Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.4%

Time: 13.1s

Alternatives: 8

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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.1 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5.1e-29)
   (* (sqrt (* 2.0 (/ PI k))) (sqrt n))
   (pow (/ k (pow (* PI (* 2.0 n)) (- 1.0 k))) -0.5)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5.1e-29) {
		tmp = sqrt((2.0 * (((double) M_PI) / k))) * sqrt(n);
	} else {
		tmp = pow((k / pow((((double) M_PI) * (2.0 * n)), (1.0 - k))), -0.5);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 5.1e-29) {
		tmp = Math.sqrt((2.0 * (Math.PI / k))) * Math.sqrt(n);
	} else {
		tmp = Math.pow((k / Math.pow((Math.PI * (2.0 * n)), (1.0 - k))), -0.5);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 5.1e-29:
		tmp = math.sqrt((2.0 * (math.pi / k))) * math.sqrt(n)
	else:
		tmp = math.pow((k / math.pow((math.pi * (2.0 * n)), (1.0 - k))), -0.5)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5.1e-29)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi / k))) * sqrt(n));
	else
		tmp = Float64(k / (Float64(pi * Float64(2.0 * n)) ^ Float64(1.0 - k))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 5.1e-29)
		tmp = sqrt((2.0 * (pi / k))) * sqrt(n);
	else
		tmp = (k / ((pi * (2.0 * n)) ^ (1.0 - k))) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5.1e-29], N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Power[N[(k / N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.09999999999999986e-29

   1. Initial program 99.3%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

      3. associate-*l/ 97.7%

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

      4. *-un-lft-identity 97.7%

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

      5. associate-*r* 97.7%

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

      6. div-sub 97.7%

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

      7. metadata-eval 97.7%

         \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}}}\right)}^{3} \]

      8. div-inv 97.7%

         \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \color{blue}{k \cdot \frac{1}{2}}\right)}}{\sqrt{k}}}\right)}^{3} \]

      9. metadata-eval 97.7%

         \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot \color{blue}{0.5}\right)}}{\sqrt{k}}}\right)}^{3} \]

   3. Applied egg-rr 97.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\sqrt{k}}}\right)}^{3}} \]

   4. Taylor expanded in k around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-/l* 73.6%

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

   6. Simplified 73.6%

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

      1. expm1-log1p-u 69.9%

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

      2. expm1-udef 52.3%

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

      3. sqrt-unprod 52.3%

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

      4. associate-/r/ 52.3%

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

   8. Applied egg-rr 52.3%

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

      1. expm1-def 70.0%

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

      2. expm1-log1p 73.9%

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

      3. *-commutative 73.9%

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

      4. associate-*r* 73.9%

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

      5. *-commutative 73.9%

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

      6. associate-*l/ 73.8%

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

      7. *-lft-identity 73.8%

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

      8. times-frac 73.8%

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

      9. /-rgt-identity 73.8%

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

   10. Simplified 73.8%

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

       1. *-commutative 73.8%

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

       2. sqrt-prod 99.5%

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

       3. *-un-lft-identity 99.5%

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

       4. times-frac 99.5%

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

       5. metadata-eval 99.5%

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

   12. Applied egg-rr 99.5%

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

   if 5.09999999999999986e-29 < k

   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. Step-by-step derivation

      1. associate-/r/ 99.6%

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

      2. associate-*r* 99.6%

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

      3. div-sub 99.6%

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

      4. metadata-eval 99.6%

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

      5. div-inv 99.6%

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

      6. metadata-eval 99.6%

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

   3. Applied egg-rr 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

      \[\leadsto \color{blue}{{\left(\frac{k}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.1 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \]

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k + -1\right)}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.3e-30)
   (* (sqrt (* 2.0 (/ PI k))) (sqrt n))
   (pow (* k (pow (* 2.0 (* PI n)) (+ k -1.0))) -0.5)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.3e-30) {
		tmp = sqrt((2.0 * (((double) M_PI) / k))) * sqrt(n);
	} else {
		tmp = pow((k * pow((2.0 * (((double) M_PI) * n)), (k + -1.0))), -0.5);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.3e-30) {
		tmp = Math.sqrt((2.0 * (Math.PI / k))) * Math.sqrt(n);
	} else {
		tmp = Math.pow((k * Math.pow((2.0 * (Math.PI * n)), (k + -1.0))), -0.5);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.3e-30:
		tmp = math.sqrt((2.0 * (math.pi / k))) * math.sqrt(n)
	else:
		tmp = math.pow((k * math.pow((2.0 * (math.pi * n)), (k + -1.0))), -0.5)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.3e-30)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi / k))) * sqrt(n));
	else
		tmp = Float64(k * (Float64(2.0 * Float64(pi * n)) ^ Float64(k + -1.0))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.3e-30)
		tmp = sqrt((2.0 * (pi / k))) * sqrt(n);
	else
		tmp = (k * ((2.0 * (pi * n)) ^ (k + -1.0))) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.3e-30], N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Power[N[(k * N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(k + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.3000000000000003e-30

   1. Initial program 99.3%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

      3. associate-*l/ 97.7%

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

      4. *-un-lft-identity 97.7%

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

      5. associate-*r* 97.7%

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

      6. div-sub 97.7%

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

      7. metadata-eval 97.7%

         \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}}}\right)}^{3} \]

      8. div-inv 97.7%

         \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \color{blue}{k \cdot \frac{1}{2}}\right)}}{\sqrt{k}}}\right)}^{3} \]

      9. metadata-eval 97.7%

         \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot \color{blue}{0.5}\right)}}{\sqrt{k}}}\right)}^{3} \]

   3. Applied egg-rr 97.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\sqrt{k}}}\right)}^{3}} \]

   4. Taylor expanded in k around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-/l* 73.6%

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

   6. Simplified 73.6%

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

      1. expm1-log1p-u 69.9%

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

      2. expm1-udef 52.3%

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

      3. sqrt-unprod 52.3%

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

      4. associate-/r/ 52.3%

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

   8. Applied egg-rr 52.3%

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

      1. expm1-def 70.0%

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

      2. expm1-log1p 73.9%

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

      3. *-commutative 73.9%

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

      4. associate-*r* 73.9%

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

      5. *-commutative 73.9%

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

      6. associate-*l/ 73.8%

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

      7. *-lft-identity 73.8%

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

      8. times-frac 73.8%

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

      9. /-rgt-identity 73.8%

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

   10. Simplified 73.8%

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

       1. *-commutative 73.8%

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

       2. sqrt-prod 99.5%

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

       3. *-un-lft-identity 99.5%

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

       4. times-frac 99.5%

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

       5. metadata-eval 99.5%

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

   12. Applied egg-rr 99.5%

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

   if 3.3000000000000003e-30 < k

   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. Step-by-step derivation

      1. associate-/r/ 99.6%

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

      2. associate-*r* 99.6%

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

      3. div-sub 99.6%

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

      4. metadata-eval 99.6%

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

      5. div-inv 99.6%

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

      6. metadata-eval 99.6%

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

   3. Applied egg-rr 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. sub-neg 99.6%

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

      6. +-commutative 99.6%

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

      7. distribute-neg-in 99.6%

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

      8. remove-double-neg 99.6%

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

      9. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

      1. inv-pow 99.6%

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

      2. sqrt-unprod 99.6%

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

      3. sqrt-pow2 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*l* 99.6%

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

      6. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(k \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k + -1\right)}\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k + -1\right)}\right)}^{-0.5}\\ \end{array} \]

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.2e-29)
   (* (sqrt (* 2.0 (/ PI k))) (sqrt n))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.2e-29) {
		tmp = sqrt((2.0 * (((double) M_PI) / k))) * sqrt(n);
	} else {
		tmp = sqrt((pow((((double) M_PI) * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.2e-29) {
		tmp = Math.sqrt((2.0 * (Math.PI / k))) * Math.sqrt(n);
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 4.2e-29:
		tmp = math.sqrt((2.0 * (math.pi / k))) * math.sqrt(n)
	else:
		tmp = math.sqrt((math.pow((math.pi * (2.0 * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.2e-29)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi / k))) * sqrt(n));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(2.0 * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 4.2e-29)
		tmp = sqrt((2.0 * (pi / k))) * sqrt(n);
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.2e-29], N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.19999999999999979e-29

   1. Initial program 99.3%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

      3. associate-*l/ 97.7%

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

      4. *-un-lft-identity 97.7%

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

      5. associate-*r* 97.7%

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

      6. div-sub 97.7%

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

      7. metadata-eval 97.7%

         \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}}}\right)}^{3} \]

      8. div-inv 97.7%

         \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \color{blue}{k \cdot \frac{1}{2}}\right)}}{\sqrt{k}}}\right)}^{3} \]

      9. metadata-eval 97.7%

         \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot \color{blue}{0.5}\right)}}{\sqrt{k}}}\right)}^{3} \]

   3. Applied egg-rr 97.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\sqrt{k}}}\right)}^{3}} \]

   4. Taylor expanded in k around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-/l* 73.6%

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

   6. Simplified 73.6%

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

      1. expm1-log1p-u 69.9%

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

      2. expm1-udef 52.3%

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

      3. sqrt-unprod 52.3%

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

      4. associate-/r/ 52.3%

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

   8. Applied egg-rr 52.3%

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

      1. expm1-def 70.0%

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

      2. expm1-log1p 73.9%

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

      3. *-commutative 73.9%

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

      4. associate-*r* 73.9%

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

      5. *-commutative 73.9%

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

      6. associate-*l/ 73.8%

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

      7. *-lft-identity 73.8%

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

      8. times-frac 73.8%

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

      9. /-rgt-identity 73.8%

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

   10. Simplified 73.8%

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

       1. *-commutative 73.8%

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

       2. sqrt-prod 99.5%

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

       3. *-un-lft-identity 99.5%

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

       4. times-frac 99.5%

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

       5. metadata-eval 99.5%

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

   12. Applied egg-rr 99.5%

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

   if 4.19999999999999979e-29 < k

   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. Step-by-step derivation

      1. add-sqr-sqrt 99.6%

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

      2. sqrt-unprod 99.6%

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

      3. *-commutative 99.6%

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

      4. div-inv 99.6%

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

      5. *-commutative 99.6%

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

      6. div-inv 99.6%

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

      7. frac-times 99.6%

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

   3. Applied egg-rr 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 4: 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/ 99.5%

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

   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. sqr-pow 99.3%

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

   4. pow-sqr 99.5%

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

   5. associate-*l* 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l/ 99.5%

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

   8. associate-/l* 99.5%

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

   9. metadata-eval 99.5%

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

   10. /-rgt-identity 99.5%

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

   11. div-sub 99.5%

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

   12. 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. Final simplification 99.5%

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

Alternative 5: 37.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[Abs[N[(N[(n / k), $MachinePrecision] * N[(2.0 * Pi), $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. add-cube-cbrt 98.8%

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

   2. pow3 98.8%

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

   3. associate-*l/ 98.7%

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

   4. *-un-lft-identity 98.7%

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

   5. associate-*r* 98.7%

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

   6. div-sub 98.7%

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

   7. metadata-eval 98.7%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}}}\right)}^{3} \]

   8. div-inv 98.7%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \color{blue}{k \cdot \frac{1}{2}}\right)}}{\sqrt{k}}}\right)}^{3} \]

   9. metadata-eval 98.7%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot \color{blue}{0.5}\right)}}{\sqrt{k}}}\right)}^{3} \]

3. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\sqrt{k}}}\right)}^{3}} \]

4. Taylor expanded in k around 0 39.4%

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

   1. *-commutative 39.4%

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

   2. associate-/l* 39.4%

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

6. Simplified 39.4%

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

   1. expm1-log1p-u 37.5%

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

   2. expm1-udef 33.5%

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

   3. sqrt-unprod 33.5%

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

   4. associate-/r/ 33.5%

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

8. Applied egg-rr 33.5%

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

   1. expm1-def 37.6%

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

   2. expm1-log1p 39.5%

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

   3. *-commutative 39.5%

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

   4. associate-*r* 39.5%

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

   5. *-commutative 39.5%

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

   6. associate-*l/ 39.5%

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

   7. *-lft-identity 39.5%

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

   8. times-frac 39.5%

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

   9. /-rgt-identity 39.5%

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

10. Simplified 39.5%

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

    1. add-sqr-sqrt 39.5%

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

    2. pow1/2 39.5%

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

    3. pow1/2 39.5%

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

    4. pow-prod-down 26.9%

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

    5. pow2 26.9%

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

    6. associate-*r/ 27.0%

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

    7. *-commutative 27.0%

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

    8. associate-*l* 27.0%

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

    9. *-commutative 27.0%

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

    10. *-un-lft-identity 27.0%

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

    11. times-frac 27.0%

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

    12. metadata-eval 27.0%

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

    13. associate-/l* 27.0%

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

12. Applied egg-rr 27.0%

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

    1. unpow1/2 27.0%

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

    2. unpow2 27.0%

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

    3. rem-sqrt-square 39.5%

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

    4. *-commutative 39.5%

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

    5. associate-/r/ 39.6%

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

    6. associate-*l* 39.6%

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

    7. *-commutative 39.6%

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

14. Simplified 39.6%

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

15. Final simplification 39.6%

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

Alternative 6: 49.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $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. add-cube-cbrt 98.8%

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

   2. pow3 98.8%

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

   3. associate-*l/ 98.7%

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

   4. *-un-lft-identity 98.7%

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

   5. associate-*r* 98.7%

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

   6. div-sub 98.7%

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

   7. metadata-eval 98.7%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}}}\right)}^{3} \]

   8. div-inv 98.7%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \color{blue}{k \cdot \frac{1}{2}}\right)}}{\sqrt{k}}}\right)}^{3} \]

   9. metadata-eval 98.7%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot \color{blue}{0.5}\right)}}{\sqrt{k}}}\right)}^{3} \]

3. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\sqrt{k}}}\right)}^{3}} \]

4. Taylor expanded in k around 0 39.4%

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

   1. *-commutative 39.4%

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

   2. associate-/l* 39.4%

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

6. Simplified 39.4%

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

   1. expm1-log1p-u 37.5%

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

   2. expm1-udef 33.5%

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

   3. sqrt-unprod 33.5%

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

   4. associate-/r/ 33.5%

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

8. Applied egg-rr 33.5%

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

   1. expm1-def 37.6%

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

   2. expm1-log1p 39.5%

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

   3. *-commutative 39.5%

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

   4. associate-*r* 39.5%

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

   5. *-commutative 39.5%

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

   6. associate-*l/ 39.5%

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

   7. *-lft-identity 39.5%

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

   8. times-frac 39.5%

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

   9. /-rgt-identity 39.5%

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

10. Simplified 39.5%

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

    1. *-commutative 39.5%

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

    2. sqrt-prod 51.5%

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

    3. *-un-lft-identity 51.5%

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

    4. times-frac 51.5%

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

    5. metadata-eval 51.5%

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

12. Applied egg-rr 51.5%

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

13. Final simplification 51.5%

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

Alternative 7: 37.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-cube-cbrt 98.8%

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

   2. pow3 98.8%

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

   3. associate-*l/ 98.7%

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

   4. *-un-lft-identity 98.7%

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

   5. associate-*r* 98.7%

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

   6. div-sub 98.7%

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

   7. metadata-eval 98.7%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}}}\right)}^{3} \]

   8. div-inv 98.7%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \color{blue}{k \cdot \frac{1}{2}}\right)}}{\sqrt{k}}}\right)}^{3} \]

   9. metadata-eval 98.7%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot \color{blue}{0.5}\right)}}{\sqrt{k}}}\right)}^{3} \]

3. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\sqrt{k}}}\right)}^{3}} \]

4. Taylor expanded in k around 0 39.4%

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

   1. *-commutative 39.4%

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

   2. associate-/l* 39.4%

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

6. Simplified 39.4%

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

   1. expm1-log1p-u 37.5%

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

   2. expm1-udef 33.5%

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

   3. sqrt-unprod 33.5%

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

   4. associate-/r/ 33.5%

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

8. Applied egg-rr 33.5%

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

   1. expm1-def 37.6%

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

   2. expm1-log1p 39.5%

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

   3. *-commutative 39.5%

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

   4. associate-*r* 39.5%

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

   5. *-commutative 39.5%

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

   6. associate-*l/ 39.5%

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

   7. *-lft-identity 39.5%

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

   8. times-frac 39.5%

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

   9. /-rgt-identity 39.5%

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

10. Simplified 39.5%

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

11. Final simplification 39.5%

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

Alternative 8: 37.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(n / k), $MachinePrecision] * N[(2.0 * Pi), $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. add-cube-cbrt 98.8%

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

   2. pow3 98.8%

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

   3. associate-*l/ 98.7%

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

   4. *-un-lft-identity 98.7%

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

   5. associate-*r* 98.7%

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

   6. div-sub 98.7%

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

   7. metadata-eval 98.7%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}}}\right)}^{3} \]

   8. div-inv 98.7%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \color{blue}{k \cdot \frac{1}{2}}\right)}}{\sqrt{k}}}\right)}^{3} \]

   9. metadata-eval 98.7%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot \color{blue}{0.5}\right)}}{\sqrt{k}}}\right)}^{3} \]

3. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\sqrt{k}}}\right)}^{3}} \]

4. Taylor expanded in k around 0 39.4%

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

   1. *-commutative 39.4%

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

   2. associate-/l* 39.4%

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

6. Simplified 39.4%

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

   1. expm1-log1p-u 37.5%

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

   2. expm1-udef 33.5%

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

   3. sqrt-unprod 33.5%

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

   4. associate-/r/ 33.5%

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

8. Applied egg-rr 33.5%

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

   1. expm1-def 37.6%

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

   2. expm1-log1p 39.5%

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

   3. *-commutative 39.5%

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

   4. associate-*l* 39.5%

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

   5. *-commutative 39.5%

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

10. Simplified 39.5%

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

11. Final simplification 39.5%

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

Reproduce

herbie shell --seed 2023305 
(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))))