Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 11.9s

Alternatives: 10

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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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-*l/ 99.7%

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

   2. *-lft-identity 99.7%

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

   3. sqr-pow 99.5%

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

   4. pow-sqr 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l/ 99.7%

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

   7. associate-/l* 99.7%

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

   8. metadata-eval 99.7%

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

   9. /-rgt-identity 99.7%

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

   10. div-sub 99.7%

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

   11. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 41.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.76 \cdot 10^{+189}:\\ \;\;\;\;{\left(0.5 \cdot \frac{k}{\pi \cdot n}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(n \cdot \frac{\pi}{k}\right)}^{2} \cdot 4\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.76e+189)
   (pow (* 0.5 (/ k (* PI n))) -0.5)
   (pow (* (pow (* n (/ PI k)) 2.0) 4.0) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.76e+189) {
		tmp = pow((0.5 * (k / (((double) M_PI) * n))), -0.5);
	} else {
		tmp = pow((pow((n * (((double) M_PI) / k)), 2.0) * 4.0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.76e+189) {
		tmp = Math.pow((0.5 * (k / (Math.PI * n))), -0.5);
	} else {
		tmp = Math.pow((Math.pow((n * (Math.PI / k)), 2.0) * 4.0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.76e+189:
		tmp = math.pow((0.5 * (k / (math.pi * n))), -0.5)
	else:
		tmp = math.pow((math.pow((n * (math.pi / k)), 2.0) * 4.0), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.76e+189)
		tmp = Float64(0.5 * Float64(k / Float64(pi * n))) ^ -0.5;
	else
		tmp = Float64((Float64(n * Float64(pi / k)) ^ 2.0) * 4.0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.76e+189)
		tmp = (0.5 * (k / (pi * n))) ^ -0.5;
	else
		tmp = (((n * (pi / k)) ^ 2.0) * 4.0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.76e+189], N[Power[N[(0.5 * N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[Power[N[(N[Power[N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 4.0), $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.75999999999999999e189

   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. expm1-log1p-u 95.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{k}}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

      2. expm1-udef 79.0%

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

      3. inv-pow 79.0%

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

      4. sqrt-pow2 79.0%

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

      5. metadata-eval 79.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left({k}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   3. Applied egg-rr 79.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-0.5}\right)} - 1\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   4. Step-by-step derivation

      1. expm1-def 95.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

      2. expm1-log1p 99.6%

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

   5. Simplified 99.6%

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

      1. metadata-eval 99.6%

         \[\leadsto {k}^{\color{blue}{\left(0.25 \cdot -2\right)}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

      2. pow-pow 99.3%

         \[\leadsto \color{blue}{{\left({k}^{0.25}\right)}^{-2}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

      3. add-sqr-sqrt 99.2%

         \[\leadsto \color{blue}{\sqrt{{\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \sqrt{{\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}} \]

      4. sqrt-unprod 86.5%

         \[\leadsto \color{blue}{\sqrt{\left({\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \left({\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)}} \]

      5. *-commutative 86.5%

         \[\leadsto \sqrt{\color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left({k}^{0.25}\right)}^{-2}\right)} \cdot \left({\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)} \]

      6. *-commutative 86.5%

         \[\leadsto \sqrt{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left({k}^{0.25}\right)}^{-2}\right) \cdot \color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left({k}^{0.25}\right)}^{-2}\right)}} \]

      7. swap-sqr 86.4%

         \[\leadsto \sqrt{\color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \left({\left({k}^{0.25}\right)}^{-2} \cdot {\left({k}^{0.25}\right)}^{-2}\right)}} \]

   7. Applied egg-rr 86.9%

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

      1. pow1/2 86.9%

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

      2. pow-flip 87.1%

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

      3. *-commutative 87.1%

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

      4. associate-*r* 87.1%

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

      5. metadata-eval 87.1%

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

   9. Applied egg-rr 87.1%

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

   10. Taylor expanded in k around 0 49.5%

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

   if 1.75999999999999999e189 < 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. *-commutative 100.0%

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

      2. div-sub 100.0%

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

      3. metadata-eval 100.0%

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

      4. div-inv 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. frac-times 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in k around 0 2.4%

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

      1. associate-/l* 2.4%

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

      2. associate-/r/ 2.4%

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

   6. Simplified 2.4%

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

      1. pow1/2 2.4%

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

      2. metadata-eval 2.4%

         \[\leadsto {\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{\color{blue}{\left(0.25 + 0.25\right)}} \]

      3. pow-prod-up 2.4%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{0.25} \cdot {\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{0.25}} \]

      4. pow-prod-down 11.3%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right) \cdot \left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)\right)}^{0.25}} \]

      5. pow2 11.3%

         \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{2}\right)}}^{0.25} \]

      6. *-commutative 11.3%

         \[\leadsto {\left({\left(2 \cdot \color{blue}{\left(\pi \cdot \frac{n}{k}\right)}\right)}^{2}\right)}^{0.25} \]

   8. Applied egg-rr 11.3%

      \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{2}\right)}^{0.25}} \]
   9. Step-by-step derivation

      1. unpow2 11.3%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right) \cdot \left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)\right)}}^{0.25} \]

      2. *-commutative 11.3%

         \[\leadsto {\left(\color{blue}{\left(\left(\pi \cdot \frac{n}{k}\right) \cdot 2\right)} \cdot \left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)\right)}^{0.25} \]

      3. *-commutative 11.3%

         \[\leadsto {\left(\left(\left(\pi \cdot \frac{n}{k}\right) \cdot 2\right) \cdot \color{blue}{\left(\left(\pi \cdot \frac{n}{k}\right) \cdot 2\right)}\right)}^{0.25} \]

      4. swap-sqr 11.3%

         \[\leadsto {\color{blue}{\left(\left(\left(\pi \cdot \frac{n}{k}\right) \cdot \left(\pi \cdot \frac{n}{k}\right)\right) \cdot \left(2 \cdot 2\right)\right)}}^{0.25} \]

      5. unpow2 11.3%

         \[\leadsto {\left(\color{blue}{{\left(\pi \cdot \frac{n}{k}\right)}^{2}} \cdot \left(2 \cdot 2\right)\right)}^{0.25} \]

      6. associate-*r/ 11.3%

         \[\leadsto {\left({\color{blue}{\left(\frac{\pi \cdot n}{k}\right)}}^{2} \cdot \left(2 \cdot 2\right)\right)}^{0.25} \]

      7. associate-*l/ 11.3%

         \[\leadsto {\left({\color{blue}{\left(\frac{\pi}{k} \cdot n\right)}}^{2} \cdot \left(2 \cdot 2\right)\right)}^{0.25} \]

      8. *-commutative 11.3%

         \[\leadsto {\left({\color{blue}{\left(n \cdot \frac{\pi}{k}\right)}}^{2} \cdot \left(2 \cdot 2\right)\right)}^{0.25} \]

      9. metadata-eval 11.3%

         \[\leadsto {\left({\left(n \cdot \frac{\pi}{k}\right)}^{2} \cdot \color{blue}{4}\right)}^{0.25} \]

   10. Simplified 11.3%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.76 \cdot 10^{+189}:\\ \;\;\;\;{\left(0.5 \cdot \frac{k}{\pi \cdot n}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(n \cdot \frac{\pi}{k}\right)}^{2} \cdot 4\right)}^{0.25}\\ \end{array} \]

Alternative 3: 44.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.85 \cdot 10^{+132}:\\ \;\;\;\;{\left(0.5 \cdot \frac{k}{\pi \cdot n}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{\pi \cdot n}{\frac{k}{2}}\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.85e+132)
   (pow (* 0.5 (/ k (* PI n))) -0.5)
   (pow (pow (/ (* PI n) (/ k 2.0)) 3.0) 0.16666666666666666)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.85e+132) {
		tmp = pow((0.5 * (k / (((double) M_PI) * n))), -0.5);
	} else {
		tmp = pow(pow(((((double) M_PI) * n) / (k / 2.0)), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.85e+132) {
		tmp = Math.pow((0.5 * (k / (Math.PI * n))), -0.5);
	} else {
		tmp = Math.pow(Math.pow(((Math.PI * n) / (k / 2.0)), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.85e+132:
		tmp = math.pow((0.5 * (k / (math.pi * n))), -0.5)
	else:
		tmp = math.pow(math.pow(((math.pi * n) / (k / 2.0)), 3.0), 0.16666666666666666)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.85e+132)
		tmp = Float64(0.5 * Float64(k / Float64(pi * n))) ^ -0.5;
	else
		tmp = (Float64(Float64(pi * n) / Float64(k / 2.0)) ^ 3.0) ^ 0.16666666666666666;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.85e+132)
		tmp = (0.5 * (k / (pi * n))) ^ -0.5;
	else
		tmp = (((pi * n) / (k / 2.0)) ^ 3.0) ^ 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.85e+132], N[Power[N[(0.5 * N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[Power[N[Power[N[(N[(Pi * n), $MachinePrecision] / N[(k / 2.0), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.8499999999999999e132

   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. Step-by-step derivation

      1. expm1-log1p-u 94.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{k}}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

      2. expm1-udef 82.3%

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

      3. inv-pow 82.3%

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

      4. sqrt-pow2 82.3%

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

      5. metadata-eval 82.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left({k}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   3. Applied egg-rr 82.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-0.5}\right)} - 1\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   4. Step-by-step derivation

      1. expm1-def 94.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

      2. expm1-log1p 99.5%

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

   5. Simplified 99.5%

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

      1. metadata-eval 99.5%

         \[\leadsto {k}^{\color{blue}{\left(0.25 \cdot -2\right)}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

      2. pow-pow 99.3%

         \[\leadsto \color{blue}{{\left({k}^{0.25}\right)}^{-2}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

      3. add-sqr-sqrt 99.1%

         \[\leadsto \color{blue}{\sqrt{{\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \sqrt{{\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}} \]

      4. sqrt-unprod 85.1%

         \[\leadsto \color{blue}{\sqrt{\left({\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \left({\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)}} \]

      5. *-commutative 85.1%

         \[\leadsto \sqrt{\color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left({k}^{0.25}\right)}^{-2}\right)} \cdot \left({\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)} \]

      6. *-commutative 85.1%

         \[\leadsto \sqrt{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left({k}^{0.25}\right)}^{-2}\right) \cdot \color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left({k}^{0.25}\right)}^{-2}\right)}} \]

      7. swap-sqr 85.1%

         \[\leadsto \sqrt{\color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \left({\left({k}^{0.25}\right)}^{-2} \cdot {\left({k}^{0.25}\right)}^{-2}\right)}} \]

   7. Applied egg-rr 85.6%

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

      1. pow1/2 85.6%

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

      2. pow-flip 85.7%

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

      3. *-commutative 85.7%

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

      4. associate-*r* 85.7%

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

      5. metadata-eval 85.7%

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

   9. Applied egg-rr 85.7%

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

   10. Taylor expanded in k around 0 54.3%

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

   if 2.8499999999999999e132 < 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. *-commutative 100.0%

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

      2. div-sub 100.0%

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

      3. metadata-eval 100.0%

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

      4. div-inv 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. frac-times 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in k around 0 2.4%

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

      1. associate-/l* 2.4%

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

      2. associate-/r/ 2.4%

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

   6. Simplified 2.4%

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

      1. *-commutative 2.4%

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

      2. clear-num 2.4%

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

      3. un-div-inv 2.4%

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

   8. Applied egg-rr 2.4%

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

      1. pow1/2 2.4%

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

      2. div-inv 2.4%

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

      3. clear-num 2.4%

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

      4. metadata-eval 2.4%

         \[\leadsto {\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      5. pow-pow 5.0%

         \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

      6. sqr-pow 5.0%

         \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left({\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

      7. pow-prod-down 12.9%

         \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5} \cdot {\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

   10. Applied egg-rr 12.9%

       \[\leadsto \color{blue}{{\left({\left(\pi \cdot \left(\frac{n}{k} \cdot 2\right)\right)}^{3}\right)}^{0.16666666666666666}} \]
   11. Step-by-step derivation

       1. *-commutative 12.9%

          \[\leadsto {\left({\left(\pi \cdot \color{blue}{\left(2 \cdot \frac{n}{k}\right)}\right)}^{3}\right)}^{0.16666666666666666} \]

       2. associate-*r* 12.9%

          \[\leadsto {\left({\color{blue}{\left(\left(\pi \cdot 2\right) \cdot \frac{n}{k}\right)}}^{3}\right)}^{0.16666666666666666} \]

       3. associate-*r/ 12.9%

          \[\leadsto {\left({\color{blue}{\left(\frac{\left(\pi \cdot 2\right) \cdot n}{k}\right)}}^{3}\right)}^{0.16666666666666666} \]

       4. associate-*l/ 12.9%

          \[\leadsto {\left({\color{blue}{\left(\frac{\pi \cdot 2}{k} \cdot n\right)}}^{3}\right)}^{0.16666666666666666} \]

       5. associate-/l* 12.9%

          \[\leadsto {\left({\left(\color{blue}{\frac{\pi}{\frac{k}{2}}} \cdot n\right)}^{3}\right)}^{0.16666666666666666} \]

       6. associate-*l/ 12.9%

          \[\leadsto {\left({\color{blue}{\left(\frac{\pi \cdot n}{\frac{k}{2}}\right)}}^{3}\right)}^{0.16666666666666666} \]

       7. *-commutative 12.9%

          \[\leadsto {\left({\left(\frac{\color{blue}{n \cdot \pi}}{\frac{k}{2}}\right)}^{3}\right)}^{0.16666666666666666} \]

   12. Simplified 12.9%

       \[\leadsto \color{blue}{{\left({\left(\frac{n \cdot \pi}{\frac{k}{2}}\right)}^{3}\right)}^{0.16666666666666666}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.85 \cdot 10^{+132}:\\ \;\;\;\;{\left(0.5 \cdot \frac{k}{\pi \cdot n}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{\pi \cdot n}{\frac{k}{2}}\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \]

Alternative 4: 88.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double k, double n) {
	return Math.pow((k / Math.pow((n * (2.0 * Math.PI)), (1.0 - k))), -0.5);
}

Python

def code(k, n):
	return math.pow((k / math.pow((n * (2.0 * math.pi)), (1.0 - k))), -0.5)

Julia

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

MATLAB

function tmp = code(k, n)
	tmp = (k / ((n * (2.0 * pi)) ^ (1.0 - k))) ^ -0.5;
end

Wolfram

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

Derivation

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. expm1-log1p-u 96.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{k}}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. expm1-udef 70.5%

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

   3. inv-pow 70.5%

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

   4. sqrt-pow2 70.5%

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

   5. metadata-eval 70.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left({k}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

3. Applied egg-rr 70.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-0.5}\right)} - 1\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
4. Step-by-step derivation

   1. expm1-def 96.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. expm1-log1p 99.7%

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

5. Simplified 99.7%

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

   1. metadata-eval 99.7%

      \[\leadsto {k}^{\color{blue}{\left(0.25 \cdot -2\right)}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. pow-pow 99.5%

      \[\leadsto \color{blue}{{\left({k}^{0.25}\right)}^{-2}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   3. add-sqr-sqrt 99.4%

      \[\leadsto \color{blue}{\sqrt{{\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \sqrt{{\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}} \]

   4. sqrt-unprod 89.3%

      \[\leadsto \color{blue}{\sqrt{\left({\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \left({\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)}} \]

   5. *-commutative 89.3%

      \[\leadsto \sqrt{\color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left({k}^{0.25}\right)}^{-2}\right)} \cdot \left({\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)} \]

   6. *-commutative 89.3%

      \[\leadsto \sqrt{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left({k}^{0.25}\right)}^{-2}\right) \cdot \color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left({k}^{0.25}\right)}^{-2}\right)}} \]

   7. swap-sqr 89.3%

      \[\leadsto \sqrt{\color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \left({\left({k}^{0.25}\right)}^{-2} \cdot {\left({k}^{0.25}\right)}^{-2}\right)}} \]

7. Applied egg-rr 89.6%

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

   1. pow1/2 89.6%

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

   2. pow-flip 89.7%

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

   3. *-commutative 89.7%

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

   4. associate-*r* 89.7%

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

   5. metadata-eval 89.7%

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

9. Applied egg-rr 89.7%

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

10. Final simplification 89.7%

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

Alternative 5: 87.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. *-commutative 99.6%

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

   2. div-sub 99.6%

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

   3. metadata-eval 99.6%

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

   4. div-inv 99.7%

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

   5. add-sqr-sqrt 99.4%

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

   6. sqrt-unprod 89.5%

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

   7. frac-times 89.4%

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

3. Applied egg-rr 89.5%

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

4. Final simplification 89.5%

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

Alternative 6: 39.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. expm1-log1p-u 96.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{k}}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. expm1-udef 70.5%

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

   3. inv-pow 70.5%

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

   4. sqrt-pow2 70.5%

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

   5. metadata-eval 70.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left({k}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

3. Applied egg-rr 70.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-0.5}\right)} - 1\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
4. Step-by-step derivation

   1. expm1-def 96.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. expm1-log1p 99.7%

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

5. Simplified 99.7%

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

   1. metadata-eval 99.7%

      \[\leadsto {k}^{\color{blue}{\left(0.25 \cdot -2\right)}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. pow-pow 99.5%

      \[\leadsto \color{blue}{{\left({k}^{0.25}\right)}^{-2}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   3. add-sqr-sqrt 99.4%

      \[\leadsto \color{blue}{\sqrt{{\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \sqrt{{\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}} \]

   4. sqrt-unprod 89.3%

      \[\leadsto \color{blue}{\sqrt{\left({\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \left({\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)}} \]

   5. *-commutative 89.3%

      \[\leadsto \sqrt{\color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left({k}^{0.25}\right)}^{-2}\right)} \cdot \left({\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)} \]

   6. *-commutative 89.3%

      \[\leadsto \sqrt{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left({k}^{0.25}\right)}^{-2}\right) \cdot \color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left({k}^{0.25}\right)}^{-2}\right)}} \]

   7. swap-sqr 89.3%

      \[\leadsto \sqrt{\color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \left({\left({k}^{0.25}\right)}^{-2} \cdot {\left({k}^{0.25}\right)}^{-2}\right)}} \]

7. Applied egg-rr 89.6%

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

   1. pow1/2 89.6%

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

   2. pow-flip 89.7%

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

   3. *-commutative 89.7%

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

   4. associate-*r* 89.7%

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

   5. metadata-eval 89.7%

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

9. Applied egg-rr 89.7%

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

10. Taylor expanded in k around 0 39.7%

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

11. Final simplification 39.7%

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

Alternative 7: 38.3% accurate, 1.5× 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 99.6%

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

   1. *-commutative 99.6%

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

   2. div-sub 99.6%

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

   3. metadata-eval 99.6%

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

   4. div-inv 99.7%

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

   5. add-sqr-sqrt 99.4%

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

   6. sqrt-unprod 89.5%

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

   7. frac-times 89.4%

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

3. Applied egg-rr 89.5%

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

4. Taylor expanded in k around 0 39.5%

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

   1. associate-/l* 39.5%

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

   2. associate-/r/ 39.5%

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

6. Simplified 39.5%

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

7. Final simplification 39.5%

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

Alternative 8: 38.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. *-commutative 99.6%

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

   2. div-sub 99.6%

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

   3. metadata-eval 99.6%

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

   4. div-inv 99.7%

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

   5. add-sqr-sqrt 99.4%

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

   6. sqrt-unprod 89.5%

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

   7. frac-times 89.4%

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

3. Applied egg-rr 89.5%

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

4. Taylor expanded in k around 0 39.5%

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

   1. associate-/l* 39.5%

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

   2. associate-/r/ 39.5%

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

6. Simplified 39.5%

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

   1. *-commutative 39.5%

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

   2. clear-num 39.5%

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

   3. un-div-inv 39.5%

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

8. Applied egg-rr 39.5%

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

9. Final simplification 39.5%

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

Alternative 9: 38.3% accurate, 1.5× speedup

Math

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

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(Float64(pi * 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[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

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. *-commutative 99.6%

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

   2. div-sub 99.6%

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

   3. metadata-eval 99.6%

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

   4. div-inv 99.7%

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

   5. add-sqr-sqrt 99.4%

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

   6. sqrt-unprod 89.5%

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

   7. frac-times 89.4%

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

3. Applied egg-rr 89.5%

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

4. Taylor expanded in k around 0 39.5%

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

5. Final simplification 39.5%

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

Alternative 10: 38.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. *-commutative 99.6%

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

   2. div-sub 99.6%

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

   3. metadata-eval 99.6%

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

   4. div-inv 99.7%

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

   5. add-sqr-sqrt 99.4%

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

   6. sqrt-unprod 89.5%

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

   7. frac-times 89.4%

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

3. Applied egg-rr 89.5%

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

4. Taylor expanded in k around 0 39.5%

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

   1. associate-/l* 39.5%

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

   2. associate-/r/ 39.5%

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

6. Simplified 39.5%

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

   1. *-commutative 39.5%

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

   2. clear-num 39.5%

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

   3. un-div-inv 39.5%

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

8. Applied egg-rr 39.5%

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

9. Taylor expanded in k around 0 39.5%

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

    1. associate-*r/ 39.5%

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

    2. *-commutative 39.5%

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

    3. associate-*r* 39.5%

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

    4. associate-*r/ 39.5%

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

    5. *-commutative 39.5%

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

    6. associate-/l* 39.5%

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

    7. associate-*l/ 39.5%

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

    8. *-commutative 39.5%

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

11. Simplified 39.5%

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

12. Final simplification 39.5%

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

Reproduce

herbie shell --seed 2023271 
(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))))