Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 21.2s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ \frac{\sqrt{t\_0}}{\sqrt{k} \cdot {t\_0}^{\left(k \cdot 0.5\right)}} \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* 2.0 PI) n)))
   (/ (sqrt t_0) (* (sqrt k) (pow t_0 (* k 0.5))))))

C

double code(double k, double n) {
	double t_0 = (2.0 * ((double) M_PI)) * n;
	return sqrt(t_0) / (sqrt(k) * pow(t_0, (k * 0.5)));
}

Java

public static double code(double k, double n) {
	double t_0 = (2.0 * Math.PI) * n;
	return Math.sqrt(t_0) / (Math.sqrt(k) * Math.pow(t_0, (k * 0.5)));
}

Python

def code(k, n):
	t_0 = (2.0 * math.pi) * n
	return math.sqrt(t_0) / (math.sqrt(k) * math.pow(t_0, (k * 0.5)))

Julia

function code(k, n)
	t_0 = Float64(Float64(2.0 * pi) * n)
	return Float64(sqrt(t_0) / Float64(sqrt(k) * (t_0 ^ Float64(k * 0.5))))
end

MATLAB

function tmp = code(k, n)
	t_0 = (2.0 * pi) * n;
	tmp = sqrt(t_0) / (sqrt(k) * (t_0 ^ (k * 0.5)));
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision]}, N[(N[Sqrt[t$95$0], $MachinePrecision] / N[(N[Sqrt[k], $MachinePrecision] * N[Power[t$95$0, N[(k * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. 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. *-un-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. associate-*r* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

   6. pow-div 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*r* 99.7%

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

   2. associate-*r* 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.2e-52)
   (/ (sqrt (* 2.0 n)) (sqrt (/ k PI)))
   (sqrt (/ (pow (* (* 2.0 PI) n) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.2e-52) {
		tmp = sqrt((2.0 * n)) / sqrt((k / ((double) M_PI)));
	} else {
		tmp = sqrt((pow(((2.0 * ((double) M_PI)) * n), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.2e-52) {
		tmp = Math.sqrt((2.0 * n)) / Math.sqrt((k / Math.PI));
	} else {
		tmp = Math.sqrt((Math.pow(((2.0 * Math.PI) * n), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.2e-52:
		tmp = math.sqrt((2.0 * n)) / math.sqrt((k / math.pi))
	else:
		tmp = math.sqrt((math.pow(((2.0 * math.pi) * n), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.2e-52)
		tmp = Float64(sqrt(Float64(2.0 * n)) / sqrt(Float64(k / pi)));
	else
		tmp = sqrt(Float64((Float64(Float64(2.0 * pi) * n) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.2e-52)
		tmp = sqrt((2.0 * n)) / sqrt((k / pi));
	else
		tmp = sqrt(((((2.0 * pi) * n) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.2e-52], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.20000000000000009e-52

   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 66.4%

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

      1. associate-/l* 66.3%

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

   5. Simplified 66.3%

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

      1. pow1 66.3%

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

      2. *-commutative 66.3%

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

      3. sqrt-unprod 66.6%

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

   7. Applied egg-rr 66.6%

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

      1. unpow1 66.6%

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

   9. Simplified 66.6%

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

       1. clear-num 66.6%

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

       2. un-div-inv 66.7%

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

   11. Applied egg-rr 66.7%

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

       1. associate-*r/ 66.7%

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

       2. *-commutative 66.7%

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

       3. sqrt-div 99.5%

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

   13. Applied egg-rr 99.5%

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

   if 2.20000000000000009e-52 < k

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 99.6%

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

      2. sqrt-unprod 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*r* 99.6%

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

      5. div-sub 99.6%

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

      6. metadata-eval 99.6%

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

      7. div-inv 99.6%

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

      8. *-commutative 99.6%

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

   4. Applied egg-rr 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.6e+42)
   (* (sqrt (* 2.0 n)) (sqrt (/ PI k)))
   (sqrt (* 2.0 (+ -1.0 (fma n (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.6e+42) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	} else {
		tmp = sqrt((2.0 * (-1.0 + fma(n, (((double) M_PI) / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.6e+42)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	else
		tmp = sqrt(Float64(2.0 * Float64(-1.0 + fma(n, Float64(pi / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.6e+42], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(-1.0 + N[(n * N[(Pi / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.5999999999999999e42

   1. Initial program 99.0%

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

   3. Taylor expanded in k around 0 62.0%

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

      1. associate-/l* 62.0%

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

   5. Simplified 62.0%

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

      1. pow1 62.0%

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

      2. *-commutative 62.0%

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

      3. sqrt-unprod 62.3%

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

   7. Applied egg-rr 62.3%

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

      1. unpow1 62.3%

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

   9. Simplified 62.3%

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

       1. associate-*r* 62.3%

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

       2. *-commutative 62.3%

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

       3. sqrt-prod 86.4%

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

   11. Applied egg-rr 86.4%

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

   if 2.5999999999999999e42 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.6%

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

      1. associate-/l* 2.6%

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

   5. Simplified 2.6%

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

      1. pow1 2.6%

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

      2. *-commutative 2.6%

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

      3. sqrt-unprod 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. unpow1 2.6%

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

   9. Simplified 2.6%

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

       1. clear-num 2.6%

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

       2. un-div-inv 2.6%

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

   11. Applied egg-rr 2.6%

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

       1. expm1-log1p-u 2.6%

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

       2. expm1-undefine 28.1%

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

       3. associate-/r/ 28.1%

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

       4. *-commutative 28.1%

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

   13. Applied egg-rr 28.1%

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

       1. sub-neg 28.1%

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

       2. metadata-eval 28.1%

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

       3. +-commutative 28.1%

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

       4. log1p-undefine 28.1%

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

       5. rem-exp-log 28.1%

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

       6. +-commutative 28.1%

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

       7. *-commutative 28.1%

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

       8. associate-*l/ 28.1%

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

       9. associate-*r/ 28.1%

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

       10. fma-define 28.1%

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

   15. Simplified 28.1%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{+169}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\pi \cdot n}{k \cdot \frac{k}{\pi \cdot n}} \cdot 4\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.9e+169)
   (* (sqrt (* 2.0 n)) (sqrt (/ PI k)))
   (pow (* (/ (* PI n) (* k (/ k (* PI n)))) 4.0) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.9e+169) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	} else {
		tmp = pow((((((double) M_PI) * n) / (k * (k / (((double) M_PI) * n)))) * 4.0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.9e+169) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((Math.PI / k));
	} else {
		tmp = Math.pow((((Math.PI * n) / (k * (k / (Math.PI * n)))) * 4.0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.9e+169:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((math.pi / k))
	else:
		tmp = math.pow((((math.pi * n) / (k * (k / (math.pi * n)))) * 4.0), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.9e+169)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	else
		tmp = Float64(Float64(Float64(pi * n) / Float64(k * Float64(k / Float64(pi * n)))) * 4.0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.9e+169)
		tmp = sqrt((2.0 * n)) * sqrt((pi / k));
	else
		tmp = (((pi * n) / (k * (k / (pi * n)))) * 4.0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.9e+169], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(Pi * n), $MachinePrecision] / N[(k * N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.9000000000000001e169

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 47.9%

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

      1. associate-/l* 47.9%

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

   5. Simplified 47.9%

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

      1. pow1 47.9%

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

      2. *-commutative 47.9%

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

      3. sqrt-unprod 48.1%

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

   7. Applied egg-rr 48.1%

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

      1. unpow1 48.1%

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

   9. Simplified 48.1%

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

       1. associate-*r* 48.1%

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

       2. *-commutative 48.1%

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

       3. sqrt-prod 66.5%

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

   11. Applied egg-rr 66.5%

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

   if 2.9000000000000001e169 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.5%

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

      1. associate-/l* 2.5%

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

   5. Simplified 2.5%

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

      1. pow1 2.5%

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

      2. *-commutative 2.5%

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

      3. sqrt-unprod 2.5%

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

   7. Applied egg-rr 2.5%

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

      1. unpow1 2.5%

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

   9. Simplified 2.5%

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

       1. pow1/2 2.5%

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

       2. associate-*r/ 2.5%

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

       3. associate-/l* 2.5%

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

       4. metadata-eval 2.5%

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

       5. pow-prod-up 2.5%

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

       6. pow-prod-down 10.0%

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

       7. associate-/l* 10.0%

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

       8. associate-*r/ 10.0%

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

       9. *-commutative 10.0%

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

       10. associate-/l* 10.0%

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

       11. associate-*r/ 10.0%

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

       12. *-commutative 10.0%

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

       13. swap-sqr 11.5%

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

       14. pow2 11.5%

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

       15. metadata-eval 11.5%

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

   11. Applied egg-rr 11.5%

       \[\leadsto \color{blue}{{\left({\left(n \cdot \frac{\pi}{k}\right)}^{2} \cdot 4\right)}^{0.25}} \]
   12. Step-by-step derivation

       1. unpow2 11.5%

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

       2. clear-num 11.5%

          \[\leadsto {\left(\left(\left(n \cdot \color{blue}{\frac{1}{\frac{k}{\pi}}}\right) \cdot \left(n \cdot \frac{\pi}{k}\right)\right) \cdot 4\right)}^{0.25} \]

       3. div-inv 11.5%

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

       4. clear-num 11.5%

          \[\leadsto {\left(\left(\color{blue}{\frac{1}{\frac{\frac{k}{\pi}}{n}}} \cdot \left(n \cdot \frac{\pi}{k}\right)\right) \cdot 4\right)}^{0.25} \]

       5. associate-*r/ 11.5%

          \[\leadsto {\left(\left(\frac{1}{\frac{\frac{k}{\pi}}{n}} \cdot \color{blue}{\frac{n \cdot \pi}{k}}\right) \cdot 4\right)}^{0.25} \]

       6. frac-times 25.3%

          \[\leadsto {\left(\color{blue}{\frac{1 \cdot \left(n \cdot \pi\right)}{\frac{\frac{k}{\pi}}{n} \cdot k}} \cdot 4\right)}^{0.25} \]

       7. *-un-lft-identity 25.3%

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

       8. *-commutative 25.3%

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

       9. associate-/l/ 25.3%

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

       10. *-commutative 25.3%

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

   13. Applied egg-rr 25.3%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{+169}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\pi \cdot n}{k \cdot \frac{k}{\pi \cdot n}} \cdot 4\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. 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. associate-*l* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 6: 44.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{+168}:\\ \;\;\;\;{\left(\frac{\frac{k}{\pi}}{2 \cdot n}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\pi \cdot n}{k \cdot \frac{k}{\pi \cdot n}} \cdot 4\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 9.2e+168)
   (pow (/ (/ k PI) (* 2.0 n)) -0.5)
   (pow (* (/ (* PI n) (* k (/ k (* PI n)))) 4.0) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 9.2e+168) {
		tmp = pow(((k / ((double) M_PI)) / (2.0 * n)), -0.5);
	} else {
		tmp = pow((((((double) M_PI) * n) / (k * (k / (((double) M_PI) * n)))) * 4.0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 9.2e+168) {
		tmp = Math.pow(((k / Math.PI) / (2.0 * n)), -0.5);
	} else {
		tmp = Math.pow((((Math.PI * n) / (k * (k / (Math.PI * n)))) * 4.0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 9.2e+168:
		tmp = math.pow(((k / math.pi) / (2.0 * n)), -0.5)
	else:
		tmp = math.pow((((math.pi * n) / (k * (k / (math.pi * n)))) * 4.0), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 9.2e+168)
		tmp = Float64(Float64(k / pi) / Float64(2.0 * n)) ^ -0.5;
	else
		tmp = Float64(Float64(Float64(pi * n) / Float64(k * Float64(k / Float64(pi * n)))) * 4.0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 9.2e+168)
		tmp = ((k / pi) / (2.0 * n)) ^ -0.5;
	else
		tmp = (((pi * n) / (k * (k / (pi * n)))) * 4.0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 9.2e+168], N[Power[N[(N[(k / Pi), $MachinePrecision] / N[(2.0 * n), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[Power[N[(N[(N[(Pi * n), $MachinePrecision] / N[(k * N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.1999999999999997e168

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 48.1%

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

      1. associate-/l* 48.1%

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

   5. Simplified 48.1%

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

      1. pow1 48.1%

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

      2. *-commutative 48.1%

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

      3. sqrt-unprod 48.3%

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

   7. Applied egg-rr 48.3%

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

      1. unpow1 48.3%

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

   9. Simplified 48.3%

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

       1. clear-num 48.3%

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

       2. un-div-inv 48.3%

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

   11. Applied egg-rr 48.3%

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

       1. associate-*r/ 48.3%

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

       2. *-commutative 48.3%

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

       3. div-inv 48.3%

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

       4. clear-num 48.3%

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

       5. associate-*r/ 48.3%

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

       6. clear-num 48.3%

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

       7. metadata-eval 48.3%

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

       8. add-sqr-sqrt 48.1%

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

       9. frac-times 48.2%

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

       10. sqrt-unprod 49.9%

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

       11. add-sqr-sqrt 50.1%

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

       12. inv-pow 50.1%

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

       13. sqrt-pow2 50.2%

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

       14. *-commutative 50.2%

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

       15. associate-/r* 50.2%

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

       16. metadata-eval 50.2%

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

   13. Applied egg-rr 50.2%

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

   if 9.1999999999999997e168 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.5%

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

      1. associate-/l* 2.5%

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

   5. Simplified 2.5%

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

      1. pow1 2.5%

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

      2. *-commutative 2.5%

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

      3. sqrt-unprod 2.5%

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

   7. Applied egg-rr 2.5%

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

      1. unpow1 2.5%

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

   9. Simplified 2.5%

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

       1. pow1/2 2.5%

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

       2. associate-*r/ 2.5%

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

       3. associate-/l* 2.5%

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

       4. metadata-eval 2.5%

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

       5. pow-prod-up 2.5%

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

       6. pow-prod-down 9.9%

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

       7. associate-/l* 9.9%

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

       8. associate-*r/ 9.9%

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

       9. *-commutative 9.9%

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

       10. associate-/l* 9.9%

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

       11. associate-*r/ 9.9%

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

       12. *-commutative 9.9%

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

       13. swap-sqr 11.4%

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

       14. pow2 11.4%

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

       15. metadata-eval 11.4%

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

   11. Applied egg-rr 11.4%

       \[\leadsto \color{blue}{{\left({\left(n \cdot \frac{\pi}{k}\right)}^{2} \cdot 4\right)}^{0.25}} \]
   12. Step-by-step derivation

       1. unpow2 11.4%

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

       2. clear-num 11.4%

          \[\leadsto {\left(\left(\left(n \cdot \color{blue}{\frac{1}{\frac{k}{\pi}}}\right) \cdot \left(n \cdot \frac{\pi}{k}\right)\right) \cdot 4\right)}^{0.25} \]

       3. div-inv 11.4%

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

       4. clear-num 11.4%

          \[\leadsto {\left(\left(\color{blue}{\frac{1}{\frac{\frac{k}{\pi}}{n}}} \cdot \left(n \cdot \frac{\pi}{k}\right)\right) \cdot 4\right)}^{0.25} \]

       5. associate-*r/ 11.4%

          \[\leadsto {\left(\left(\frac{1}{\frac{\frac{k}{\pi}}{n}} \cdot \color{blue}{\frac{n \cdot \pi}{k}}\right) \cdot 4\right)}^{0.25} \]

       6. frac-times 24.9%

          \[\leadsto {\left(\color{blue}{\frac{1 \cdot \left(n \cdot \pi\right)}{\frac{\frac{k}{\pi}}{n} \cdot k}} \cdot 4\right)}^{0.25} \]

       7. *-un-lft-identity 24.9%

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

       8. *-commutative 24.9%

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

       9. associate-/l/ 24.9%

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

       10. *-commutative 24.9%

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

   13. Applied egg-rr 24.9%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{+168}:\\ \;\;\;\;{\left(\frac{\frac{k}{\pi}}{2 \cdot n}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\pi \cdot n}{k \cdot \frac{k}{\pi \cdot n}} \cdot 4\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 36.7%

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

   1. associate-/l* 36.7%

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

5. Simplified 36.7%

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

   1. pow1 36.7%

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

   2. *-commutative 36.7%

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

   3. sqrt-unprod 36.9%

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

7. Applied egg-rr 36.9%

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

   1. unpow1 36.9%

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

9. Simplified 36.9%

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

    1. clear-num 36.8%

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

    2. un-div-inv 36.9%

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

11. Applied egg-rr 36.9%

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

    1. associate-*r/ 36.9%

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

    2. *-commutative 36.9%

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

    3. div-inv 36.8%

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

    4. clear-num 36.9%

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

    5. associate-*r/ 36.8%

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

    6. clear-num 36.8%

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

    7. metadata-eval 36.8%

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

    8. add-sqr-sqrt 36.7%

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

    9. frac-times 36.8%

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

    10. sqrt-unprod 38.1%

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

    11. add-sqr-sqrt 38.2%

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

    12. inv-pow 38.2%

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

    13. sqrt-pow2 38.3%

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

    14. *-commutative 38.3%

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

    15. associate-/r* 38.3%

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

    16. metadata-eval 38.3%

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

13. Applied egg-rr 38.3%

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

14. Final simplification 38.3%

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

Alternative 8: 37.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 36.7%

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

   1. associate-/l* 36.7%

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

5. Simplified 36.7%

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

   1. pow1 36.7%

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

   2. *-commutative 36.7%

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

   3. sqrt-unprod 36.9%

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

7. Applied egg-rr 36.9%

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

   1. unpow1 36.9%

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

9. Simplified 36.9%

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

10. Final simplification 36.9%

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

Alternative 9: 37.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 36.7%

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

   1. associate-/l* 36.7%

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

5. Simplified 36.7%

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

   1. pow1 36.7%

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

   2. *-commutative 36.7%

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

   3. sqrt-unprod 36.9%

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

7. Applied egg-rr 36.9%

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

   1. unpow1 36.9%

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

9. Simplified 36.9%

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

    1. clear-num 36.8%

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

    2. un-div-inv 36.9%

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

11. Applied egg-rr 36.9%

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

12. Final simplification 36.9%

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

Reproduce

herbie shell --seed 2024085 
(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))))