Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 7.0s

Alternatives: 8

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 2.2107459172483446e+223
  2. n = -2.0234927353485714e+106

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. mult-flip-rev N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.5%

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

Alternative 2: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around inf

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. associate-/r* N/A

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

   4. mult-flip N/A

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

   5. div-flip N/A

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

   6. /-rgt-identity N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 90.9%

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

6. Applied rewrites 99.4%

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

Alternative 3: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-sqrt.f64 49.9

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

   6. Applied rewrites 49.9%

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

   if 1 < k

   1. Initial program 99.4%

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

   2. Taylor expanded in k around inf

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. associate-/r* N/A

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

      4. mult-flip N/A

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

      5. div-flip N/A

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

      6. /-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 90.9%

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in k around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 53.0

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

   9. Applied rewrites 53.0%

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

Alternative 4: 74.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 8 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{\frac{\pi \cdot k}{n} \cdot 2} \cdot n}{k}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{k \cdot n}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 8e-43)
   (/ (* (sqrt (* (/ (* PI k) n) 2.0)) n) k)
   (* (sqrt (/ (+ PI PI) (* k n))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 8e-43) {
		tmp = (sqrt((((((double) M_PI) * k) / n) * 2.0)) * 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 <= 8e-43) {
		tmp = (Math.sqrt((((Math.PI * k) / n) * 2.0)) * n) / k;
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (k * n))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 8e-43:
		tmp = (math.sqrt((((math.pi * k) / n) * 2.0)) * 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 <= 8e-43)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(pi * k) / n) * 2.0)) * 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 <= 8e-43)
		tmp = (sqrt((((pi * k) / n) * 2.0)) * n) / k;
	else
		tmp = sqrt(((pi + pi) / (k * n))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 8e-43], N[(N[(N[Sqrt[N[(N[(N[(Pi * k), $MachinePrecision] / n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * n), $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 < 8.00000000000000062e-43

   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. metadata-eval N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-div N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 99.4

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow1/2 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-+.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. sqrt-unprod N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-*.f64 38.3

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

   6. Applied rewrites 38.3%

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

   7. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 51.5

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

   9. Applied rewrites 51.5%

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

   if 8.00000000000000062e-43 < n

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 38.3

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

   4. Applied rewrites 38.3%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 49.7

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

   7. Applied rewrites 49.7%

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

Alternative 5: 61.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 5000:\\ \;\;\;\;\sqrt{\frac{\pi \cdot n}{k} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{k \cdot n}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 5000.0)
   (sqrt (* (/ (* PI n) k) 2.0))
   (* (sqrt (/ (+ PI PI) (* k n))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 5000.0) {
		tmp = sqrt((((((double) M_PI) * n) / k) * 2.0));
	} 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 <= 5000.0) {
		tmp = Math.sqrt((((Math.PI * n) / k) * 2.0));
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (k * n))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 5000.0:
		tmp = math.sqrt((((math.pi * n) / k) * 2.0))
	else:
		tmp = math.sqrt(((math.pi + math.pi) / (k * n))) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 5000.0)
		tmp = sqrt(Float64(Float64(Float64(pi * n) / k) * 2.0));
	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 <= 5000.0)
		tmp = sqrt((((pi * n) / k) * 2.0));
	else
		tmp = sqrt(((pi + pi) / (k * n))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 5000.0], N[Sqrt[N[(N[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision] * 2.0), $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 < 5e3

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 38.3

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

   4. Applied rewrites 38.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 38.3

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

   6. Applied rewrites 38.3%

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

   if 5e3 < n

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 38.3

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

   4. Applied rewrites 38.3%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-*.f64 49.7

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

   7. Applied rewrites 49.7%

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

Alternative 6: 49.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. mult-flip-rev N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in k around 0

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

   1. unpow1/2 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lift-+.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-sqrt.f64 49.9

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

6. Applied rewrites 49.9%

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

Alternative 7: 38.3% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. unpow1/2 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-PI.f64 38.3

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

4. Applied rewrites 38.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2-rev N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-PI.f64 38.3

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

6. Applied rewrites 38.3%

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

Alternative 8: 38.3% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. unpow1/2 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-PI.f64 38.3

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

4. Applied rewrites 38.3%

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

Reproduce

herbie shell --seed 2025140 
(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))))