Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 10.4s

Alternatives: 8

Speedup: 1.1×

• could not determine a ground truth

  1. k = 4.6153314453442134e+153
  2. n = -0.037436418659901174

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, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot n\right)\\ \frac{\sqrt{t\_0}}{\sqrt{k \cdot {t\_0}^{k}}} \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* 2.0 n)))) (/ (sqrt t_0) (sqrt (* k (pow t_0 k))))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (2.0 * n);
	return sqrt(t_0) / sqrt((k * pow(t_0, k)));
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (2.0 * n);
	return Math.sqrt(t_0) / Math.sqrt((k * Math.pow(t_0, k)));
}

Python

def code(k, n):
	t_0 = math.pi * (2.0 * n)
	return math.sqrt(t_0) / math.sqrt((k * math.pow(t_0, k)))

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(2.0 * n))
	return Float64(sqrt(t_0) / sqrt(Float64(k * (t_0 ^ k))))
end

MATLAB

function tmp = code(k, n)
	t_0 = pi * (2.0 * n);
	tmp = sqrt(t_0) / sqrt((k * (t_0 ^ k)));
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(k * N[Power[t$95$0, k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.6%

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

   1. *-commutative N/A

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

   2. un-div-inv 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. div-sub N/A

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

   4. metadata-eval N/A

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

   5. pow-sub N/A

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

   6. associate-/l/ N/A

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

   7. /-lowering-/.f64 N/A

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

   8. unpow1/2 N/A

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

   9. sqrt-lowering-sqrt.f64 N/A

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

   10. associate-*l* N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. PI-lowering-PI.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. sqrt-lowering-sqrt.f64 N/A

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

   16. pow-lowering-pow.f64 N/A

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

4. Applied egg-rr 99.7%

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

   1. /-lowering-/.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. associate-/r/ N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 N/A

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

   9. PI-lowering-PI.f64 N/A

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

   10. associate-/r/ N/A

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

   11. metadata-eval N/A

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

   12. *-lowering-*.f64 N/A

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

   13. pow-unpow N/A

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

   14. unpow1/2 N/A

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

   15. sqrt-unprod N/A

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

   16. sqrt-lowering-sqrt.f64 N/A

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

   17. *-lowering-*.f64 N/A

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

   18. pow-lowering-pow.f64 N/A

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

6. Applied egg-rr 99.7%

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

Alternative 2: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(k \cdot -0.5\right)}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.0)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (/ (pow (* PI (* 2.0 n)) (* k -0.5)) (sqrt k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.0) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = pow((((double) M_PI) * (2.0 * n)), (k * -0.5)) / sqrt(k);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.0) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((2.0 * n));
	} else {
		tmp = Math.pow((Math.PI * (2.0 * n)), (k * -0.5)) / Math.sqrt(k);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.0:
		tmp = math.sqrt((math.pi / k)) * math.sqrt((2.0 * n))
	else:
		tmp = math.pow((math.pi * (2.0 * n)), (k * -0.5)) / math.sqrt(k)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.0)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = Float64((Float64(pi * Float64(2.0 * n)) ^ Float64(k * -0.5)) / sqrt(k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.0)
		tmp = sqrt((pi / k)) * sqrt((2.0 * n));
	else
		tmp = ((pi * (2.0 * n)) ^ (k * -0.5)) / sqrt(k);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.0], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(k * -0.5), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 99.2%

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

   3. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 75.3

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

   5. Simplified 75.3%

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

      1. sqrt-unprod N/A

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

      2. associate-*l/ N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. sqrt-undiv N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. associate-/r/ N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. PI-lowering-PI.f64 N/A

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

      15. associate-/r/ N/A

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

      16. metadata-eval N/A

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

      17. *-lowering-*.f64 N/A

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

      18. sqrt-lowering-sqrt.f64 98.4

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

   7. Applied egg-rr 98.4%

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

      1. sqrt-prod N/A

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

      2. pow1/2 N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. PI-lowering-PI.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. *-lowering-*.f64 98.5

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

   9. Applied egg-rr 98.5%

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

   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. Add Preprocessing

   3. Taylor expanded in k around inf

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

      1. *-lowering-*.f64 100.0

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   5. Simplified 100.0%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. /-lowering-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. associate-/r/ N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. PI-lowering-PI.f64 N/A

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

      13. associate-/r/ N/A

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

      14. metadata-eval N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 N/A

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

      18. sqrt-lowering-sqrt.f64 100.0

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

   7. Applied egg-rr 100.0%

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

Alternative 3: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}{\sqrt{k}} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (/ (pow (* PI (* 2.0 n)) (fma k -0.5 0.5)) (sqrt k)))

C

double code(double k, double n) {
	return pow((((double) M_PI) * (2.0 * n)), fma(k, -0.5, 0.5)) / sqrt(k);
}

Julia

function code(k, n)
	return Float64((Float64(pi * Float64(2.0 * n)) ^ fma(k, -0.5, 0.5)) / sqrt(k))
end

Wolfram

code[k_, n_] := N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(k * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. *-commutative N/A

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

   2. un-div-inv 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. div-sub N/A

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

   4. metadata-eval N/A

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

   5. pow-sub N/A

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

   6. associate-/l/ N/A

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

   7. /-lowering-/.f64 N/A

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

   8. unpow1/2 N/A

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

   9. sqrt-lowering-sqrt.f64 N/A

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

   10. associate-*l* N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. PI-lowering-PI.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. sqrt-lowering-sqrt.f64 N/A

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

   16. pow-lowering-pow.f64 N/A

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

4. Applied egg-rr 99.7%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}{\sqrt{k}}} \]
7. Add Preprocessing

Alternative 4: 61.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.5:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k} \cdot \frac{1}{k}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 0.5)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (sqrt (sqrt (* (/ 1.0 k) (/ 1.0 k))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 0.5) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = sqrt(sqrt(((1.0 / k) * (1.0 / k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 0.5) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((2.0 * n));
	} else {
		tmp = Math.sqrt(Math.sqrt(((1.0 / k) * (1.0 / k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 0.5:
		tmp = math.sqrt((math.pi / k)) * math.sqrt((2.0 * n))
	else:
		tmp = math.sqrt(math.sqrt(((1.0 / k) * (1.0 / k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 0.5)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = sqrt(sqrt(Float64(Float64(1.0 / k) * Float64(1.0 / k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 0.5)
		tmp = sqrt((pi / k)) * sqrt((2.0 * n));
	else
		tmp = sqrt(sqrt(((1.0 / k) * (1.0 / k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 0.5], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Sqrt[N[(N[(1.0 / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.5

   1. Initial program 99.2%

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

   3. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. PI-lowering-PI.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 75.3

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

   5. Simplified 75.3%

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

      1. sqrt-unprod N/A

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

      2. associate-*l/ N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. sqrt-undiv N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. associate-/r/ N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. PI-lowering-PI.f64 N/A

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

      15. associate-/r/ N/A

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

      16. metadata-eval N/A

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

      17. *-lowering-*.f64 N/A

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

      18. sqrt-lowering-sqrt.f64 98.4

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

   7. Applied egg-rr 98.4%

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

      1. sqrt-prod N/A

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

      2. pow1/2 N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. PI-lowering-PI.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. *-lowering-*.f64 98.5

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

   9. Applied egg-rr 98.5%

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

   if 0.5 < 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. Add Preprocessing

   3. Taylor expanded in k around inf

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

      1. *-lowering-*.f64 100.0

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   5. Simplified 100.0%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   7. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]

      2. /-lowering-/.f64 3.3

         \[\leadsto \sqrt{\color{blue}{\frac{1}{k}}} \]

   8. Simplified 3.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k}} \cdot \sqrt{\frac{1}{k}}}} \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k} \cdot \frac{1}{k}}}} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k} \cdot \frac{1}{k}}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\sqrt{\color{blue}{\frac{1}{k} \cdot \frac{1}{k}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\sqrt{\color{blue}{\frac{1}{k}} \cdot \frac{1}{k}}} \]

      6. /-lowering-/.f64 29.2

         \[\leadsto \sqrt{\sqrt{\frac{1}{k} \cdot \color{blue}{\frac{1}{k}}}} \]

   10. Applied egg-rr 29.2%

       \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k} \cdot \frac{1}{k}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 50.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. PI-lowering-PI.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 39.8

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

5. Simplified 39.8%

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

   1. sqrt-unprod N/A

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

   2. associate-*l/ N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. sqrt-undiv N/A

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

   6. /-lowering-/.f64 N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. associate-/r/ N/A

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

   12. *-commutative N/A

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

   13. *-lowering-*.f64 N/A

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

   14. PI-lowering-PI.f64 N/A

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

   15. associate-/r/ N/A

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f64 N/A

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

   18. sqrt-lowering-sqrt.f64 51.7

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

7. Applied egg-rr 51.7%

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

   1. sqrt-prod N/A

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

   2. pow1/2 N/A

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

   3. associate-*l/ N/A

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

   4. sqrt-div N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. PI-lowering-PI.f64 N/A

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

   9. pow1/2 N/A

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

   10. sqrt-lowering-sqrt.f64 N/A

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

   11. *-lowering-*.f64 51.8

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

9. Applied egg-rr 51.8%

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

Alternative 6: 50.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. PI-lowering-PI.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 39.8

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

5. Simplified 39.8%

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

   1. sqrt-unprod N/A

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

   2. associate-/l* N/A

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

   3. associate-*l* N/A

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

   4. sqrt-prod N/A

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

   5. pow1/2 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. pow1/2 N/A

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

   8. sqrt-lowering-sqrt.f64 N/A

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

   9. sqrt-lowering-sqrt.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. PI-lowering-PI.f64 51.8

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

7. Applied egg-rr 51.8%

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

8. Final simplification 51.8%

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

Alternative 7: 39.3% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. PI-lowering-PI.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 39.8

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

5. Simplified 39.8%

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

   1. sqrt-unprod N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. /-lowering-/.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. associate-/r/ N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 N/A

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

   13. PI-lowering-PI.f64 N/A

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

   14. associate-/r/ N/A

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 39.9

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

7. Applied egg-rr 39.9%

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

Alternative 8: 5.1% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{k}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (/ 1.0 k)))

C

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

Fortran

real(8) function code(k, n)
    real(8), intent (in) :: k
    real(8), intent (in) :: n
    code = sqrt((1.0d0 / k))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around inf

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

   1. *-lowering-*.f64 52.6

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

5. Simplified 52.6%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
7. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]

   2. /-lowering-/.f64 5.4

      \[\leadsto \sqrt{\color{blue}{\frac{1}{k}}} \]

8. Simplified 5.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024201 
(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))))