Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 4.8s

Alternatives: 7

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 1.5931988282589807e+40
  2. n = -1.3483469403651068e+99

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. lower-/.f64 N/A

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

3. Applied rewrites 99.4%

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in k around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.4

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

8. Applied rewrites 99.4%

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

Alternative 2: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 98.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in k around 0

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

      1. *-lft-identity N/A

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

      2. lift-+.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-sqrt.f64 N/A

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

   6. Applied rewrites 96.2%

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

   if 1 < k

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in k around inf

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

      1. lower-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 99.5

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

   8. Applied rewrites 99.5%

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

Alternative 3: 50.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. lower-/.f64 N/A

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in k around 0

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

   1. *-lft-identity N/A

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

   2. lift-+.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. unpow-prod-down N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. sqrt-unprod N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. lower-sqrt.f64 N/A

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

6. Applied rewrites 50.4%

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

Alternative 4: 50.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. sqrt-unprod N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 38.8

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

4. Applied rewrites 38.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l/ N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-/.f64 N/A

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

   11. count-2-rev N/A

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

   12. lift-+.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 38.8

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

6. Applied rewrites 38.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 38.8

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

8. Applied rewrites 38.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqrt-prod N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 50.5

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

10. Applied rewrites 50.5%

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

Alternative 5: 38.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. sqrt-unprod N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 38.8

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

4. Applied rewrites 38.8%

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

   1. lift-/.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-/.f64 38.8

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

6. Applied rewrites 38.8%

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

Alternative 6: 38.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. sqrt-unprod N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 38.8

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

4. Applied rewrites 38.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l/ N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-/.f64 N/A

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

   11. count-2-rev N/A

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

   12. lift-+.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 38.8

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

6. Applied rewrites 38.8%

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

Alternative 7: 38.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. sqrt-unprod N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 38.8

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

4. Applied rewrites 38.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l/ N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-/.f64 N/A

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

   11. count-2-rev N/A

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

   12. lift-+.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 38.8

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

6. Applied rewrites 38.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 38.8

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

8. Applied rewrites 38.8%

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

Reproduce

herbie shell --seed 2025115 
(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))))