Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 4.2s

Alternatives: 5

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 5.0156831481551905e+53
  2. n = -1.190804355551078e+20

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(k, n):
	return math.pow((n * (math.pi + math.pi)), (0.5 * (1.0 - k))) / math.sqrt(k)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lower-/.f64 99.4

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.4

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

   9. lift-*.f64 N/A

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

   10. count-2-rev N/A

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

   11. lower-+.f64 99.4

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

   12. lift-/.f64 N/A

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

   13. div-flip N/A

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

   14. associate-/r/ N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f64 99.4

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

3. Applied rewrites 99.4%

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

Alternative 2: 98.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.0)
   (/ (sqrt (* (+ PI PI) n)) (sqrt k))
   (/ (pow (* n (+ PI PI)) (* -0.5 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((n * (((double) M_PI) + ((double) M_PI))), (-0.5 * k)) / sqrt(k);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.0) {
		tmp = Math.sqrt(((Math.PI + Math.PI) * n)) / Math.sqrt(k);
	} else {
		tmp = Math.pow((n * (Math.PI + Math.PI)), (-0.5 * 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((n * (math.pi + math.pi)), (-0.5 * 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((Float64(n * Float64(pi + pi)) ^ Float64(-0.5 * k)) / sqrt(k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.0)
		tmp = sqrt(((pi + pi) * n)) / sqrt(k);
	else
		tmp = ((n * (pi + pi)) ^ (-0.5 * 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[(n * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision], N[(-0.5 * k), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 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. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.8

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

   4. Applied rewrites 49.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-sqrt.f64 49.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 49.8

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

   6. Applied rewrites 49.8%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 99.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.4

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

      9. lift-*.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 99.4

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

      12. lift-/.f64 N/A

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

      13. div-flip N/A

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

      14. associate-/r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 99.4

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in k around inf

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

      1. lower-*.f64 53.0

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

   6. Applied rewrites 53.0%

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

Alternative 3: 61.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 8.855 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n \cdot \pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{\pi}{k \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 8.855e+27)
   (sqrt (* 2.0 (/ (* n PI) k)))
   (* n (sqrt (* 2.0 (/ PI (* k n)))))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 8.855e+27) {
		tmp = sqrt((2.0 * ((n * ((double) M_PI)) / k)));
	} else {
		tmp = n * sqrt((2.0 * (((double) M_PI) / (k * n))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 8.855e+27) {
		tmp = Math.sqrt((2.0 * ((n * Math.PI) / k)));
	} else {
		tmp = n * Math.sqrt((2.0 * (Math.PI / (k * n))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 8.855e+27:
		tmp = math.sqrt((2.0 * ((n * math.pi) / k)))
	else:
		tmp = n * math.sqrt((2.0 * (math.pi / (k * n))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 8.855e+27)
		tmp = sqrt(Float64(2.0 * Float64(Float64(n * pi) / k)));
	else
		tmp = Float64(n * sqrt(Float64(2.0 * Float64(pi / Float64(k * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 8.855e+27)
		tmp = sqrt((2.0 * ((n * pi) / k)));
	else
		tmp = n * sqrt((2.0 * (pi / (k * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 8.855e+27], N[Sqrt[N[(2.0 * N[(N[(n * Pi), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(n * N[Sqrt[N[(2.0 * N[(Pi / N[(k * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 8.8550000000000005e27

   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. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.8

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

   4. Applied rewrites 49.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-sqrt.f64 49.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 49.8

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

   6. Applied rewrites 49.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 37.9

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

   8. Applied rewrites 37.9%

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

   9. Taylor expanded in k around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-PI.f64 37.9

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

   11. Applied rewrites 37.9%

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

   if 8.8550000000000005e27 < 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. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.8

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

   4. Applied rewrites 49.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-sqrt.f64 49.8

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 49.8

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

   6. Applied rewrites 49.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 37.9

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

   8. Applied rewrites 37.9%

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

   9. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-PI.f64 N/A

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

       6. lower-*.f64 49.1

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

   11. Applied rewrites 49.1%

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

Alternative 4: 49.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 49.8

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

4. Applied rewrites 49.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. count-2-rev N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-lft-in N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-sqrt.f64 49.8

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 49.8

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

6. Applied rewrites 49.8%

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

Alternative 5: 37.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 49.8

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

4. Applied rewrites 49.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. count-2-rev N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-lft-in N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-sqrt.f64 49.8

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 49.8

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

6. Applied rewrites 49.8%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 37.9

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

8. Applied rewrites 37.9%

   \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]
9. 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. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 37.8

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

10. Applied rewrites 37.8%

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

Reproduce

herbie shell --seed 2025143 
(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))))