Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 11.4s

Alternatives: 10

Speedup: 1.1×

• could not determine a ground truth

  1. k = 1.665200541351124e+256
  2. n = -8.415826476049506e-11

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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. pow1/2 N/A

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

   2. pow-flip N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 99.5

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

4. Applied egg-rr 99.5%

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

6. Applied egg-rr 99.5%

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

   1. *-commutative N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 99.5

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

8. Applied egg-rr 99.5%

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

Alternative 2: 67.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (((1.0 / sqrt(k)) * ((n * (pi * 2.0)) ^ ((1.0 - k) / 2.0))) <= 0.0)
		tmp = (1.0 / ((k * k) * (k * k))) ^ 0.125;
	else
		tmp = sqrt((pi / k)) * sqrt((2.0 * n));
	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[(Pi * 2.0), $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[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $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. 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

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

       5. pow-pow N/A

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

       6. inv-pow N/A

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

       7. lift-/.f64 N/A

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

       8. metadata-eval N/A

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

       9. pow-prod-up N/A

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

       10. unpow-prod-down N/A

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

       11. lift-*.f64 N/A

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

       12. metadata-eval N/A

           \[\leadsto {\left(\frac{1}{k} \cdot \frac{1}{k}\right)}^{\color{blue}{\left(2 \cdot \frac{1}{8}\right)}} \]

       13. metadata-eval N/A

           \[\leadsto {\left(\frac{1}{k} \cdot \frac{1}{k}\right)}^{\left(2 \cdot \color{blue}{{\frac{1}{2}}^{3}}\right)} \]

       14. pow-sqr N/A

           \[\leadsto \color{blue}{{\left(\frac{1}{k} \cdot \frac{1}{k}\right)}^{\left({\frac{1}{2}}^{3}\right)} \cdot {\left(\frac{1}{k} \cdot \frac{1}{k}\right)}^{\left({\frac{1}{2}}^{3}\right)}} \]

       15. pow-prod-down N/A

           \[\leadsto \color{blue}{{\left(\left(\frac{1}{k} \cdot \frac{1}{k}\right) \cdot \left(\frac{1}{k} \cdot \frac{1}{k}\right)\right)}^{\left({\frac{1}{2}}^{3}\right)}} \]

       16. lower-pow.f64 N/A

           \[\leadsto \color{blue}{{\left(\left(\frac{1}{k} \cdot \frac{1}{k}\right) \cdot \left(\frac{1}{k} \cdot \frac{1}{k}\right)\right)}^{\left({\frac{1}{2}}^{3}\right)}} \]

   12. Applied egg-rr 77.9%

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

   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 99.2%

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

   3. Taylor expanded in k around 0

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

      1. 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 48.2

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

   5. Simplified 48.2%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   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. pow1/2 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. pow1/2 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 66.2

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

   7. Applied egg-rr 66.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. sqrt-unprod N/A

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

      16. lift-*.f64 N/A

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

      17. unpow1/2 N/A

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

      18. lower-*.f64 N/A

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

      19. pow1/2 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. unpow1/2 N/A

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

   9. Applied egg-rr 66.2%

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

4. Final simplification 68.9%

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

Alternative 3: 49.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {t\_0}^{\left(\frac{1 - k}{2}\right)} \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\frac{\pi \cdot \left(2 \cdot n\right)}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{k \cdot t\_0}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* n (* PI 2.0))))
   (if (<= (* (/ 1.0 (sqrt k)) (pow t_0 (/ (- 1.0 k) 2.0))) 5e+150)
     (sqrt (/ (* PI (* 2.0 n)) k))
     (/ (sqrt (* k t_0)) k))))

C

double code(double k, double n) {
	double t_0 = n * (((double) M_PI) * 2.0);
	double tmp;
	if (((1.0 / sqrt(k)) * pow(t_0, ((1.0 - k) / 2.0))) <= 5e+150) {
		tmp = sqrt(((((double) M_PI) * (2.0 * n)) / k));
	} else {
		tmp = sqrt((k * t_0)) / k;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = n * (Math.PI * 2.0);
	double tmp;
	if (((1.0 / Math.sqrt(k)) * Math.pow(t_0, ((1.0 - k) / 2.0))) <= 5e+150) {
		tmp = Math.sqrt(((Math.PI * (2.0 * n)) / k));
	} else {
		tmp = Math.sqrt((k * t_0)) / k;
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = n * (math.pi * 2.0)
	tmp = 0
	if ((1.0 / math.sqrt(k)) * math.pow(t_0, ((1.0 - k) / 2.0))) <= 5e+150:
		tmp = math.sqrt(((math.pi * (2.0 * n)) / k))
	else:
		tmp = math.sqrt((k * t_0)) / k
	return tmp

Julia

function code(k, n)
	t_0 = Float64(n * Float64(pi * 2.0))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(k)) * (t_0 ^ Float64(Float64(1.0 - k) / 2.0))) <= 5e+150)
		tmp = sqrt(Float64(Float64(pi * Float64(2.0 * n)) / k));
	else
		tmp = Float64(sqrt(Float64(k * t_0)) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = n * (pi * 2.0);
	tmp = 0.0;
	if (((1.0 / sqrt(k)) * (t_0 ^ ((1.0 - k) / 2.0))) <= 5e+150)
		tmp = sqrt(((pi * (2.0 * n)) / k));
	else
		tmp = sqrt((k * t_0)) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e+150], N[Sqrt[N[(N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(k * t$95$0), $MachinePrecision]], $MachinePrecision] / k), $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)))) < 5.00000000000000009e150

   1. Initial program 99.2%

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

   3. Taylor expanded in k around 0

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

      1. 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 61.2

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

   5. Simplified 61.2%

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 61.4

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 61.4

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

   7. Applied egg-rr 61.4%

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

   if 5.00000000000000009e150 < (*.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 99.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)\right) + \sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{k}} \]

   5. Simplified 70.2%

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

   6. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 38.5

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

   8. Simplified 38.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 38.5

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

   10. Applied egg-rr 38.4%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\frac{\pi \cdot \left(2 \cdot n\right)}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{k \cdot \left(n \cdot \left(\pi \cdot 2\right)\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 98.9%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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. 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 70.6

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

   5. Simplified 70.6%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   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. pow1/2 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. pow1/2 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 97.1

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

   7. Applied egg-rr 97.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. sqrt-unprod N/A

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

      16. lift-*.f64 N/A

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

      17. unpow1/2 N/A

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

      18. lower-*.f64 N/A

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

      19. pow1/2 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. unpow1/2 N/A

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

   9. Applied egg-rr 97.3%

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

   if 1 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around inf

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

      1. lower-*.f64 99.2

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

   5. Simplified 99.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 99.2

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 99.2

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.2

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

   7. Applied egg-rr 99.2%

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

Alternative 5: 61.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.5:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot k\right)}^{-0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 0.5) (* (sqrt (/ PI k)) (sqrt (* 2.0 n))) (pow (* k k) -0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 0.5) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = pow((k * k), -0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 0.5) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((2.0 * n));
	} else {
		tmp = Math.pow((k * k), -0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 0.5:
		tmp = math.sqrt((math.pi / k)) * math.sqrt((2.0 * n))
	else:
		tmp = math.pow((k * k), -0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 0.5)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = Float64(k * k) ^ -0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 0.5)
		tmp = sqrt((pi / k)) * sqrt((2.0 * n));
	else
		tmp = (k * k) ^ -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 0.5], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(k * k), $MachinePrecision], -0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.5

   1. Initial program 98.9%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. 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 70.6

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

   5. Simplified 70.6%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   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. pow1/2 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. pow1/2 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 97.1

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

   7. Applied egg-rr 97.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. sqrt-unprod N/A

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

      16. lift-*.f64 N/A

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

      17. unpow1/2 N/A

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

      18. lower-*.f64 N/A

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

      19. pow1/2 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. unpow1/2 N/A

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

   9. Applied egg-rr 97.3%

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

   if 0.5 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around inf

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

      1. lower-*.f64 99.2

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

   5. Simplified 99.2%

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

   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 3.2

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

   8. Simplified 3.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   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 3.2

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

   10. Applied egg-rr 3.2%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{k}}} \]
   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 25.0

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

   12. Applied egg-rr 25.0%

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

Alternative 6: 60.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 0.5

   1. Initial program 98.9%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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. 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 70.6

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

   5. Simplified 70.6%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   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. pow1/2 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. pow1/2 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 97.1

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

   7. Applied egg-rr 97.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. sqrt-unprod N/A

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

      16. lift-*.f64 N/A

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

      17. unpow1/2 N/A

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

      18. lower-*.f64 N/A

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

      19. pow1/2 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. unpow1/2 N/A

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

   9. Applied egg-rr 97.3%

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

   if 0.5 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around inf

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

      1. lower-*.f64 99.2

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

   5. Simplified 99.2%

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

   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 3.2

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

   8. Simplified 3.2%

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

      1. lift-/.f64 3.2

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

      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 23.5

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

   10. Applied egg-rr 23.5%

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

Alternative 7: 49.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 38.2

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

5. Simplified 38.2%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
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. pow1/2 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. pow1/2 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 52.1

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

7. Applied egg-rr 52.1%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. sqrt-prod N/A

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

   10. pow1/2 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. associate-*l* N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. sqrt-unprod N/A

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

   16. lift-*.f64 N/A

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

   17. unpow1/2 N/A

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

   18. lower-*.f64 N/A

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

   19. pow1/2 N/A

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

   20. lower-sqrt.f64 N/A

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

   21. unpow1/2 N/A

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

9. Applied egg-rr 52.2%

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

Alternative 8: 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 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 38.2

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

5. Simplified 38.2%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
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. pow1/2 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. pow1/2 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 52.1

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

7. Applied egg-rr 52.1%

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

8. Final simplification 52.1%

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

Alternative 9: 38.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 38.2

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

5. Simplified 38.2%

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

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-/.f64 38.3

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. associate-*l* N/A

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

   19. lower-*.f64 N/A

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

   20. lower-*.f64 38.3

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

7. Applied egg-rr 38.3%

   \[\leadsto \color{blue}{\sqrt{\frac{\pi \cdot \left(2 \cdot n\right)}{k}}} \]
8. 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 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 51.0

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

5. Simplified 51.0%

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

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 5.3

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

8. Simplified 5.3%

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

Reproduce

herbie shell --seed 2024211 
(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))))