Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.2%

Time: 18.5s

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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.6e-46)
   (/ (sqrt (* PI (* n 2.0))) (sqrt k))
   (sqrt (/ (pow (* n (* PI 2.0)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.6e-46) {
		tmp = sqrt((((double) M_PI) * (n * 2.0))) / sqrt(k);
	} else {
		tmp = sqrt((pow((n * (((double) M_PI) * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.6e-46) {
		tmp = Math.sqrt((Math.PI * (n * 2.0))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow((n * (Math.PI * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.6e-46:
		tmp = math.sqrt((math.pi * (n * 2.0))) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow((n * (math.pi * 2.0)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.6e-46)
		tmp = Float64(sqrt(Float64(pi * Float64(n * 2.0))) / sqrt(k));
	else
		tmp = sqrt(Float64((Float64(n * Float64(pi * 2.0)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.6e-46)
		tmp = sqrt((pi * (n * 2.0))) / sqrt(k);
	else
		tmp = sqrt((((n * (pi * 2.0)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.6e-46], N[(N[Sqrt[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.6e-46

   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 99.0%

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

      1. associate-*l/ 99.3%

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

      2. *-un-lft-identity 99.3%

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

      3. *-commutative 99.3%

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

      4. *-commutative 99.3%

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

      5. sqrt-prod 99.5%

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

      6. clear-num 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. associate-/l* 99.5%

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

      2. *-lft-identity 99.5%

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

      3. associate-*r* 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-*l* 99.5%

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

   7. Simplified 99.5%

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

   if 1.6e-46 < k

   1. Initial program 99.7%

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

   3. Taylor expanded in n around 0 98.8%

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

      1. associate-*r* 98.8%

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

      2. exp-prod 98.9%

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

      3. +-commutative 98.9%

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

      4. log-prod 98.9%

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

      5. associate-*r* 98.9%

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

      6. *-commutative 98.9%

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

      7. *-commutative 98.9%

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

      8. *-commutative 98.9%

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

      9. exp-to-pow 99.7%

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

      10. unpow1/2 99.7%

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

   5. Simplified 99.7%

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

      1. add-sqr-sqrt 99.7%

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

      2. sqrt-unprod 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-un-lft-identity 99.7%

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

      5. associate-*l/ 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. frac-times 99.7%

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

   7. Applied egg-rr 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in n around 0 95.9%

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

   1. associate-*r* 95.9%

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

   2. exp-prod 95.9%

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

   3. +-commutative 95.9%

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

   4. log-prod 96.2%

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

   5. associate-*r* 96.2%

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

   6. *-commutative 96.2%

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

   7. *-commutative 96.2%

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

   8. *-commutative 96.2%

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

   9. exp-to-pow 99.5%

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

   10. unpow1/2 99.5%

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

5. Simplified 99.5%

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

   1. associate-*l/ 99.6%

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

   2. *-un-lft-identity 99.6%

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

   3. *-commutative 99.6%

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

   4. *-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. associate-*r* 99.6%

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

   7. *-commutative 99.6%

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

   8. associate-*l* 99.6%

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

7. Applied egg-rr 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   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 99.6%

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

   5. *-commutative 99.6%

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

   6. associate-*l* 99.6%

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

   7. associate-*r/ 99.6%

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

   8. *-commutative 99.6%

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

   9. associate-/l* 99.6%

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

   10. metadata-eval 99.6%

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

   11. /-rgt-identity 99.6%

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

   12. div-sub 99.6%

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

   13. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 50.4% 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 99.5%

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

3. Taylor expanded in k around 0 50.6%

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

   1. expm1-log1p-u 47.8%

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

   2. expm1-udef 46.9%

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

   3. associate-*l/ 46.9%

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

   4. *-un-lft-identity 46.9%

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

   5. *-commutative 46.9%

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

   6. *-commutative 46.9%

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

   7. sqrt-prod 46.9%

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

   8. sqrt-undiv 36.7%

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

5. Applied egg-rr 36.7%

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

   1. expm1-def 37.7%

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

   2. expm1-log1p 39.5%

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

   3. associate-/l* 39.5%

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

   4. *-commutative 39.5%

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

7. Simplified 39.5%

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

8. Taylor expanded in k around 0 39.5%

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

   1. *-commutative 39.5%

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

   2. associate-*r/ 39.5%

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

   3. associate-*l/ 39.5%

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

   4. *-commutative 39.5%

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

   5. associate-*l* 39.5%

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

   6. *-commutative 39.5%

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

   7. associate-*l/ 39.5%

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

10. Simplified 39.5%

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

    1. div-inv 39.5%

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

    2. *-commutative 39.5%

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

    3. associate-*l* 39.5%

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

    4. div-inv 39.5%

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

    5. associate-*l* 39.5%

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

    6. *-commutative 39.5%

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

    7. associate-*r* 39.5%

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

    8. sqrt-prod 50.8%

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

12. Applied egg-rr 50.8%

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

    1. *-commutative 50.8%

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

    2. associate-*l/ 50.8%

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

    3. associate-*r/ 50.8%

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

14. Simplified 50.8%

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

15. Final simplification 50.8%

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

Alternative 5: 50.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 50.6%

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

   1. expm1-log1p-u 47.8%

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

   2. expm1-udef 46.9%

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

   3. associate-*l/ 46.9%

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

   4. *-un-lft-identity 46.9%

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

   5. *-commutative 46.9%

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

   6. *-commutative 46.9%

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

   7. sqrt-prod 46.9%

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

   8. sqrt-undiv 36.7%

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

5. Applied egg-rr 36.7%

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

   1. expm1-def 37.7%

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

   2. expm1-log1p 39.5%

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

   3. associate-/l* 39.5%

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

   4. *-commutative 39.5%

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

7. Simplified 39.5%

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

8. Taylor expanded in k around 0 39.5%

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

   1. *-commutative 39.5%

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

   2. associate-*r/ 39.5%

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

   3. associate-*l/ 39.5%

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

   4. *-commutative 39.5%

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

   5. associate-*l* 39.5%

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

   6. *-commutative 39.5%

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

   7. associate-*l/ 39.5%

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

10. Simplified 39.5%

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

    1. div-inv 39.5%

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

    2. *-commutative 39.5%

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

    3. associate-*l* 39.5%

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

    4. div-inv 39.5%

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

    5. associate-*l* 39.5%

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

    6. *-commutative 39.5%

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

    7. associate-*r* 39.5%

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

    8. sqrt-prod 50.8%

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

12. Applied egg-rr 50.8%

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

13. Final simplification 50.8%

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

Alternative 6: 50.4% 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 99.5%

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

3. Taylor expanded in k around 0 50.6%

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

   1. associate-*l/ 50.7%

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

   2. *-un-lft-identity 50.7%

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

   3. *-commutative 50.7%

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

   4. *-commutative 50.7%

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

   5. sqrt-prod 50.8%

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

   6. clear-num 50.7%

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

5. Applied egg-rr 50.7%

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

   1. associate-/l* 50.8%

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

   2. *-lft-identity 50.8%

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

   3. associate-*r* 50.8%

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

   4. *-commutative 50.8%

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

   5. associate-*l* 50.8%

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

7. Simplified 50.8%

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

8. Final simplification 50.8%

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

Alternative 7: 39.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   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 99.6%

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

   5. *-commutative 99.6%

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

   6. associate-*l* 99.6%

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

   7. associate-*r/ 99.6%

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

   8. *-commutative 99.6%

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

   9. associate-/l* 99.6%

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

   10. metadata-eval 99.6%

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

   11. /-rgt-identity 99.6%

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

   12. div-sub 99.6%

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

   13. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. associate-*r* 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

   5. sub-neg 99.5%

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

   6. div-inv 99.5%

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

   7. metadata-eval 99.5%

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

   8. distribute-rgt-neg-in 99.5%

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

   9. metadata-eval 99.5%

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

   10. inv-pow 99.5%

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

   11. sqrt-pow2 99.6%

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

   12. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in k around 0 50.7%

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

   1. *-commutative 50.7%

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

   2. metadata-eval 50.7%

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

   3. pow-flip 50.6%

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

   4. pow1/2 50.6%

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

   5. sqrt-prod 50.6%

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

   6. associate-*r* 50.6%

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

   7. sqrt-prod 50.6%

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

   8. sqrt-prod 50.7%

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

   9. *-commutative 50.7%

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

   10. div-inv 50.8%

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

   11. clear-num 50.7%

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

   12. sqrt-undiv 39.7%

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

   13. *-commutative 39.7%

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

   14. associate-*r* 39.7%

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

   15. *-commutative 39.7%

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

   16. associate-*l* 39.7%

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

9. Applied egg-rr 39.7%

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

10. Final simplification 39.7%

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

Alternative 8: 39.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 50.6%

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

   1. expm1-log1p-u 47.8%

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

   2. expm1-udef 46.9%

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

   3. associate-*l/ 46.9%

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

   4. *-un-lft-identity 46.9%

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

   5. *-commutative 46.9%

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

   6. *-commutative 46.9%

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

   7. sqrt-prod 46.9%

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

   8. sqrt-undiv 36.7%

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

5. Applied egg-rr 36.7%

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

   1. expm1-def 37.7%

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

   2. expm1-log1p 39.5%

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

   3. associate-/l* 39.5%

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

   4. *-commutative 39.5%

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

7. Simplified 39.5%

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

   1. *-un-lft-identity 39.5%

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

   2. times-frac 39.4%

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

9. Applied egg-rr 39.4%

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

    1. clear-num 39.4%

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

    2. sqrt-div 39.6%

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

    3. metadata-eval 39.6%

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

    4. div-inv 39.6%

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

    5. frac-times 39.7%

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

    6. *-un-lft-identity 39.7%

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

    7. metadata-eval 39.7%

       \[\leadsto \frac{1}{\sqrt{\frac{k}{n \cdot \pi} \cdot \color{blue}{0.5}}} \]

11. Applied egg-rr 39.7%

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{k}{n \cdot \pi} \cdot 0.5}}} \]
12. Step-by-step derivation

    1. metadata-eval 39.7%

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

    2. times-frac 39.7%

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

    3. *-commutative 39.7%

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

    4. *-rgt-identity 39.7%

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

    5. associate-*r* 39.7%

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

    6. *-commutative 39.7%

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

    7. associate-/r* 39.7%

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

    8. *-commutative 39.7%

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

13. Simplified 39.7%

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

14. Final simplification 39.7%

    \[\leadsto \frac{1}{\sqrt{\frac{\frac{k}{n \cdot 2}}{\pi}}} \]
15. Add Preprocessing

Alternative 9: 38.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 50.6%

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

   1. expm1-log1p-u 47.8%

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

   2. expm1-udef 46.9%

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

   3. associate-*l/ 46.9%

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

   4. *-un-lft-identity 46.9%

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

   5. *-commutative 46.9%

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

   6. *-commutative 46.9%

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

   7. sqrt-prod 46.9%

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

   8. sqrt-undiv 36.7%

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

5. Applied egg-rr 36.7%

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

   1. expm1-def 37.7%

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

   2. expm1-log1p 39.5%

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

   3. associate-/l* 39.5%

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

   4. *-commutative 39.5%

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

7. Simplified 39.5%

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

   1. *-un-lft-identity 39.5%

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

   2. times-frac 39.4%

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

9. Applied egg-rr 39.4%

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

    1. pow1/2 39.4%

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

    2. associate-/r* 39.5%

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

    3. div-inv 39.5%

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

    4. clear-num 39.5%

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

    5. /-rgt-identity 39.5%

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

    6. *-commutative 39.5%

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

11. Applied egg-rr 39.5%

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

12. Final simplification 39.5%

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

Alternative 10: 38.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 50.6%

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

   1. expm1-log1p-u 47.8%

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

   2. expm1-udef 46.9%

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

   3. associate-*l/ 46.9%

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

   4. *-un-lft-identity 46.9%

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

   5. *-commutative 46.9%

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

   6. *-commutative 46.9%

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

   7. sqrt-prod 46.9%

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

   8. sqrt-undiv 36.7%

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

5. Applied egg-rr 36.7%

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

   1. expm1-def 37.7%

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

   2. expm1-log1p 39.5%

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

   3. associate-/l* 39.5%

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

   4. *-commutative 39.5%

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

7. Simplified 39.5%

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

8. Taylor expanded in k around 0 39.5%

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

   1. *-commutative 39.5%

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

   2. associate-*r/ 39.5%

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

   3. associate-*l/ 39.5%

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

   4. *-commutative 39.5%

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

   5. associate-*l* 39.5%

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

   6. *-commutative 39.5%

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

   7. associate-*l/ 39.5%

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

10. Simplified 39.5%

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

11. Final simplification 39.5%

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

Reproduce

herbie shell --seed 2024014 
(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))))