Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.0%

Time: 14.9s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% 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: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\frac{2}{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5.5e-9)
   (* (sqrt n) (sqrt (/ 2.0 (/ k PI))))
   (sqrt (/ (pow (* n (* 2.0 PI)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5.5e-9) {
		tmp = sqrt(n) * sqrt((2.0 / (k / ((double) M_PI))));
	} else {
		tmp = sqrt((pow((n * (2.0 * ((double) M_PI))), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 5.5e-9) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 / (k / Math.PI)));
	} else {
		tmp = Math.sqrt((Math.pow((n * (2.0 * Math.PI)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 5.5e-9:
		tmp = math.sqrt(n) * math.sqrt((2.0 / (k / math.pi)))
	else:
		tmp = math.sqrt((math.pow((n * (2.0 * math.pi)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5.5e-9)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 / Float64(k / pi))));
	else
		tmp = sqrt(Float64((Float64(n * Float64(2.0 * pi)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 5.5e-9)
		tmp = sqrt(n) * sqrt((2.0 / (k / pi)));
	else
		tmp = sqrt((((n * (2.0 * pi)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5.5e-9], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 / N[(k / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.4999999999999996e-9

   1. Initial program 98.3%

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

   3. Taylor expanded in k around 0 98.2%

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

      1. expm1-log1p-u 92.4%

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

      2. expm1-udef 77.9%

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

      3. associate-*l/ 77.9%

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

      4. *-un-lft-identity 77.9%

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

      5. sqrt-unprod 77.9%

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

      6. *-commutative 77.9%

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

      7. *-commutative 77.9%

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

      8. associate-*r* 77.9%

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

      9. *-commutative 77.9%

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

      10. sqrt-undiv 55.7%

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

      11. *-commutative 55.7%

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

      12. associate-*r* 55.7%

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

   5. Applied egg-rr 55.7%

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

      1. expm1-def 70.2%

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

      2. expm1-log1p 74.1%

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

      3. associate-/l* 74.0%

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

      4. *-commutative 74.0%

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

   7. Simplified 74.0%

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

   8. Taylor expanded in k around 0 74.1%

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

      1. associate-*r/ 74.1%

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

      2. associate-*l* 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-*l* 74.1%

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

   10. Simplified 74.1%

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

       1. sqrt-prod 98.9%

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

   12. Applied egg-rr 98.9%

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

       1. associate-*r/ 98.9%

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

       2. associate-/l* 98.8%

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

   14. Simplified 98.8%

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

   if 5.4999999999999996e-9 < k

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*r* 99.8%

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

      6. div-sub 99.8%

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

      7. metadata-eval 99.8%

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

      8. div-inv 99.8%

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

      9. *-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\frac{2}{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. associate-*l/ 99.1%

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

   2. *-lft-identity 99.1%

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

   3. sqr-pow 99.0%

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

   4. pow-sqr 99.1%

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

   5. *-commutative 99.1%

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

   6. associate-*l* 99.1%

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

   7. associate-*r/ 99.1%

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

   8. *-commutative 99.1%

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

   9. associate-/l* 99.1%

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

   10. metadata-eval 99.1%

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

   11. /-rgt-identity 99.1%

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

   12. div-sub 99.1%

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

   13. metadata-eval 99.1%

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

3. Simplified 99.1%

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

   1. div-inv 99.1%

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

   2. associate-*r* 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-*l* 99.1%

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

   5. div-inv 99.1%

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

   6. metadata-eval 99.1%

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

   7. pow1/2 99.1%

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

   8. pow-flip 99.2%

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

   9. metadata-eval 99.2%

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

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. associate-*l/ 99.1%

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

   2. *-lft-identity 99.1%

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

   3. sqr-pow 99.0%

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

   4. pow-sqr 99.1%

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

   5. *-commutative 99.1%

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

   6. associate-*l* 99.1%

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

   7. associate-*r/ 99.1%

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

   8. *-commutative 99.1%

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

   9. associate-/l* 99.1%

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

   10. metadata-eval 99.1%

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

   11. /-rgt-identity 99.1%

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

   12. div-sub 99.1%

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

   13. metadata-eval 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 4: 50.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 48.6%

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

   1. expm1-log1p-u 45.8%

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

   2. expm1-udef 47.1%

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

   3. associate-*l/ 47.1%

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

   4. *-un-lft-identity 47.1%

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

   5. sqrt-unprod 47.1%

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

   6. *-commutative 47.1%

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

   7. *-commutative 47.1%

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

   8. associate-*r* 47.1%

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

   9. *-commutative 47.1%

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

   10. sqrt-undiv 36.5%

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

   11. *-commutative 36.5%

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

   12. associate-*r* 36.5%

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

5. Applied egg-rr 36.5%

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

   1. expm1-def 35.2%

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

   2. expm1-log1p 37.1%

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

   3. associate-/l* 37.0%

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

   4. *-commutative 37.0%

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

7. Simplified 37.0%

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

8. Taylor expanded in k around 0 37.1%

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

   1. associate-*r/ 37.1%

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

   2. associate-*l* 37.1%

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

   3. *-commutative 37.1%

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

   4. associate-*l* 37.1%

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

10. Simplified 37.1%

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

    1. sqrt-prod 48.9%

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

12. Applied egg-rr 48.9%

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

    1. associate-*r/ 48.9%

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

    2. associate-/l* 48.9%

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

14. Simplified 48.9%

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

15. Final simplification 48.9%

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

Alternative 5: 39.0% accurate, 2.0× speedup

Math

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

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 / n) / Float64(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), $MachinePrecision] / N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 48.6%

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

   1. expm1-log1p-u 45.8%

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

   2. expm1-udef 47.1%

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

   3. associate-*l/ 47.1%

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

   4. *-un-lft-identity 47.1%

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

   5. sqrt-unprod 47.1%

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

   6. *-commutative 47.1%

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

   7. *-commutative 47.1%

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

   8. associate-*r* 47.1%

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

   9. *-commutative 47.1%

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

   10. sqrt-undiv 36.5%

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

   11. *-commutative 36.5%

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

   12. associate-*r* 36.5%

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

5. Applied egg-rr 36.5%

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

   1. expm1-def 35.2%

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

   2. expm1-log1p 37.1%

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

   3. associate-/l* 37.0%

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

   4. *-commutative 37.0%

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

7. Simplified 37.0%

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

   1. clear-num 37.0%

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

   2. sqrt-div 38.1%

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

   3. metadata-eval 38.1%

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

   4. div-inv 38.1%

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

   5. associate-/r* 38.3%

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

   6. metadata-eval 38.3%

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

9. Applied egg-rr 38.3%

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

    1. associate-*l/ 38.3%

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

    2. associate-/l* 38.3%

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

    3. metadata-eval 38.3%

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

    4. associate-/l* 38.3%

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

    5. *-commutative 38.3%

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

    6. /-rgt-identity 38.3%

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

11. Simplified 38.3%

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

12. Final simplification 38.3%

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

Alternative 6: 38.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 48.6%

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

   1. expm1-log1p-u 45.8%

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

   2. expm1-udef 47.1%

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

   3. associate-*l/ 47.1%

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

   4. *-un-lft-identity 47.1%

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

   5. sqrt-unprod 47.1%

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

   6. *-commutative 47.1%

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

   7. *-commutative 47.1%

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

   8. associate-*r* 47.1%

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

   9. *-commutative 47.1%

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

   10. sqrt-undiv 36.5%

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

   11. *-commutative 36.5%

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

   12. associate-*r* 36.5%

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

5. Applied egg-rr 36.5%

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

   1. expm1-def 35.2%

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

   2. expm1-log1p 37.1%

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

   3. associate-/l* 37.0%

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

   4. *-commutative 37.0%

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

7. Simplified 37.0%

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

8. Taylor expanded in k around 0 37.1%

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

   1. associate-*r/ 37.1%

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

   2. associate-*l* 37.1%

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

   3. *-commutative 37.1%

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

   4. associate-*l* 37.1%

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

10. Simplified 37.1%

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

11. Taylor expanded in n around 0 37.1%

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

    1. *-commutative 37.1%

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

    2. associate-*r/ 37.1%

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

13. Simplified 37.1%

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

14. Final simplification 37.1%

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

Alternative 7: 38.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 48.6%

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

   1. expm1-log1p-u 45.8%

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

   2. expm1-udef 47.1%

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

   3. associate-*l/ 47.1%

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

   4. *-un-lft-identity 47.1%

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

   5. sqrt-unprod 47.1%

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

   6. *-commutative 47.1%

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

   7. *-commutative 47.1%

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

   8. associate-*r* 47.1%

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

   9. *-commutative 47.1%

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

   10. sqrt-undiv 36.5%

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

   11. *-commutative 36.5%

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

   12. associate-*r* 36.5%

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

5. Applied egg-rr 36.5%

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

   1. expm1-def 35.2%

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

   2. expm1-log1p 37.1%

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

   3. associate-/l* 37.0%

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

   4. *-commutative 37.0%

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

7. Simplified 37.0%

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

8. Taylor expanded in k around 0 37.1%

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

   1. associate-*r/ 37.1%

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

   2. associate-*l* 37.1%

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

   3. *-commutative 37.1%

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

   4. associate-*l* 37.1%

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

10. Simplified 37.1%

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

11. Final simplification 37.1%

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

Alternative 8: 38.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 48.6%

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

   1. expm1-log1p-u 45.8%

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

   2. expm1-udef 47.1%

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

   3. associate-*l/ 47.1%

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

   4. *-un-lft-identity 47.1%

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

   5. sqrt-unprod 47.1%

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

   6. *-commutative 47.1%

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

   7. *-commutative 47.1%

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

   8. associate-*r* 47.1%

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

   9. *-commutative 47.1%

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

   10. sqrt-undiv 36.5%

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

   11. *-commutative 36.5%

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

   12. associate-*r* 36.5%

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

5. Applied egg-rr 36.5%

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

   1. expm1-def 35.2%

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

   2. expm1-log1p 37.1%

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

   3. associate-/l* 37.0%

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

   4. *-commutative 37.0%

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

7. Simplified 37.0%

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

8. Taylor expanded in k around 0 37.1%

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

   1. associate-*r/ 37.1%

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

   2. associate-*l* 37.1%

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

   3. *-commutative 37.1%

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

   4. associate-*l* 37.1%

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

10. Simplified 37.1%

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

    1. *-commutative 37.1%

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

    2. associate-*r* 37.1%

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

    3. associate-/r/ 37.0%

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

    4. associate-*r/ 37.0%

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

    5. div-inv 37.0%

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

    6. clear-num 37.1%

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

    7. associate-*r/ 37.1%

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

    8. associate-*r* 37.1%

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

12. Applied egg-rr 37.1%

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

13. Final simplification 37.1%

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

Reproduce

herbie shell --seed 2024019 
(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))))