Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.5%

Time: 15.4s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-20}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2e-20)
   (* (pow k -0.5) (sqrt (* PI (* 2.0 n))))
   (/ 1.0 (sqrt (/ k (pow (* 2.0 (* PI n)) (- 1.0 k)))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2e-20) {
		tmp = pow(k, -0.5) * sqrt((((double) M_PI) * (2.0 * n)));
	} else {
		tmp = 1.0 / sqrt((k / pow((2.0 * (((double) M_PI) * n)), (1.0 - k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2e-20) {
		tmp = Math.pow(k, -0.5) * Math.sqrt((Math.PI * (2.0 * n)));
	} else {
		tmp = 1.0 / Math.sqrt((k / Math.pow((2.0 * (Math.PI * n)), (1.0 - k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2e-20:
		tmp = math.pow(k, -0.5) * math.sqrt((math.pi * (2.0 * n)))
	else:
		tmp = 1.0 / math.sqrt((k / math.pow((2.0 * (math.pi * n)), (1.0 - k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2e-20)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(pi * Float64(2.0 * n))));
	else
		tmp = Float64(1.0 / sqrt(Float64(k / (Float64(2.0 * Float64(pi * n)) ^ Float64(1.0 - k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2e-20)
		tmp = (k ^ -0.5) * sqrt((pi * (2.0 * n)));
	else
		tmp = 1.0 / sqrt((k / ((2.0 * (pi * n)) ^ (1.0 - k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2e-20], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(k / N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999989e-20

   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 68.4%

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

      1. *-commutative 68.4%

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

      2. associate-/l* 68.4%

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

   5. Simplified 68.4%

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

      1. pow1 68.4%

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

      2. sqrt-unprod 68.6%

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

   7. Applied egg-rr 68.6%

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

      1. unpow1 68.6%

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

      2. associate-*r/ 68.6%

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

      3. associate-/l* 68.6%

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

      4. *-commutative 68.6%

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

      5. *-commutative 68.6%

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

      6. associate-/l* 68.6%

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

      7. *-commutative 68.6%

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

   9. Simplified 68.6%

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

       1. associate-*r/ 68.6%

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

       2. *-commutative 68.6%

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

       3. *-commutative 68.6%

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

       4. sqrt-div 99.4%

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

       5. div-inv 99.4%

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

       6. *-commutative 99.4%

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

       7. inv-pow 99.4%

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

       8. sqrt-pow2 99.5%

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

       9. metadata-eval 99.5%

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

       10. *-commutative 99.5%

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

       11. associate-*l* 99.5%

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

   11. Applied egg-rr 99.5%

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

   if 1.99999999999999989e-20 < k

   1. Initial program 99.5%

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

      1. associate-*l/ 99.5%

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

      2. *-lft-identity 99.5%

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

      3. associate-*l* 99.5%

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

      4. div-sub 99.5%

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

      5. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. div-inv 99.5%

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

      2. div-inv 99.5%

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

      3. metadata-eval 99.5%

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

      4. inv-pow 99.5%

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

      5. sqrt-pow2 99.5%

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

      6. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   8. Applied egg-rr 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-20}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-20}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.45e-20)
   (* (pow k -0.5) (sqrt (* PI (* 2.0 n))))
   (sqrt (/ (pow (* n (* 2.0 PI)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.45e-20) {
		tmp = pow(k, -0.5) * sqrt((((double) M_PI) * (2.0 * n)));
	} else {
		tmp = sqrt((pow((n * (2.0 * ((double) M_PI))), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.45e-20) {
		tmp = Math.pow(k, -0.5) * Math.sqrt((Math.PI * (2.0 * n)));
	} else {
		tmp = Math.sqrt((Math.pow((n * (2.0 * Math.PI)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.45e-20:
		tmp = math.pow(k, -0.5) * math.sqrt((math.pi * (2.0 * n)))
	else:
		tmp = math.sqrt((math.pow((n * (2.0 * math.pi)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.45e-20)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(pi * Float64(2.0 * n))));
	else
		tmp = sqrt(Float64((Float64(n * Float64(2.0 * pi)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.45e-20)
		tmp = (k ^ -0.5) * sqrt((pi * (2.0 * n)));
	else
		tmp = sqrt((((n * (2.0 * pi)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.45e-20], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.45e-20

   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 68.4%

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

      1. *-commutative 68.4%

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

      2. associate-/l* 68.4%

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

   5. Simplified 68.4%

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

      1. pow1 68.4%

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

      2. sqrt-unprod 68.6%

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

   7. Applied egg-rr 68.6%

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

      1. unpow1 68.6%

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

      2. associate-*r/ 68.6%

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

      3. associate-/l* 68.6%

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

      4. *-commutative 68.6%

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

      5. *-commutative 68.6%

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

      6. associate-/l* 68.6%

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

      7. *-commutative 68.6%

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

   9. Simplified 68.6%

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

       1. associate-*r/ 68.6%

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

       2. *-commutative 68.6%

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

       3. *-commutative 68.6%

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

       4. sqrt-div 99.4%

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

       5. div-inv 99.4%

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

       6. *-commutative 99.4%

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

       7. inv-pow 99.4%

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

       8. sqrt-pow2 99.5%

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

       9. metadata-eval 99.5%

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

       10. *-commutative 99.5%

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

       11. associate-*l* 99.5%

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

   11. Applied egg-rr 99.5%

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

   if 1.45e-20 < k

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 99.5%

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

      2. sqrt-unprod 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*r* 99.5%

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

      5. div-sub 99.5%

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

      6. metadata-eval 99.5%

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

      7. div-inv 99.5%

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

      8. *-commutative 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-20}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+101}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, \pi \cdot \frac{2}{k}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.5e+101)
   (* (pow k -0.5) (sqrt (* PI (* 2.0 n))))
   (sqrt (+ -1.0 (fma n (* PI (/ 2.0 k)) 1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.5e+101) {
		tmp = pow(k, -0.5) * sqrt((((double) M_PI) * (2.0 * n)));
	} else {
		tmp = sqrt((-1.0 + fma(n, (((double) M_PI) * (2.0 / k)), 1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.5e+101)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(pi * Float64(2.0 * n))));
	else
		tmp = sqrt(Float64(-1.0 + fma(n, Float64(pi * Float64(2.0 / k)), 1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.5e+101], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-1.0 + N[(n * N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.49999999999999997e101

   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 53.6%

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

      1. *-commutative 53.6%

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

      2. associate-/l* 53.6%

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

   5. Simplified 53.6%

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

      1. pow1 53.6%

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

      2. sqrt-unprod 53.7%

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

   7. Applied egg-rr 53.7%

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

      1. unpow1 53.7%

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

      2. associate-*r/ 53.7%

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

      3. associate-/l* 53.7%

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

      4. *-commutative 53.7%

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

      5. *-commutative 53.7%

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

      6. associate-/l* 53.7%

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

      7. *-commutative 53.7%

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

   9. Simplified 53.7%

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

       1. associate-*r/ 53.7%

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

       2. *-commutative 53.7%

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

       3. *-commutative 53.7%

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

       4. sqrt-div 75.8%

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

       5. div-inv 75.8%

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

       6. *-commutative 75.8%

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

       7. inv-pow 75.8%

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

       8. sqrt-pow2 75.8%

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

       9. metadata-eval 75.8%

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

       10. *-commutative 75.8%

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

       11. associate-*l* 75.8%

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

   11. Applied egg-rr 75.8%

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

   if 1.49999999999999997e101 < 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. Step-by-step derivation

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. div-sub 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. div-inv 100.0%

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

      2. div-inv 100.0%

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

      3. metadata-eval 100.0%

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

      4. inv-pow 100.0%

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

      5. sqrt-pow2 100.0%

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

      6. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in k around 0 2.7%

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

      1. *-commutative 2.7%

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

      2. *-commutative 2.7%

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

      3. associate-/l* 2.7%

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

   9. Simplified 2.7%

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

       1. sqrt-unprod 2.7%

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

       2. clear-num 2.7%

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

       3. un-div-inv 2.7%

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

   11. Applied egg-rr 2.7%

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

       1. associate-/r/ 2.7%

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

       2. *-commutative 2.7%

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

       3. *-commutative 2.7%

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

       4. expm1-log1p-u 2.7%

          \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(n \cdot \frac{\pi}{k}\right) \cdot 2\right)\right)}} \]

       5. expm1-undefine 32.4%

          \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\left(n \cdot \frac{\pi}{k}\right) \cdot 2\right)} - 1}} \]

       6. *-commutative 32.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \left(n \cdot \frac{\pi}{k}\right)}\right)} - 1} \]

       7. *-commutative 32.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(2 \cdot \color{blue}{\left(\frac{\pi}{k} \cdot n\right)}\right)} - 1} \]

       8. associate-/r/ 32.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(2 \cdot \color{blue}{\frac{\pi}{\frac{k}{n}}}\right)} - 1} \]

       9. associate-*r/ 32.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \pi}{\frac{k}{n}}}\right)} - 1} \]

       10. div-inv 32.4%

           \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\color{blue}{\left(2 \cdot \pi\right) \cdot \frac{1}{\frac{k}{n}}}\right)} - 1} \]

       11. *-commutative 32.4%

           \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot \frac{1}{\frac{k}{n}}\right)} - 1} \]

       12. clear-num 32.4%

           \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\left(\pi \cdot 2\right) \cdot \color{blue}{\frac{n}{k}}\right)} - 1} \]

   13. Applied egg-rr 32.4%

       \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\left(\pi \cdot 2\right) \cdot \frac{n}{k}\right)} - 1}} \]
   14. Step-by-step derivation

       1. sub-neg 32.4%

          \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\left(\pi \cdot 2\right) \cdot \frac{n}{k}\right)} + \left(-1\right)}} \]

       2. metadata-eval 32.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\left(\pi \cdot 2\right) \cdot \frac{n}{k}\right)} + \color{blue}{-1}} \]

       3. +-commutative 32.4%

          \[\leadsto \sqrt{\color{blue}{-1 + e^{\mathsf{log1p}\left(\left(\pi \cdot 2\right) \cdot \frac{n}{k}\right)}}} \]

       4. log1p-undefine 32.4%

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

       5. rem-exp-log 32.4%

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

       6. +-commutative 32.4%

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

       7. associate-*r/ 32.4%

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

       8. *-commutative 32.4%

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

       9. associate-*r* 32.4%

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

       10. *-commutative 32.4%

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

       11. associate-/l* 32.4%

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

       12. fma-define 32.4%

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

       13. *-lft-identity 32.4%

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

       14. metadata-eval 32.4%

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

       15. associate-/r* 32.4%

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

       16. neg-mul-1 32.4%

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

       17. fma-define 32.4%

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

   15. Simplified 32.4%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{+101}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, \pi \cdot \frac{2}{k}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double k, double n) {
	return Math.pow((2.0 * (Math.PI * n)), (0.5 - (0.5 * k))) * Math.pow(k, -0.5);
}

Python

def code(k, n):
	return math.pow((2.0 * (math.pi * n)), (0.5 - (0.5 * k))) * math.pow(k, -0.5)

Julia

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

MATLAB

function tmp = code(k, n)
	tmp = ((2.0 * (pi * n)) ^ (0.5 - (0.5 * k))) * (k ^ -0.5);
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. associate-*l* 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. div-inv 99.5%

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

   2. div-inv 99.5%

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

   3. metadata-eval 99.5%

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

   4. inv-pow 99.5%

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

   5. sqrt-pow2 99.5%

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

   6. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double k, double n) {
	return Math.pow((2.0 * (Math.PI * n)), (0.5 - (k / 2.0))) / Math.sqrt(k);
}

Python

def code(k, n):
	return math.pow((2.0 * (math.pi * n)), (0.5 - (k / 2.0))) / math.sqrt(k)

Julia

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

MATLAB

function tmp = code(k, n)
	tmp = ((2.0 * (pi * n)) ^ (0.5 - (k / 2.0))) / sqrt(k);
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. associate-*l* 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

3. Simplified 99.4%

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

Alternative 6: 49.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double k, double n) {
	return Math.pow(k, -0.5) * Math.sqrt((Math.PI * (2.0 * n)));
}

Python

def code(k, n):
	return math.pow(k, -0.5) * math.sqrt((math.pi * (2.0 * n)))

Julia

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

MATLAB

function tmp = code(k, n)
	tmp = (k ^ -0.5) * sqrt((pi * (2.0 * n)));
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 36.3%

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

   1. *-commutative 36.3%

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

   2. associate-/l* 36.3%

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

5. Simplified 36.3%

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

   1. pow1 36.3%

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

   2. sqrt-unprod 36.4%

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

7. Applied egg-rr 36.4%

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

   1. unpow1 36.4%

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

   2. associate-*r/ 36.4%

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

   3. associate-/l* 36.4%

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

   4. *-commutative 36.4%

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

   5. *-commutative 36.4%

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

   6. associate-/l* 36.4%

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

   7. *-commutative 36.4%

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

9. Simplified 36.4%

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

    1. associate-*r/ 36.4%

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

    2. *-commutative 36.4%

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

    3. *-commutative 36.4%

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

    4. sqrt-div 51.0%

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

    5. div-inv 51.0%

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

    6. *-commutative 51.0%

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

    7. inv-pow 51.0%

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

    8. sqrt-pow2 51.0%

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

    9. metadata-eval 51.0%

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

    10. *-commutative 51.0%

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

    11. associate-*l* 51.0%

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

11. Applied egg-rr 51.0%

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

12. Final simplification 51.0%

    \[\leadsto {k}^{-0.5} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)} \]
13. Add Preprocessing

Alternative 7: 49.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 36.3%

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

   1. *-commutative 36.3%

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

   2. associate-/l* 36.3%

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

5. Simplified 36.3%

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

   1. pow1 36.3%

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

   2. sqrt-unprod 36.4%

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

7. Applied egg-rr 36.4%

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

   1. unpow1 36.4%

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

   2. associate-*r/ 36.4%

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

   3. associate-/l* 36.4%

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

   4. *-commutative 36.4%

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

   5. *-commutative 36.4%

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

   6. associate-/l* 36.4%

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

   7. *-commutative 36.4%

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

9. Simplified 36.4%

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

    1. pow1/2 36.4%

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

    2. associate-*l* 36.3%

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

    3. unpow-prod-down 51.0%

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

    4. pow1/2 51.0%

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

11. Applied egg-rr 51.0%

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

    1. unpow1/2 51.0%

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

13. Simplified 51.0%

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

Alternative 8: 38.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. associate-*l* 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. div-inv 99.5%

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

   2. div-inv 99.5%

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

   3. metadata-eval 99.5%

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

   4. inv-pow 99.5%

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

   5. sqrt-pow2 99.5%

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

   6. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

8. Applied egg-rr 85.3%

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

9. Taylor expanded in k around 0 36.8%

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

    1. associate-*r* 36.8%

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

    2. *-commutative 36.8%

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

    3. *-commutative 36.8%

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

11. Simplified 36.8%

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

12. Final simplification 36.8%

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

Alternative 9: 38.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. associate-*l* 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. div-inv 99.5%

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

   2. div-inv 99.5%

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

   3. metadata-eval 99.5%

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

   4. inv-pow 99.5%

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

   5. sqrt-pow2 99.5%

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

   6. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in k around 0 36.3%

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

   1. *-commutative 36.3%

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

   2. *-commutative 36.3%

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

   3. associate-/l* 36.3%

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

9. Simplified 36.3%

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

    1. sqrt-unprod 36.4%

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

    2. clear-num 36.3%

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

    3. un-div-inv 36.4%

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

11. Applied egg-rr 36.4%

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

    1. clear-num 36.4%

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

    2. associate-/r/ 36.3%

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

    3. clear-num 36.4%

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

13. Applied egg-rr 36.4%

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

14. Final simplification 36.4%

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

Reproduce

herbie shell --seed 2024103 
(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))))