Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.5%

Time: 4.4s

Alternatives: 8

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 1.5593394419603236e+262
  2. n = -1.314844970515926e+102

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: 99.5% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 99.5

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

3. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-div N/A

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

   5. metadata-eval N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift--.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. associate-*l/ N/A

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

   13. count-2-rev N/A

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

   14. lower-/.f64 N/A

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

5. Applied rewrites 99.5%

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

Alternative 3: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(k, n)
	t_0 = (pi + pi) * n;
	tmp = 0.0;
	if (k <= 1.0)
		tmp = sqrt((1.0 / k)) * sqrt(t_0);
	else
		tmp = (t_0 ^ (-0.5 * k)) / sqrt(k);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[k, 1.0], N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[Power[t$95$0, 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 98.9%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 99.0

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in k around 0

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

      1. count-2-rev N/A

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

      2. *-lft-identity N/A

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

      3. sqrt-unprod N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-sqrt.f64 97.1

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. count-2-rev N/A

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

      14. lift-+.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lift-PI.f64 97.1

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

   6. Applied rewrites 97.1%

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

   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. 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 100.0

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

   3. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. count-2-rev N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in k around inf

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

      1. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

Alternative 4: 50.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 99.5

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in k around 0

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

   1. count-2-rev N/A

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

   2. *-lft-identity N/A

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

   3. sqrt-unprod N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-sqrt.f64 50.0

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. count-2-rev N/A

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

   14. lift-+.f64 N/A

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

   15. lift-PI.f64 N/A

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

   16. lift-PI.f64 50.0

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

6. Applied rewrites 50.0%

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

Alternative 5: 50.0% 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 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-div N/A

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

   5. metadata-eval N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift--.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. associate-*l/ N/A

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

   13. count-2-rev N/A

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

   14. lower-/.f64 N/A

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in k around 0

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

   1. count-2-rev N/A

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

   2. sqrt-unprod N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lift-+.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-PI.f64 50.0

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

8. Applied rewrites 50.0%

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

Alternative 6: 38.5% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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 38.5

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

4. Applied rewrites 38.5%

   \[\leadsto \color{blue}{\sqrt{\frac{\pi \cdot n}{k} \cdot 2}} \]
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. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-PI.f64 38.5

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

6. Applied rewrites 38.5%

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

Alternative 7: 38.5% 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 99.5%

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

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 38.5

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

4. Applied rewrites 38.5%

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

   1. associate-*l/ 38.5

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

   2. count-2-rev 38.5

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l/ N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-/.f64 38.5

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

6. Applied rewrites 38.5%

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

Alternative 8: 38.5% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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 38.5

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

4. Applied rewrites 38.5%

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

   1. associate-*l/ 38.5

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

   2. count-2-rev 38.5

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l/ N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-/.f64 38.5

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

6. Applied rewrites 38.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2-rev N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. count-2-rev N/A

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

   10. lift-+.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-/.f64 38.5

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

8. Applied rewrites 38.5%

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

Reproduce

herbie shell --seed 2025123 
(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))))