Migdal et al, Equation (51)

Percentage Accurate: 98.6% → 99.6%

Time: 14.6s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. k = 4.94066e-324
  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: 98.6% 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: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

   1. associate-*r* 98.0%

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

   2. add-sqr-sqrt 81.9%

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

   3. sqrt-unprod 79.2%

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

   4. *-commutative 79.2%

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

   5. *-commutative 79.2%

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

   6. swap-sqr 79.2%

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

   7. pow2 79.2%

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

   8. metadata-eval 79.2%

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

4. Applied egg-rr 79.2%

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

   1. *-commutative 79.2%

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

   2. metadata-eval 79.2%

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

   3. unpow2 79.2%

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

   4. swap-sqr 79.2%

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

   5. rem-sqrt-square 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.5%

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

6. Simplified 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. inv-pow 99.5%

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

   3. sqrt-pow2 99.6%

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

   4. metadata-eval 99.6%

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

8. Applied egg-rr 99.6%

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

   1. *-lft-identity 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.7e-68)
   (* (sqrt (* n 2.0)) (sqrt (/ PI k)))
   (sqrt (/ (pow (* PI (* n 2.0)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.7e-68) {
		tmp = sqrt((n * 2.0)) * sqrt((((double) M_PI) / k));
	} else {
		tmp = sqrt((pow((((double) M_PI) * (n * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.7e-68) {
		tmp = Math.sqrt((n * 2.0)) * Math.sqrt((Math.PI / k));
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (n * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.7e-68:
		tmp = math.sqrt((n * 2.0)) * math.sqrt((math.pi / k))
	else:
		tmp = math.sqrt((math.pow((math.pi * (n * 2.0)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.7e-68)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(pi / k)));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(n * 2.0)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.7e-68)
		tmp = sqrt((n * 2.0)) * sqrt((pi / k));
	else
		tmp = sqrt((((pi * (n * 2.0)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.7e-68], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.70000000000000002e-68

   1. Initial program 99.2%

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

   3. Taylor expanded in k around 0 65.7%

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

      1. *-commutative 65.7%

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

      2. associate-/l* 65.7%

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

   5. Simplified 65.7%

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

      1. pow1 65.7%

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

      2. sqrt-unprod 65.9%

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

      3. associate-*r/ 65.9%

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

      4. *-commutative 65.9%

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

   7. Applied egg-rr 65.9%

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

      1. unpow1 65.9%

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

      2. *-commutative 65.9%

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

      3. *-commutative 65.9%

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

      4. *-commutative 65.9%

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

      5. associate-/l* 65.9%

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

   9. Simplified 65.9%

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

       1. associate-*r* 65.9%

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

       2. *-commutative 65.9%

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

       3. sqrt-prod 99.4%

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

   11. Applied egg-rr 99.4%

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

   if 3.70000000000000002e-68 < k

   1. Initial program 97.3%

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

      1. add-sqr-sqrt 97.3%

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

      2. sqrt-unprod 97.3%

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

      3. *-commutative 97.3%

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

      4. associate-*r* 97.3%

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

      5. div-sub 97.3%

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

      6. metadata-eval 97.3%

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

      7. div-inv 97.4%

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

      8. *-commutative 97.4%

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

   4. Applied egg-rr 97.3%

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

   6. Simplified 97.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2e+62)
   (/ (sqrt (* PI (* n 2.0))) (sqrt k))
   (sqrt (* 2.0 (+ -1.0 (fma n (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2e+62) {
		tmp = sqrt((((double) M_PI) * (n * 2.0))) / sqrt(k);
	} else {
		tmp = sqrt((2.0 * (-1.0 + fma(n, (((double) M_PI) / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2e+62)
		tmp = Float64(sqrt(Float64(pi * Float64(n * 2.0))) / sqrt(k));
	else
		tmp = sqrt(Float64(2.0 * Float64(-1.0 + fma(n, Float64(pi / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2e+62], N[(N[Sqrt[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(-1.0 + N[(n * N[(Pi / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.00000000000000007e62

   1. Initial program 96.5%

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

   3. Taylor expanded in k around 0 53.0%

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

      1. *-commutative 53.0%

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

      2. associate-/l* 53.0%

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

   5. Simplified 53.0%

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

      1. pow1 53.0%

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

      2. sqrt-unprod 53.2%

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

      3. associate-*r/ 53.1%

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

      4. *-commutative 53.1%

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

   7. Applied egg-rr 53.1%

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

      1. unpow1 53.1%

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

      2. *-commutative 53.1%

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

      3. *-commutative 53.1%

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

      4. *-commutative 53.1%

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

      5. associate-/l* 53.2%

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

   9. Simplified 53.2%

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

       1. associate-*r/ 53.1%

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

       2. associate-*r/ 53.1%

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

       3. add-sqr-sqrt 53.0%

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

       4. add-sqr-sqrt 53.0%

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

       5. frac-times 53.1%

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

       6. *-un-lft-identity 53.1%

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

       7. associate-*l/ 53.0%

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

       8. *-un-lft-identity 53.0%

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

       9. associate-*l/ 53.0%

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

       10. sqrt-unprod 72.3%

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

       11. add-sqr-sqrt 72.6%

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

       12. associate-*l/ 72.8%

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

   11. Applied egg-rr 72.8%

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

   if 2.00000000000000007e62 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 1.8%

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

      1. *-commutative 1.8%

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

      2. associate-/l* 1.8%

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

   5. Simplified 1.8%

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

      1. pow1 1.8%

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

      2. sqrt-unprod 1.8%

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

      3. associate-*r/ 1.8%

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

      4. *-commutative 1.8%

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

   7. Applied egg-rr 1.8%

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

      1. unpow1 1.8%

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

      2. *-commutative 1.8%

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

      3. *-commutative 1.8%

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

      4. *-commutative 1.8%

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

      5. associate-/l* 1.8%

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

   9. Simplified 1.8%

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

   10. Taylor expanded in n around 0 1.8%

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

       1. *-commutative 1.8%

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

       2. associate-/l* 1.8%

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

   12. Simplified 1.8%

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

       1. *-commutative 1.8%

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

       2. div-inv 1.8%

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

       3. associate-*r* 1.8%

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

       4. *-commutative 1.8%

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

       5. expm1-log1p-u 1.8%

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

       6. expm1-undefine 36.8%

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

       7. div-inv 36.8%

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

       8. clear-num 36.8%

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

       9. un-div-inv 36.8%

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

   14. Applied egg-rr 36.8%

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

       1. sub-neg 36.8%

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

       2. metadata-eval 36.8%

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

       3. +-commutative 36.8%

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

       4. log1p-undefine 36.8%

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

       5. rem-exp-log 36.8%

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

       6. +-commutative 36.8%

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

       7. associate-/r/ 36.8%

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

       8. associate-*l/ 36.8%

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

       9. associate-/l* 36.8%

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

       10. fma-define 36.8%

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

   16. Simplified 36.8%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% 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 98.0%

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

   1. associate-*l/ 98.1%

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

   2. *-lft-identity 98.1%

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

   3. associate-*l* 98.1%

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

   4. div-sub 98.1%

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

   5. metadata-eval 98.1%

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

3. Simplified 98.1%

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

5. Final simplification 98.1%

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

Alternative 5: 30.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

   1. associate-*r* 98.0%

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

   2. add-sqr-sqrt 81.9%

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

   3. sqrt-unprod 79.2%

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

   4. *-commutative 79.2%

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

   5. *-commutative 79.2%

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

   6. swap-sqr 79.2%

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

   7. pow2 79.2%

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

   8. metadata-eval 79.2%

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

4. Applied egg-rr 79.2%

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

   1. *-commutative 79.2%

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

   2. metadata-eval 79.2%

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

   3. unpow2 79.2%

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

   4. swap-sqr 79.2%

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

   5. rem-sqrt-square 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.5%

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

6. Simplified 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. inv-pow 99.5%

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

   3. sqrt-pow2 99.6%

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

   4. metadata-eval 99.6%

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

8. Applied egg-rr 99.6%

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

   1. *-lft-identity 99.6%

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

10. Simplified 99.6%

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

11. Taylor expanded in k around 0 31.8%

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

    1. *-commutative 31.8%

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

    2. *-commutative 31.8%

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

    3. associate-*r* 31.8%

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

    4. *-commutative 31.8%

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

13. Simplified 31.8%

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

14. Final simplification 31.8%

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

Alternative 6: 39.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in k around 0 31.2%

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

   1. *-commutative 31.2%

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

   2. associate-/l* 31.2%

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

5. Simplified 31.2%

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

   1. pow1 31.2%

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

   2. sqrt-unprod 31.3%

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

   3. associate-*r/ 31.3%

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

   4. *-commutative 31.3%

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

7. Applied egg-rr 31.3%

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

   1. unpow1 31.3%

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

   2. *-commutative 31.3%

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

   3. *-commutative 31.3%

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

   4. *-commutative 31.3%

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

   5. associate-/l* 31.3%

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

9. Simplified 31.3%

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

10. Taylor expanded in n around 0 31.3%

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

    1. *-commutative 31.3%

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

    2. associate-/l* 31.3%

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

12. Simplified 31.3%

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

    1. pow1/2 31.3%

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

    2. *-commutative 31.3%

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

    3. associate-*r/ 31.3%

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

    4. associate-*l/ 31.3%

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

    5. associate-*r* 31.3%

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

    6. *-commutative 31.3%

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

    7. associate-*r/ 31.3%

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

    8. associate-*l* 31.3%

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

    9. unpow-prod-down 42.6%

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

    10. pow1/2 42.6%

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

14. Applied egg-rr 42.6%

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

    1. unpow1/2 42.6%

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

16. Simplified 42.6%

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

17. Final simplification 42.6%

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

Alternative 7: 39.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in k around 0 31.2%

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

   1. *-commutative 31.2%

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

   2. associate-/l* 31.2%

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

5. Simplified 31.2%

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

   1. pow1 31.2%

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

   2. sqrt-unprod 31.3%

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

   3. associate-*r/ 31.3%

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

   4. *-commutative 31.3%

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

7. Applied egg-rr 31.3%

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

   1. unpow1 31.3%

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

   2. *-commutative 31.3%

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

   3. *-commutative 31.3%

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

   4. *-commutative 31.3%

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

   5. associate-/l* 31.3%

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

9. Simplified 31.3%

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

    1. associate-*r* 31.3%

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

    2. *-commutative 31.3%

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

    3. sqrt-prod 42.6%

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

11. Applied egg-rr 42.6%

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

12. Final simplification 42.6%

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

Alternative 8: 39.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in k around 0 31.2%

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

   1. *-commutative 31.2%

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

   2. associate-/l* 31.2%

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

5. Simplified 31.2%

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

   1. pow1 31.2%

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

   2. sqrt-unprod 31.3%

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

   3. associate-*r/ 31.3%

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

   4. *-commutative 31.3%

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

7. Applied egg-rr 31.3%

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

   1. unpow1 31.3%

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

   2. *-commutative 31.3%

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

   3. *-commutative 31.3%

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

   4. *-commutative 31.3%

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

   5. associate-/l* 31.3%

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

9. Simplified 31.3%

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

    1. associate-*r/ 31.3%

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

    2. associate-*r/ 31.3%

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

    3. add-sqr-sqrt 31.2%

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

    4. add-sqr-sqrt 31.2%

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

    5. frac-times 31.2%

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

    6. *-un-lft-identity 31.2%

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

    7. associate-*l/ 31.2%

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

    8. *-un-lft-identity 31.2%

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

    9. associate-*l/ 31.2%

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

    10. sqrt-unprod 42.3%

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

    11. add-sqr-sqrt 42.5%

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

    12. associate-*l/ 42.6%

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

11. Applied egg-rr 42.6%

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

12. Final simplification 42.6%

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

Alternative 9: 30.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in k around 0 31.2%

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

   1. *-commutative 31.2%

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

   2. associate-/l* 31.2%

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

5. Simplified 31.2%

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

   1. pow1 31.2%

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

   2. sqrt-unprod 31.3%

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

   3. associate-*r/ 31.3%

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

   4. *-commutative 31.3%

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

7. Applied egg-rr 31.3%

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

   1. unpow1 31.3%

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

   2. *-commutative 31.3%

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

   3. *-commutative 31.3%

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

   4. *-commutative 31.3%

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

   5. associate-/l* 31.3%

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

9. Simplified 31.3%

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

10. Taylor expanded in n around 0 31.3%

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

    1. *-commutative 31.3%

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

    2. associate-/l* 31.3%

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

12. Simplified 31.3%

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

    1. *-commutative 31.3%

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

    2. associate-*r/ 31.3%

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

    3. associate-*l/ 31.3%

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

    4. associate-*r* 31.3%

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

    5. sqrt-undiv 42.6%

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

    6. clear-num 42.5%

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

    7. sqrt-undiv 31.6%

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

14. Applied egg-rr 31.6%

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

15. Final simplification 31.6%

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

Alternative 10: 29.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in k around 0 31.2%

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

   1. *-commutative 31.2%

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

   2. associate-/l* 31.2%

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

5. Simplified 31.2%

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

   1. pow1 31.2%

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

   2. sqrt-unprod 31.3%

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

   3. associate-*r/ 31.3%

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

   4. *-commutative 31.3%

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

7. Applied egg-rr 31.3%

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

   1. unpow1 31.3%

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

   2. *-commutative 31.3%

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

   3. *-commutative 31.3%

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

   4. *-commutative 31.3%

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

   5. associate-/l* 31.3%

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

9. Simplified 31.3%

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

10. Final simplification 31.3%

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

Alternative 11: 29.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in k around 0 31.2%

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

   1. *-commutative 31.2%

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

   2. associate-/l* 31.2%

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

5. Simplified 31.2%

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

   1. pow1 31.2%

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

   2. sqrt-unprod 31.3%

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

   3. associate-*r/ 31.3%

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

   4. *-commutative 31.3%

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

7. Applied egg-rr 31.3%

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

   1. unpow1 31.3%

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

   2. *-commutative 31.3%

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

   3. *-commutative 31.3%

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

   4. *-commutative 31.3%

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

   5. associate-/l* 31.3%

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

9. Simplified 31.3%

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

10. Taylor expanded in n around 0 31.3%

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

    1. *-commutative 31.3%

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

    2. associate-/l* 31.3%

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

12. Simplified 31.3%

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

13. Final simplification 31.3%

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

Alternative 12: 29.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in k around 0 31.2%

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

   1. *-commutative 31.2%

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

   2. associate-/l* 31.2%

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

5. Simplified 31.2%

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

   1. pow1 31.2%

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

   2. sqrt-unprod 31.3%

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

   3. associate-*r/ 31.3%

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

   4. *-commutative 31.3%

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

7. Applied egg-rr 31.3%

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

   1. unpow1 31.3%

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

   2. *-commutative 31.3%

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

   3. *-commutative 31.3%

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

   4. *-commutative 31.3%

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

   5. associate-/l* 31.3%

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

9. Simplified 31.3%

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

    1. clear-num 31.3%

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

    2. un-div-inv 31.3%

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

11. Applied egg-rr 31.3%

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

12. Final simplification 31.3%

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

Reproduce

herbie shell --seed 2024058 
(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))))