites 49.3%

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

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      3. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\mathsf{PI}\left(\right) + \pi}{n \cdot k}} \cdot n \]

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. div-add N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. frac-add N/A

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

      12. lower-/.f64 N/A

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

   9. Applied rewrites 36.9%

      \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{\pi}{k}, n, \frac{\pi \cdot n}{k}\right)}{n \cdot n}} \cdot n \]

   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)))) < 5.00000000000000016e285

   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. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

      11. lift-PI.f64 37.2

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

   4. Applied rewrites 37.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\mathsf{PI}\left(\right) + \pi\right) \cdot n}{k}} \]

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-+.f64 N/A

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

      11. lift-PI.f64 N/A

          \[\leadsto \sqrt{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot \frac{n}{k}} \]

      12. lift-PI.f64 N/A

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

      13. lower-/.f64 37.2

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

   6. Applied rewrites 37.2%

      \[\leadsto \sqrt{\left(\pi + \pi\right) \cdot \frac{n}{k}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

         \[\leadsto \sqrt{\left(\mathsf{PI}\left(\right) + \pi\right) \cdot \frac{n}{k}} \]

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-PI.f64 N/A

         \[\leadsto \sqrt{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot \frac{n}{k}} \]

      9. lift-PI.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\mathsf{PI}\left(\right) + \pi\right) \cdot n}{k}} \]

      12. lift-PI.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. count-2-rev N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

   8. Applied rewrites 49.0%

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

   if 5.00000000000000016e285 < (*.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. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

      11. lift-PI.f64 37.2

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

   4. Applied rewrites 37.2%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \mathsf{PI}\left(\right)}{k \cdot n}} \cdot n \]

      9. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{k \cdot n}} \cdot n \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      11. lower-*.f64 49.3

          \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

   7. Applied rewrites 49.3%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 50.1

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

   10. Applied rewrites 50.1%

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

Alternative 9: 73.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\sqrt{\frac{\pi \cdot k}{n} \cdot 2}}{k} \cdot n\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\pi \cdot n\right) \cdot k\right) \cdot 2}{k \cdot k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\sqrt{k \cdot \frac{n}{\pi + \pi}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 2.2e+16)
   (* (/ (sqrt (* (/ (* PI k) n) 2.0)) k) n)
   (if (<= n 5.2e+39)
     (sqrt (/ (* (* (* PI n) k) 2.0) (* k k)))
     (/ n (sqrt (* k (/ n (+ PI PI))))))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 2.2e+16) {
		tmp = (sqrt((((((double) M_PI) * k) / n) * 2.0)) / k) * n;
	} else if (n <= 5.2e+39) {
		tmp = sqrt(((((((double) M_PI) * n) * k) * 2.0) / (k * k)));
	} else {
		tmp = n / sqrt((k * (n / (((double) M_PI) + ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 2.2e+16) {
		tmp = (Math.sqrt((((Math.PI * k) / n) * 2.0)) / k) * n;
	} else if (n <= 5.2e+39) {
		tmp = Math.sqrt(((((Math.PI * n) * k) * 2.0) / (k * k)));
	} else {
		tmp = n / Math.sqrt((k * (n / (Math.PI + Math.PI))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 2.2e+16:
		tmp = (math.sqrt((((math.pi * k) / n) * 2.0)) / k) * n
	elif n <= 5.2e+39:
		tmp = math.sqrt(((((math.pi * n) * k) * 2.0) / (k * k)))
	else:
		tmp = n / math.sqrt((k * (n / (math.pi + math.pi))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 2.2e+16)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(pi * k) / n) * 2.0)) / k) * n);
	elseif (n <= 5.2e+39)
		tmp = sqrt(Float64(Float64(Float64(Float64(pi * n) * k) * 2.0) / Float64(k * k)));
	else
		tmp = Float64(n / sqrt(Float64(k * Float64(n / Float64(pi + pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 2.2e+16)
		tmp = (sqrt((((pi * k) / n) * 2.0)) / k) * n;
	elseif (n <= 5.2e+39)
		tmp = sqrt(((((pi * n) * k) * 2.0) / (k * k)));
	else
		tmp = n / sqrt((k * (n / (pi + pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 2.2e+16], N[(N[(N[Sqrt[N[(N[(N[(Pi * k), $MachinePrecision] / n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 5.2e+39], N[Sqrt[N[(N[(N[(N[(Pi * n), $MachinePrecision] * k), $MachinePrecision] * 2.0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(n / N[Sqrt[N[(k * N[(n / N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 2.2e16

   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. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

      11. lift-PI.f64 37.2

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

   4. Applied rewrites 37.2%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \mathsf{PI}\left(\right)}{k \cdot n}} \cdot n \]

      9. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{k \cdot n}} \cdot n \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      11. lower-*.f64 49.3

          \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

   7. Applied rewrites 49.3%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 50.1

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

   10. Applied rewrites 50.1%

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

   if 2.2e16 < n < 5.2e39

   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. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

      11. lift-PI.f64 37.2

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

   4. Applied rewrites 37.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\mathsf{PI}\left(\right) + \pi\right) \cdot n}{k}} \]

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. count-2-rev N/A

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

      11. frac-add N/A

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

      12. *-commutative N/A

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

      13. count-2-rev N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 22.3%

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

   if 5.2e39 < n

   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. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

      11. lift-PI.f64 37.2

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

   4. Applied rewrites 37.2%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \mathsf{PI}\left(\right)}{k \cdot n}} \cdot n \]

      9. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{k \cdot n}} \cdot n \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      11. lower-*.f64 49.3

          \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

   7. Applied rewrites 49.3%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      4. div-flip N/A

         \[\leadsto \sqrt{\frac{1}{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      5. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\sqrt{\frac{k \cdot n}{\pi + \pi}}} \cdot n \]

      11. lower-*.f64 50.0

          \[\leadsto \frac{1}{\sqrt{\frac{k \cdot n}{\pi + \pi}}} \cdot n \]

   9. Applied rewrites 50.0%

      \[\leadsto \frac{1}{\sqrt{\frac{k \cdot n}{\pi + \pi}}} \cdot n \]

   11. Applied rewrites 50.0%

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

Alternative 10: 61.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.2 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{\frac{\pi \cdot n}{k} \cdot 2}\\ \mathbf{elif}\;n \leq 5.2 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{\frac{\left(\left(\pi \cdot n\right) \cdot k\right) \cdot 2}{k \cdot k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\sqrt{k \cdot \frac{n}{\pi + \pi}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 2.2e+16)
   (sqrt (* (/ (* PI n) k) 2.0))
   (if (<= n 5.2e+39)
     (sqrt (/ (* (* (* PI n) k) 2.0) (* k k)))
     (/ n (sqrt (* k (/ n (+ PI PI))))))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 2.2e+16) {
		tmp = sqrt((((((double) M_PI) * n) / k) * 2.0));
	} else if (n <= 5.2e+39) {
		tmp = sqrt(((((((double) M_PI) * n) * k) * 2.0) / (k * k)));
	} else {
		tmp = n / sqrt((k * (n / (((double) M_PI) + ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 2.2e+16) {
		tmp = Math.sqrt((((Math.PI * n) / k) * 2.0));
	} else if (n <= 5.2e+39) {
		tmp = Math.sqrt(((((Math.PI * n) * k) * 2.0) / (k * k)));
	} else {
		tmp = n / Math.sqrt((k * (n / (Math.PI + Math.PI))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 2.2e+16:
		tmp = math.sqrt((((math.pi * n) / k) * 2.0))
	elif n <= 5.2e+39:
		tmp = math.sqrt(((((math.pi * n) * k) * 2.0) / (k * k)))
	else:
		tmp = n / math.sqrt((k * (n / (math.pi + math.pi))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 2.2e+16)
		tmp = sqrt(Float64(Float64(Float64(pi * n) / k) * 2.0));
	elseif (n <= 5.2e+39)
		tmp = sqrt(Float64(Float64(Float64(Float64(pi * n) * k) * 2.0) / Float64(k * k)));
	else
		tmp = Float64(n / sqrt(Float64(k * Float64(n / Float64(pi + pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 2.2e+16)
		tmp = sqrt((((pi * n) / k) * 2.0));
	elseif (n <= 5.2e+39)
		tmp = sqrt(((((pi * n) * k) * 2.0) / (k * k)));
	else
		tmp = n / sqrt((k * (n / (pi + pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 2.2e+16], N[Sqrt[N[(N[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 5.2e+39], N[Sqrt[N[(N[(N[(N[(Pi * n), $MachinePrecision] * k), $MachinePrecision] * 2.0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(n / N[Sqrt[N[(k * N[(n / N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 2.2e16

   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. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

      11. lift-PI.f64 37.2

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

   4. Applied rewrites 37.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\mathsf{PI}\left(\right) + \pi\right) \cdot n}{k}} \]

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 37.2

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

   6. Applied rewrites 37.2%

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

   if 2.2e16 < n < 5.2e39

   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. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

      11. lift-PI.f64 37.2

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

   4. Applied rewrites 37.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\mathsf{PI}\left(\right) + \pi\right) \cdot n}{k}} \]

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. count-2-rev N/A

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

      11. frac-add N/A

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

      12. *-commutative N/A

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

      13. count-2-rev N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 22.3%

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

   if 5.2e39 < n

   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. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

      11. lift-PI.f64 37.2

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

   4. Applied rewrites 37.2%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \mathsf{PI}\left(\right)}{k \cdot n}} \cdot n \]

      9. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{k \cdot n}} \cdot n \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      11. lower-*.f64 49.3

          \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

   7. Applied rewrites 49.3%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      4. div-flip N/A

         \[\leadsto \sqrt{\frac{1}{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      5. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\sqrt{\frac{k \cdot n}{\pi + \pi}}} \cdot n \]

      11. lower-*.f64 50.0

          \[\leadsto \frac{1}{\sqrt{\frac{k \cdot n}{\pi + \pi}}} \cdot n \]

   9. Applied rewrites 50.0%

      \[\leadsto \frac{1}{\sqrt{\frac{k \cdot n}{\pi + \pi}}} \cdot n \]

   11. Applied rewrites 50.0%

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

Alternative 11: 61.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{\frac{\pi \cdot n}{k} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{\sqrt{k \cdot \frac{n}{\pi + \pi}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 2e+55)
   (sqrt (* (/ (* PI n) k) 2.0))
   (/ n (sqrt (* k (/ n (+ PI PI)))))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 2e+55) {
		tmp = sqrt((((((double) M_PI) * n) / k) * 2.0));
	} else {
		tmp = n / sqrt((k * (n / (((double) M_PI) + ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 2e+55) {
		tmp = Math.sqrt((((Math.PI * n) / k) * 2.0));
	} else {
		tmp = n / Math.sqrt((k * (n / (Math.PI + Math.PI))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 2e+55:
		tmp = math.sqrt((((math.pi * n) / k) * 2.0))
	else:
		tmp = n / math.sqrt((k * (n / (math.pi + math.pi))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 2e+55)
		tmp = sqrt(Float64(Float64(Float64(pi * n) / k) * 2.0));
	else
		tmp = Float64(n / sqrt(Float64(k * Float64(n / Float64(pi + pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 2e+55)
		tmp = sqrt((((pi * n) / k) * 2.0));
	else
		tmp = n / sqrt((k * (n / (pi + pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 2e+55], N[Sqrt[N[(N[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(n / N[Sqrt[N[(k * N[(n / N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.00000000000000002e55

   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. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

      11. lift-PI.f64 37.2

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

   4. Applied rewrites 37.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\mathsf{PI}\left(\right) + \pi\right) \cdot n}{k}} \]

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 37.2

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

   6. Applied rewrites 37.2%

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

   if 2.00000000000000002e55 < n

   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. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

      11. lift-PI.f64 37.2

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

   4. Applied rewrites 37.2%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \mathsf{PI}\left(\right)}{k \cdot n}} \cdot n \]

      9. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{k \cdot n}} \cdot n \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      11. lower-*.f64 49.3

          \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

   7. Applied rewrites 49.3%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      4. div-flip N/A

         \[\leadsto \sqrt{\frac{1}{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      5. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{\sqrt{\frac{k \cdot n}{\pi + \pi}}} \cdot n \]

      11. lower-*.f64 50.0

          \[\leadsto \frac{1}{\sqrt{\frac{k \cdot n}{\pi + \pi}}} \cdot n \]

   9. Applied rewrites 50.0%

      \[\leadsto \frac{1}{\sqrt{\frac{k \cdot n}{\pi + \pi}}} \cdot n \]

   11. Applied rewrites 50.0%

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

Alternative 12: 60.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 6 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{\frac{\pi \cdot n}{k} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 6e+16)
   (sqrt (* (/ (* PI n) k) 2.0))
   (* (sqrt (/ (+ PI PI) (* n k))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 6e+16) {
		tmp = sqrt((((((double) M_PI) * n) / k) * 2.0));
	} else {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / (n * k))) * n;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 6e+16) {
		tmp = Math.sqrt((((Math.PI * n) / k) * 2.0));
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (n * k))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 6e+16:
		tmp = math.sqrt((((math.pi * n) / k) * 2.0))
	else:
		tmp = math.sqrt(((math.pi + math.pi) / (n * k))) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 6e+16)
		tmp = sqrt(Float64(Float64(Float64(pi * n) / k) * 2.0));
	else
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / Float64(n * k))) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 6e+16)
		tmp = sqrt((((pi * n) / k) * 2.0));
	else
		tmp = sqrt(((pi + pi) / (n * k))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 6e+16], N[Sqrt[N[(N[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / N[(n * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 6e16

   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. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

      11. lift-PI.f64 37.2

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

   4. Applied rewrites 37.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\mathsf{PI}\left(\right) + \pi\right) \cdot n}{k}} \]

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 37.2

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

   6. Applied rewrites 37.2%

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

   if 6e16 < n

   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. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

      11. lift-PI.f64 37.2

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

   4. Applied rewrites 37.2%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \mathsf{PI}\left(\right)}{k \cdot n}} \cdot n \]

      9. lift-PI.f64 N/A

         \[\leadsto \sqrt{\frac{\pi + \pi}{k \cdot n}} \cdot n \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

      11. lower-*.f64 49.3

          \[\leadsto \sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n \]

   7. Applied rewrites 49.3%

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

Alternative 13: 49.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. unpow1/2 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

       \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

   11. lift-PI.f64 37.2

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

4. Applied rewrites 37.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

      \[\leadsto \sqrt{\frac{\left(\mathsf{PI}\left(\right) + \pi\right) \cdot n}{k}} \]

   5. lift-PI.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. count-2-rev N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. sqrt-undiv N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 49.1%

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

Alternative 14: 37.2% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. unpow1/2 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

       \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

   11. lift-PI.f64 37.2

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

4. Applied rewrites 37.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

      \[\leadsto \sqrt{\frac{\left(\mathsf{PI}\left(\right) + \pi\right) \cdot n}{k}} \]

   4. lift-PI.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2-rev N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-PI.f64 37.2

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

6. Applied rewrites 37.2%

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

Alternative 15: 37.2% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. unpow1/2 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

       \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

   11. lift-PI.f64 37.2

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

4. Applied rewrites 37.2%

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

Alternative 16: 37.2% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. unpow1/2 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

       \[\leadsto \sqrt{\frac{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

   11. lift-PI.f64 37.2

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

4. Applied rewrites 37.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

      \[\leadsto \sqrt{\frac{\left(\mathsf{PI}\left(\right) + \pi\right) \cdot n}{k}} \]

   4. lift-PI.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2-rev N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. count-2-rev N/A

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

   10. lift-+.f64 N/A

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

   11. lift-PI.f64 N/A

       \[\leadsto \sqrt{\left(\pi + \mathsf{PI}\left(\right)\right) \cdot \frac{n}{k}} \]

   12. lift-PI.f64 N/A

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

   13. lower-/.f64 37.2

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

6. Applied rewrites 37.2%

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

Reproduce

herbie shell --seed 2025136 
(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))))