Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 11.7s

Alternatives: 10

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 2.1343062093158676e+252
  2. n = -2.170167327858895e+239

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. associate-*l/ N/A

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

   11. neg-mul-1 N/A

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

   12. frac-2neg N/A

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

4. Applied egg-rr 99.6%

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

6. Applied egg-rr 99.6%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. sqrt-undiv N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. sqrt-prod N/A

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 67.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;{\left(\frac{1}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)}^{0.125}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* (/ 1.0 (sqrt k)) (pow (* n (* 2.0 PI)) (/ (- 1.0 k) 2.0))) 0.0)
   (pow (/ 1.0 (* (* k k) (* k k))) 0.125)
   (* (sqrt (* 2.0 n)) (sqrt (/ PI k)))))

C

double code(double k, double n) {
	double tmp;
	if (((1.0 / sqrt(k)) * pow((n * (2.0 * ((double) M_PI))), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = pow((1.0 / ((k * k) * (k * k))), 0.125);
	} else {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (((1.0 / Math.sqrt(k)) * Math.pow((n * (2.0 * Math.PI)), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = Math.pow((1.0 / ((k * k) * (k * k))), 0.125);
	} else {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((Math.PI / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if ((1.0 / math.sqrt(k)) * math.pow((n * (2.0 * math.pi)), ((1.0 - k) / 2.0))) <= 0.0:
		tmp = math.pow((1.0 / ((k * k) * (k * k))), 0.125)
	else:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((math.pi / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(k)) * (Float64(n * Float64(2.0 * pi)) ^ Float64(Float64(1.0 - k) / 2.0))) <= 0.0)
		tmp = Float64(1.0 / Float64(Float64(k * k) * Float64(k * k))) ^ 0.125;
	else
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (((1.0 / sqrt(k)) * ((n * (2.0 * pi)) ^ ((1.0 - k) / 2.0))) <= 0.0)
		tmp = (1.0 / ((k * k) * (k * k))) ^ 0.125;
	else
		tmp = sqrt((2.0 * n)) * sqrt((pi / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := 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], 0.0], N[Power[N[(1.0 / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.125], $MachinePrecision], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $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)))) < 0.0

   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. lower-*.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. lower-sqrt.f64 N/A

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

      2. lower-/.f64 3.8

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

   8. Simplified 3.8%

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lower-/.f64 3.8

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

   10. Applied egg-rr 3.8%

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

       1. metadata-eval N/A

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

       2. sqrt-div N/A

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

       3. lift-/.f64 N/A

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

       4. pow1/2 N/A

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

       5. metadata-eval N/A

          \[\leadsto {\left(\frac{1}{k}\right)}^{\color{blue}{\left(2 \cdot \frac{1}{4}\right)}} \]

       6. pow-pow N/A

          \[\leadsto \color{blue}{{\left({\left(\frac{1}{k}\right)}^{2}\right)}^{\frac{1}{4}}} \]

       7. pow2 N/A

          \[\leadsto {\color{blue}{\left(\frac{1}{k} \cdot \frac{1}{k}\right)}}^{\frac{1}{4}} \]

       8. lift-*.f64 N/A

          \[\leadsto {\color{blue}{\left(\frac{1}{k} \cdot \frac{1}{k}\right)}}^{\frac{1}{4}} \]

       9. sqr-pow N/A

          \[\leadsto \color{blue}{{\left(\frac{1}{k} \cdot \frac{1}{k}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot {\left(\frac{1}{k} \cdot \frac{1}{k}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}} \]

       10. pow-prod-down N/A

           \[\leadsto \color{blue}{{\left(\left(\frac{1}{k} \cdot \frac{1}{k}\right) \cdot \left(\frac{1}{k} \cdot \frac{1}{k}\right)\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}} \]

       11. lower-pow.f64 N/A

           \[\leadsto \color{blue}{{\left(\left(\frac{1}{k} \cdot \frac{1}{k}\right) \cdot \left(\frac{1}{k} \cdot \frac{1}{k}\right)\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}} \]

   12. Applied egg-rr 83.6%

       \[\leadsto \color{blue}{{\left(\frac{1}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)}^{0.125}} \]

   if 0.0 < (*.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 99.1%

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

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 55.5

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

   5. Simplified 55.5%

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

      1. lift-PI.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 55.5

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

   7. Applied egg-rr 55.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. div-inv N/A

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

   9. Applied egg-rr 73.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;{\left(\frac{1}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\right)}^{0.125}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 98.8%

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

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 73.7

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

   5. Simplified 73.7%

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

      1. lift-PI.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 73.7

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

   7. Applied egg-rr 73.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. div-inv N/A

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

   9. Applied egg-rr 97.2%

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

   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. lower-*.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. pow1/2 N/A

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

      2. pow-flip N/A

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

      3. metadata-eval N/A

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

      4. lift-PI.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. rem-exp-log N/A

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

      9. rem-exp-log N/A

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

      10. *-commutative N/A

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

      11. pow-unpow N/A

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

      12. pow-prod-down N/A

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

      13. metadata-eval N/A

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

      14. pow-flip N/A

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

      15. pow-prod-down N/A

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

      16. pow1/2 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. pow-unpow N/A

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

      19. lift-*.f64 N/A

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 98.4%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. associate-*l/ N/A

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

   11. neg-mul-1 N/A

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

   12. frac-2neg N/A

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

4. Applied egg-rr 99.6%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 5: 61.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.495:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot k\right)}^{-0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 0.495) (* (sqrt (* 2.0 n)) (sqrt (/ PI k))) (pow (* k k) -0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 0.495) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	} else {
		tmp = pow((k * k), -0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 0.495) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((Math.PI / k));
	} else {
		tmp = Math.pow((k * k), -0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 0.495:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((math.pi / k))
	else:
		tmp = math.pow((k * k), -0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 0.495)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	else
		tmp = Float64(k * k) ^ -0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 0.495)
		tmp = sqrt((2.0 * n)) * sqrt((pi / k));
	else
		tmp = (k * k) ^ -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 0.495], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(k * k), $MachinePrecision], -0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.495

   1. Initial program 98.8%

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

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 73.7

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

   5. Simplified 73.7%

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

      1. lift-PI.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 73.7

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

   7. Applied egg-rr 73.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. div-inv N/A

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

   9. Applied egg-rr 97.2%

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

   if 0.495 < 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. lower-*.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. lower-sqrt.f64 N/A

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

      2. lower-/.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. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lower-/.f64 3.3

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

   10. Applied egg-rr 3.3%

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

       1. pow1/2 N/A

          \[\leadsto \frac{1}{\color{blue}{{k}^{\frac{1}{2}}}} \]

       2. pow-flip N/A

          \[\leadsto \color{blue}{{k}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

       3. metadata-eval N/A

          \[\leadsto {k}^{\color{blue}{\frac{-1}{2}}} \]

       4. metadata-eval N/A

          \[\leadsto {k}^{\color{blue}{\left(2 \cdot \frac{-1}{4}\right)}} \]

       5. metadata-eval N/A

          \[\leadsto {k}^{\left(2 \cdot \color{blue}{\left(-1 \cdot \frac{1}{4}\right)}\right)} \]

       6. pow-sqr N/A

          \[\leadsto \color{blue}{{k}^{\left(-1 \cdot \frac{1}{4}\right)} \cdot {k}^{\left(-1 \cdot \frac{1}{4}\right)}} \]

       7. pow-prod-down N/A

          \[\leadsto \color{blue}{{\left(k \cdot k\right)}^{\left(-1 \cdot \frac{1}{4}\right)}} \]

       8. lower-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(k \cdot k\right)}^{\left(-1 \cdot \frac{1}{4}\right)}} \]

       9. lower-*.f64 N/A

          \[\leadsto {\color{blue}{\left(k \cdot k\right)}}^{\left(-1 \cdot \frac{1}{4}\right)} \]

       10. metadata-eval 26.9

           \[\leadsto {\left(k \cdot k\right)}^{\color{blue}{-0.25}} \]

   12. Applied egg-rr 26.9%

       \[\leadsto \color{blue}{{\left(k \cdot k\right)}^{-0.25}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 60.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.495:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k} \cdot \frac{1}{k}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 0.495)
   (* (sqrt (* 2.0 n)) (sqrt (/ PI k)))
   (sqrt (sqrt (* (/ 1.0 k) (/ 1.0 k))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 0.495) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	} 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 <= 0.495) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((Math.PI / k));
	} else {
		tmp = Math.sqrt(Math.sqrt(((1.0 / k) * (1.0 / k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 0.495:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((math.pi / k))
	else:
		tmp = math.sqrt(math.sqrt(((1.0 / k) * (1.0 / k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 0.495)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	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 <= 0.495)
		tmp = sqrt((2.0 * n)) * sqrt((pi / k));
	else
		tmp = sqrt(sqrt(((1.0 / k) * (1.0 / k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 0.495], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $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 < 0.495

   1. Initial program 98.8%

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

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 73.7

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

   5. Simplified 73.7%

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

      1. lift-PI.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 73.7

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

   7. Applied egg-rr 73.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. div-inv N/A

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

   9. Applied egg-rr 97.2%

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

   if 0.495 < 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. lower-*.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. lower-sqrt.f64 N/A

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

      2. lower-/.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. lift-/.f64 3.3

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

      2. rem-square-sqrt N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 25.2

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

   10. Applied egg-rr 25.2%

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

Alternative 7: 48.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 43.5

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

5. Simplified 43.5%

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

   1. lift-PI.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 43.5

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

7. Applied egg-rr 43.5%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lift-*.f64 N/A

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

   6. sqrt-undiv N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. associate-*l/ N/A

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

   11. div-inv N/A

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

9. Applied egg-rr 57.0%

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

Alternative 8: 48.5% 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 99.3%

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

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 43.5

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

5. Simplified 43.5%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sqrt-unprod N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. sqrt-prod N/A

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

   10. unpow1/2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow1/2 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 57.0

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

7. Applied egg-rr 57.0%

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

8. Final simplification 57.0%

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

Alternative 9: 37.1% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 43.5

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

5. Simplified 43.5%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sqrt-unprod N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-/.f64 43.6

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

7. Applied egg-rr 43.6%

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

8. Final simplification 43.6%

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

Alternative 10: 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 99.3%

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

3. Taylor expanded in k around 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. lower-*.f64 46.6

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

5. Simplified 46.6%

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

6. Taylor expanded in k around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.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 2024210 
(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))))