Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 6.9s

Alternatives: 11

Speedup: 0.9×

• Could not converge on a ground truth

  1. k = 1.5146129700500594e+106
  2. n = -7.76867215075591e+46

Specification

Math

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

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

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

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lower-/.f64 99.5%

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

3. Applied rewrites 99.5%

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

   1. lift-pow.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. pow-add N/A

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

   4. lower-unsound-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. distribute-lft-in N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. count-2-rev N/A

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

   11. lift-*.f64 N/A

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

   12. lower-unsound-*.f64 N/A

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

5. Applied rewrites 99.4%

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

Alternative 2: 99.4% accurate, 1.2× speedup

Math

\[\frac{{\left(n \cdot 6.283185307179586\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}{\sqrt{k}} \]

FPCore

(FPCore (k n)
 :precision binary64
 (/ (pow (* n 6.283185307179586) (fma k -0.5 0.5)) (sqrt k)))

C

double code(double k, double n) {
	return pow((n * 6.283185307179586), fma(k, -0.5, 0.5)) / sqrt(k);
}

Julia

function code(k, n)
	return Float64((Float64(n * 6.283185307179586) ^ fma(k, -0.5, 0.5)) / sqrt(k))
end

Wolfram

code[k_, n_] := N[(N[Power[N[(n * 6.283185307179586), $MachinePrecision], N[(k * -0.5 + 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. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lower-/.f64 99.5%

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

3. Applied rewrites 99.5%

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

4. Evaluated real constant 99.5%

   \[\leadsto \frac{{\left(n \cdot \color{blue}{6.283185307179586}\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}{\sqrt{k}} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;k \leq 2.8:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(n \cdot 6.283185307179586\right)}^{\left(-0.5 \cdot k\right)}}{\sqrt{k}}\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.8)
   (* (sqrt (/ (+ PI PI) k)) (sqrt n))
   (/ (pow (* n 6.283185307179586) (* -0.5 k)) (sqrt k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.8) {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / k)) * sqrt(n);
	} else {
		tmp = pow((n * 6.283185307179586), (-0.5 * k)) / sqrt(k);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.8) {
		tmp = Math.sqrt(((Math.PI + Math.PI) / k)) * Math.sqrt(n);
	} else {
		tmp = Math.pow((n * 6.283185307179586), (-0.5 * k)) / Math.sqrt(k);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.8:
		tmp = math.sqrt(((math.pi + math.pi) / k)) * math.sqrt(n)
	else:
		tmp = math.pow((n * 6.283185307179586), (-0.5 * k)) / math.sqrt(k)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.8)
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / k)) * sqrt(n));
	else
		tmp = Float64((Float64(n * 6.283185307179586) ^ Float64(-0.5 * k)) / sqrt(k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.8)
		tmp = sqrt(((pi + pi) / k)) * sqrt(n);
	else
		tmp = ((n * 6.283185307179586) ^ (-0.5 * k)) / sqrt(k);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.8], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(n * 6.283185307179586), $MachinePrecision], N[(-0.5 * k), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.7999999999999998

   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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2%

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

   6. Applied rewrites 37.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. count-2 N/A

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

      6. associate-*r* N/A

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

      7. sqrt-prod N/A

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

      8. lower-unsound-*.f64 N/A

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

      9. lower-unsound-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-unsound-sqrt.f64 49.1%

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

   8. Applied rewrites 49.1%

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

   if 2.7999999999999998 < 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. 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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Evaluated real constant 99.5%

      \[\leadsto \frac{{\left(n \cdot \color{blue}{6.283185307179586}\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}{\sqrt{k}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{{\left(n \cdot 6.283185307179586\right)}^{\color{blue}{\left(\frac{-1}{2} \cdot k\right)}}}{\sqrt{k}} \]
   6. Step-by-step derivation

      1. lower-*.f64 53.8%

         \[\leadsto \frac{{\left(n \cdot 6.283185307179586\right)}^{\left(-0.5 \cdot \color{blue}{k}\right)}}{\sqrt{k}} \]

   7. Applied rewrites 53.8%

      \[\leadsto \frac{{\left(n \cdot 6.283185307179586\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}}}{\sqrt{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 76.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\log \left(e^{\frac{n + n}{k} \cdot \pi}\right)}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \sqrt{2 \cdot \frac{k \cdot \pi}{n}}}{k}\\ \end{array} \]

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 (log (exp (* (/ (+ n n) k) PI))))
     (if (<= t_0 2e+298)
       (* (sqrt (/ (+ PI PI) k)) (sqrt n))
       (/ (* n (sqrt (* 2.0 (/ (* k PI) n)))) k)))))

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(log(exp((((n + n) / k) * ((double) M_PI)))));
	} else if (t_0 <= 2e+298) {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / k)) * sqrt(n);
	} else {
		tmp = (n * sqrt((2.0 * ((k * ((double) M_PI)) / n)))) / k;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = (1.0 / Math.sqrt(k)) * Math.pow(((2.0 * Math.PI) * n), ((1.0 - k) / 2.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt(Math.log(Math.exp((((n + n) / k) * Math.PI))));
	} else if (t_0 <= 2e+298) {
		tmp = Math.sqrt(((Math.PI + Math.PI) / k)) * Math.sqrt(n);
	} else {
		tmp = (n * Math.sqrt((2.0 * ((k * Math.PI) / n)))) / k;
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = (1.0 / math.sqrt(k)) * math.pow(((2.0 * math.pi) * n), ((1.0 - k) / 2.0))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt(math.log(math.exp((((n + n) / k) * math.pi))))
	elif t_0 <= 2e+298:
		tmp = math.sqrt(((math.pi + math.pi) / k)) * math.sqrt(n)
	else:
		tmp = (n * math.sqrt((2.0 * ((k * math.pi) / n)))) / k
	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 = sqrt(log(exp(Float64(Float64(Float64(n + n) / k) * pi))));
	elseif (t_0 <= 2e+298)
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / k)) * sqrt(n));
	else
		tmp = Float64(Float64(n * sqrt(Float64(2.0 * Float64(Float64(k * pi) / n)))) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = (1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt(log(exp((((n + n) / k) * pi))));
	elseif (t_0 <= 2e+298)
		tmp = sqrt(((pi + pi) / k)) * sqrt(n);
	else
		tmp = (n * sqrt((2.0 * ((k * pi) / n)))) / k;
	end
	tmp_2 = 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[Sqrt[N[Log[N[Exp[N[(N[(N[(n + n), $MachinePrecision] / k), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 2e+298], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[(N[(n * N[Sqrt[N[(2.0 * N[(N[(k * Pi), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k), $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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2%

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

   6. Applied rewrites 37.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. div-add N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 37.4%

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

   8. Applied rewrites 37.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-log-exp N/A

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

      5. log-pow-rev N/A

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

      6. lower-log.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. pow-exp N/A

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

      9. lift-*.f64 N/A

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

      10. lower-exp.f64 14.6%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 14.6%

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

   10. Applied rewrites 14.6%

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

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 1.9999999999999999e298

   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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2%

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

   6. Applied rewrites 37.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. count-2 N/A

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

      6. associate-*r* N/A

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

      7. sqrt-prod N/A

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

      8. lower-unsound-*.f64 N/A

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

      9. lower-unsound-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-unsound-sqrt.f64 49.1%

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

   8. Applied rewrites 49.1%

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

   if 1.9999999999999999e298 < (*.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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2%

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

   6. Applied rewrites 37.4%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 49.8%

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

   9. Applied rewrites 49.8%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-PI.f64 49.6%

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

   12. Applied rewrites 49.6%

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

Alternative 5: 73.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;n \cdot \sqrt{\frac{\frac{n + n}{k} \cdot \pi}{n \cdot n}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \sqrt{2 \cdot \frac{k \cdot \pi}{n}}}{k}\\ \end{array} \]

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)
     (* n (sqrt (/ (* (/ (+ n n) k) PI) (* n n))))
     (if (<= t_0 2e+298)
       (* (sqrt (/ (+ PI PI) k)) (sqrt n))
       (/ (* n (sqrt (* 2.0 (/ (* k PI) n)))) k)))))

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 = n * sqrt(((((n + n) / k) * ((double) M_PI)) / (n * n)));
	} else if (t_0 <= 2e+298) {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / k)) * sqrt(n);
	} else {
		tmp = (n * sqrt((2.0 * ((k * ((double) M_PI)) / n)))) / k;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = (1.0 / Math.sqrt(k)) * Math.pow(((2.0 * Math.PI) * n), ((1.0 - k) / 2.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = n * Math.sqrt(((((n + n) / k) * Math.PI) / (n * n)));
	} else if (t_0 <= 2e+298) {
		tmp = Math.sqrt(((Math.PI + Math.PI) / k)) * Math.sqrt(n);
	} else {
		tmp = (n * Math.sqrt((2.0 * ((k * Math.PI) / n)))) / k;
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = (1.0 / math.sqrt(k)) * math.pow(((2.0 * math.pi) * n), ((1.0 - k) / 2.0))
	tmp = 0
	if t_0 <= 0.0:
		tmp = n * math.sqrt(((((n + n) / k) * math.pi) / (n * n)))
	elif t_0 <= 2e+298:
		tmp = math.sqrt(((math.pi + math.pi) / k)) * math.sqrt(n)
	else:
		tmp = (n * math.sqrt((2.0 * ((k * math.pi) / n)))) / k
	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(n * sqrt(Float64(Float64(Float64(Float64(n + n) / k) * pi) / Float64(n * n))));
	elseif (t_0 <= 2e+298)
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / k)) * sqrt(n));
	else
		tmp = Float64(Float64(n * sqrt(Float64(2.0 * Float64(Float64(k * pi) / n)))) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = (1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = n * sqrt(((((n + n) / k) * pi) / (n * n)));
	elseif (t_0 <= 2e+298)
		tmp = sqrt(((pi + pi) / k)) * sqrt(n);
	else
		tmp = (n * sqrt((2.0 * ((k * pi) / n)))) / k;
	end
	tmp_2 = 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 * N[Sqrt[N[(N[(N[(N[(n + n), $MachinePrecision] / k), $MachinePrecision] * Pi), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+298], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[(N[(n * N[Sqrt[N[(2.0 * N[(N[(k * Pi), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k), $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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2%

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

   6. Applied rewrites 37.4%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 49.8%

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

   9. Applied rewrites 49.8%

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

       1. lift-*.f64 N/A

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

       2. count-2-rev N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/r* N/A

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

       6. lift-PI.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. associate-/r* N/A

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

       10. lift-PI.f64 N/A

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

       11. common-denominator N/A

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

       12. distribute-lft-in N/A

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

       13. lift-+.f64 N/A

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

       14. lift-PI.f64 N/A

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

       15. associate-*l/ N/A

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

       16. associate-*r/ N/A

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

       17. lift-/.f64 N/A

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

       18. lift-*.f64 N/A

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

       19. lower-/.f64 N/A

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

   11. Applied rewrites 38.0%

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

   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)))) < 1.9999999999999999e298

   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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2%

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

   6. Applied rewrites 37.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. count-2 N/A

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

      6. associate-*r* N/A

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

      7. sqrt-prod N/A

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

      8. lower-unsound-*.f64 N/A

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

      9. lower-unsound-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-unsound-sqrt.f64 49.1%

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

   8. Applied rewrites 49.1%

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

   if 1.9999999999999999e298 < (*.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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2%

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

   6. Applied rewrites 37.4%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 49.8%

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

   9. Applied rewrites 49.8%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-PI.f64 49.6%

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

   12. Applied rewrites 49.6%

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

Alternative 6: 73.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;n \leq 0.33:\\ \;\;\;\;\frac{n \cdot \sqrt{2 \cdot \frac{k \cdot \pi}{n}}}{k}\\ \mathbf{elif}\;n \leq 5.1 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{\pi \cdot \frac{\mathsf{fma}\left(k, n, k \cdot n\right)}{k \cdot k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{k \cdot n}} \cdot n\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 0.33)
   (/ (* n (sqrt (* 2.0 (/ (* k PI) n)))) k)
   (if (<= n 5.1e+39)
     (sqrt (* PI (/ (fma k n (* k n)) (* k k))))
     (* (sqrt (/ (+ PI PI) (* k n))) n))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 0.33) {
		tmp = (n * sqrt((2.0 * ((k * ((double) M_PI)) / n)))) / k;
	} else if (n <= 5.1e+39) {
		tmp = sqrt((((double) M_PI) * (fma(k, n, (k * n)) / (k * k))));
	} else {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / (k * n))) * n;
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 0.33)
		tmp = Float64(Float64(n * sqrt(Float64(2.0 * Float64(Float64(k * pi) / n)))) / k);
	elseif (n <= 5.1e+39)
		tmp = sqrt(Float64(pi * Float64(fma(k, n, Float64(k * n)) / Float64(k * k))));
	else
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / Float64(k * n))) * n);
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[n, 0.33], N[(N[(n * N[Sqrt[N[(2.0 * N[(N[(k * Pi), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[n, 5.1e+39], N[Sqrt[N[(Pi * N[(N[(k * n + N[(k * n), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / N[(k * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 0.330000000000000016

   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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2%

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

   6. Applied rewrites 37.4%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 49.8%

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

   9. Applied rewrites 49.8%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-PI.f64 49.6%

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

   12. Applied rewrites 49.6%

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

   if 0.330000000000000016 < n < 5.0999999999999998e39

   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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2%

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

   6. Applied rewrites 37.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. div-add N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 37.4%

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

   8. Applied rewrites 37.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. frac-add N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 22.7%

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

   10. Applied rewrites 22.7%

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

   if 5.0999999999999998e39 < 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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2%

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

   6. Applied rewrites 37.4%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 49.8%

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

   9. Applied rewrites 49.8%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 49.8%

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. associate-*r/ N/A

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

       7. lift-PI.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-PI.f64 N/A

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

       10. count-2-rev N/A

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

       11. lower-+.f64 49.8%

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

   11. Applied rewrites 49.8%

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

Alternative 7: 71.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;n \leq 4.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{n \cdot \sqrt{2 \cdot \frac{k \cdot \pi}{n}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{k \cdot n}} \cdot n\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 4.5e+30)
   (/ (* n (sqrt (* 2.0 (/ (* k PI) n)))) k)
   (* (sqrt (/ (+ PI PI) (* k n))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 4.5e+30) {
		tmp = (n * sqrt((2.0 * ((k * ((double) M_PI)) / n)))) / k;
	} else {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / (k * n))) * n;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 4.5e+30) {
		tmp = (n * Math.sqrt((2.0 * ((k * Math.PI) / n)))) / k;
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (k * n))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 4.5e+30:
		tmp = (n * math.sqrt((2.0 * ((k * math.pi) / n)))) / k
	else:
		tmp = math.sqrt(((math.pi + math.pi) / (k * n))) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 4.5e+30)
		tmp = Float64(Float64(n * sqrt(Float64(2.0 * Float64(Float64(k * pi) / n)))) / k);
	else
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / Float64(k * n))) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 4.5e+30)
		tmp = (n * sqrt((2.0 * ((k * pi) / n)))) / k;
	else
		tmp = sqrt(((pi + pi) / (k * n))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 4.5e+30], N[(N[(n * N[Sqrt[N[(2.0 * N[(N[(k * Pi), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / N[(k * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 4.49999999999999995e30

   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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2%

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

   6. Applied rewrites 37.4%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 49.8%

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

   9. Applied rewrites 49.8%

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

   10. Taylor expanded in k around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-PI.f64 49.6%

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

   12. Applied rewrites 49.6%

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

   if 4.49999999999999995e30 < 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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2%

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

   6. Applied rewrites 37.4%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 49.8%

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

   9. Applied rewrites 49.8%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 49.8%

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. associate-*r/ N/A

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

       7. lift-PI.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-PI.f64 N/A

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

       10. count-2-rev N/A

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

       11. lower-+.f64 49.8%

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

   11. Applied rewrites 49.8%

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

Alternative 8: 61.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;n \leq 4 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{k \cdot n}} \cdot n\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 4e+42)
   (* (sqrt (/ (+ PI PI) k)) (sqrt n))
   (* (sqrt (/ (+ PI PI) (* k n))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 4e+42) {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / k)) * sqrt(n);
	} else {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / (k * n))) * n;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 4e+42) {
		tmp = Math.sqrt(((Math.PI + Math.PI) / k)) * Math.sqrt(n);
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (k * n))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 4e+42:
		tmp = math.sqrt(((math.pi + math.pi) / k)) * math.sqrt(n)
	else:
		tmp = math.sqrt(((math.pi + math.pi) / (k * n))) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 4e+42)
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / k)) * sqrt(n));
	else
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / Float64(k * n))) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 4e+42)
		tmp = sqrt(((pi + pi) / k)) * sqrt(n);
	else
		tmp = sqrt(((pi + pi) / (k * n))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 4e+42], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / N[(k * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 4.00000000000000018e42

   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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2%

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

   6. Applied rewrites 37.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. count-2 N/A

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

      6. associate-*r* N/A

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

      7. sqrt-prod N/A

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

      8. lower-unsound-*.f64 N/A

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

      9. lower-unsound-sqrt.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. *-commutative N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 N/A

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

      18. lower-unsound-sqrt.f64 49.1%

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

   8. Applied rewrites 49.1%

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

   if 4.00000000000000018e42 < 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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2%

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

   6. Applied rewrites 37.4%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 49.8%

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

   9. Applied rewrites 49.8%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 49.8%

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. associate-*r/ N/A

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

       7. lift-PI.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-PI.f64 N/A

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

       10. count-2-rev N/A

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

       11. lower-+.f64 49.8%

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

   11. Applied rewrites 49.8%

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

Alternative 9: 49.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 49.2%

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

4. Applied rewrites 49.2%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow1/2 N/A

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

   4. lift-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. sqrt-undiv N/A

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

   13. lower-sqrt.f64 N/A

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

6. Applied rewrites 37.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. count-2 N/A

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

   6. associate-*r* N/A

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

   7. sqrt-prod N/A

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

   8. lower-unsound-*.f64 N/A

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

   9. lower-unsound-sqrt.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*l/ N/A

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

   12. *-commutative N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lift-PI.f64 N/A

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

   16. count-2-rev N/A

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

   17. lower-+.f64 N/A

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

   18. lower-unsound-sqrt.f64 49.1%

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

8. Applied rewrites 49.1%

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

Alternative 10: 49.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 49.2%

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

4. Applied rewrites 49.2%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow1/2 N/A

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

   4. lift-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. sqrt-undiv N/A

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

   13. lower-sqrt.f64 N/A

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

6. Applied rewrites 37.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. sqrt-prod N/A

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

   5. lower-unsound-*.f64 N/A

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

   6. lower-unsound-sqrt.f64 N/A

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

   7. lower-unsound-sqrt.f64 49.2%

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

8. Applied rewrites 49.2%

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

Alternative 11: 37.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 49.2%

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

4. Applied rewrites 49.2%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow1/2 N/A

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

   4. lift-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. sqrt-undiv N/A

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

   13. lower-sqrt.f64 N/A

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

6. Applied rewrites 37.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. distribute-rgt-in N/A

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

   10. lift-/.f64 N/A

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

   11. mult-flip N/A

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

   12. lift-/.f64 N/A

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

   13. mult-flip N/A

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

   14. div-add N/A

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

   15. lift-+.f64 N/A

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

   16. lower-/.f64 37.4%

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

8. Applied rewrites 37.4%

   \[\leadsto \sqrt{\pi \cdot \frac{n + n}{k}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025188 
(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))))