Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 12.2s

Alternatives: 8

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 8.866794125206053e+112
  2. n = -3.956322930195763e+133

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 * N[(Pi * 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 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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. *-lowering-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

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

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

   8. pow-unpow N/A

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

   9. unpow1/2 N/A

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

   10. sqrt-unprod N/A

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

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

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

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

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

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

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

   14. *-commutative N/A

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

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

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

   16. *-commutative N/A

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

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

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

   18. PI-lowering-PI.f64 99.7

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

6. Applied egg-rr 99.7%

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

Alternative 2: 60.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 5 \cdot 10^{-132}:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k} \cdot \frac{1}{k}}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{k}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (sqrt (* PI (* 2.0 n)))))
   (if (<=
        (* (/ 1.0 (sqrt k)) (pow (* n (* 2.0 PI)) (/ (- 1.0 k) 2.0)))
        5e-132)
     (* (sqrt (sqrt (* (/ 1.0 k) (/ 1.0 k)))) t_0)
     (* (sqrt (/ 1.0 k)) t_0))))

C

double code(double k, double n) {
	double t_0 = sqrt((((double) M_PI) * (2.0 * n)));
	double tmp;
	if (((1.0 / sqrt(k)) * pow((n * (2.0 * ((double) M_PI))), ((1.0 - k) / 2.0))) <= 5e-132) {
		tmp = sqrt(sqrt(((1.0 / k) * (1.0 / k)))) * t_0;
	} else {
		tmp = sqrt((1.0 / k)) * t_0;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.sqrt((Math.PI * (2.0 * n)));
	double tmp;
	if (((1.0 / Math.sqrt(k)) * Math.pow((n * (2.0 * Math.PI)), ((1.0 - k) / 2.0))) <= 5e-132) {
		tmp = Math.sqrt(Math.sqrt(((1.0 / k) * (1.0 / k)))) * t_0;
	} else {
		tmp = Math.sqrt((1.0 / k)) * t_0;
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.sqrt((math.pi * (2.0 * n)))
	tmp = 0
	if ((1.0 / math.sqrt(k)) * math.pow((n * (2.0 * math.pi)), ((1.0 - k) / 2.0))) <= 5e-132:
		tmp = math.sqrt(math.sqrt(((1.0 / k) * (1.0 / k)))) * t_0
	else:
		tmp = math.sqrt((1.0 / k)) * t_0
	return tmp

Julia

function code(k, n)
	t_0 = sqrt(Float64(pi * Float64(2.0 * n)))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(k)) * (Float64(n * Float64(2.0 * pi)) ^ Float64(Float64(1.0 - k) / 2.0))) <= 5e-132)
		tmp = Float64(sqrt(sqrt(Float64(Float64(1.0 / k) * Float64(1.0 / k)))) * t_0);
	else
		tmp = Float64(sqrt(Float64(1.0 / k)) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = sqrt((pi * (2.0 * n)));
	tmp = 0.0;
	if (((1.0 / sqrt(k)) * ((n * (2.0 * pi)) ^ ((1.0 - k) / 2.0))) <= 5e-132)
		tmp = sqrt(sqrt(((1.0 / k) * (1.0 / k)))) * t_0;
	else
		tmp = sqrt((1.0 / k)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-132], N[(N[Sqrt[N[Sqrt[N[(N[(1.0 / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 4.9999999999999999e-132

   1. Initial program 99.9%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 4.0

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

   5. Simplified 4.0%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   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. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. PI-lowering-PI.f64 4.0

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

   7. Applied egg-rr 4.0%

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

      1. div-inv N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. *-commutative N/A

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

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

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

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

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

      15. *-lowering-*.f64 4.0

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

   9. Applied egg-rr 4.0%

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

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

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

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

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

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

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

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

       6. /-lowering-/.f64 51.0

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

   11. Applied egg-rr 51.0%

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

   if 4.9999999999999999e-132 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   1. Initial program 99.3%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 48.6

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

   5. Simplified 48.6%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   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. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. PI-lowering-PI.f64 48.7

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

   7. Applied egg-rr 48.7%

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

      1. div-inv N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. *-commutative N/A

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

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

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

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

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

      15. *-lowering-*.f64 68.3

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

   9. Applied egg-rr 68.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 5 \cdot 10^{-132}:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k} \cdot \frac{1}{k}}} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{k}} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)} \end{array} \]

FPCore

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

C

double code(double k, double n) {
	return sqrt((1.0 / k)) * pow((n * (2.0 * ((double) M_PI))), fma(-0.5, k, 0.5));
}

Julia

function code(k, n)
	return Float64(sqrt(Float64(1.0 / k)) * (Float64(n * Float64(2.0 * pi)) ^ fma(-0.5, k, 0.5)))
end

Wolfram

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

3. Taylor expanded in k around inf

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

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

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

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

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

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

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. exp-prod N/A

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

   7. *-commutative N/A

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

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

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

   9. rem-exp-log N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. rem-exp-log N/A

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

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

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

   15. rem-exp-log N/A

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

   16. *-commutative N/A

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

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

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

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

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

   19. sub-neg N/A

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

   20. mul-1-neg N/A

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

   \[\leadsto \sqrt{\frac{1}{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)} \]
7. Add Preprocessing

Alternative 4: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[k_, n_] := N[(N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(k * -0.5 + 0.5), $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. Add Preprocessing
3. Step-by-step derivation

   1. div-sub N/A

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

   2. metadata-eval N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. metadata-eval 99.5

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

4. Applied egg-rr 99.5%

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

   1. metadata-eval N/A

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

   2. sqrt-div N/A

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

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

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

   4. /-lowering-/.f64 99.6

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

6. Applied egg-rr 99.6%

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

   1. *-commutative N/A

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

   2. sqrt-div N/A

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

   3. metadata-eval N/A

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

   4. un-div-inv N/A

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

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

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

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

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

   7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   12. sqrt-lowering-sqrt.f64 99.5

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

8. Applied egg-rr 99.5%

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

Alternative 5: 49.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $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. 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 34.7

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

5. Simplified 34.7%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
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. *-lowering-*.f64 N/A

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

   9. *-commutative N/A

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

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

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

   11. PI-lowering-PI.f64 34.8

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

7. Applied egg-rr 34.8%

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

   1. div-inv N/A

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

   2. sqrt-prod N/A

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

   3. pow1/2 N/A

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

   4. *-commutative N/A

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

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

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

   6. pow1/2 N/A

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

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

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

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

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

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

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

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

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

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

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

   15. *-lowering-*.f64 48.2

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

9. Applied egg-rr 48.2%

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

Alternative 6: 49.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(2.0 / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * 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. 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 34.7

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

5. Simplified 34.7%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
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. *-lowering-*.f64 N/A

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

   9. *-commutative N/A

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

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

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

   11. PI-lowering-PI.f64 34.8

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

7. Applied egg-rr 34.8%

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. sqrt-unprod N/A

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

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

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

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

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

   7. associate-*l/ N/A

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

   8. metadata-eval N/A

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

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

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

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

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

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

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

   12. PI-lowering-PI.f64 48.2

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

9. Applied egg-rr 48.2%

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

Alternative 7: 49.7% 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 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 34.7

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

5. Simplified 34.7%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
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 48.2

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

7. Applied egg-rr 48.2%

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

8. Final simplification 48.2%

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

Alternative 8: 38.1% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(Pi * n), $MachinePrecision] * N[(2.0 / 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. 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 34.7

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

5. Simplified 34.7%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
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. *-lowering-*.f64 N/A

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

   9. *-commutative N/A

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

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

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

   11. PI-lowering-PI.f64 34.8

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

7. Applied egg-rr 34.8%

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

   1. associate-*l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*l/ N/A

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

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

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

   5. associate-*l/ N/A

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

   6. metadata-eval N/A

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

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

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

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

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

   9. PI-lowering-PI.f64 34.8

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

9. Applied egg-rr 34.8%

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

10. Final simplification 34.8%

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

Reproduce

herbie shell --seed 2024198 
(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))))