Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.5%

Time: 49.6s

Alternatives: 12

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 99.6%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{\sqrt{k} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.8e-51)
   (/ 1.0 (* (sqrt k) (pow (* (* 2.0 PI) n) -0.5)))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.8e-51) {
		tmp = 1.0 / (sqrt(k) * pow(((2.0 * ((double) M_PI)) * n), -0.5));
	} else {
		tmp = sqrt((pow((((double) M_PI) * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.8e-51) {
		tmp = 1.0 / (Math.sqrt(k) * Math.pow(((2.0 * Math.PI) * n), -0.5));
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.8e-51:
		tmp = 1.0 / (math.sqrt(k) * math.pow(((2.0 * math.pi) * n), -0.5))
	else:
		tmp = math.sqrt((math.pow((math.pi * (2.0 * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.8e-51)
		tmp = Float64(1.0 / Float64(sqrt(k) * (Float64(Float64(2.0 * pi) * n) ^ -0.5)));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(2.0 * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.8e-51)
		tmp = 1.0 / (sqrt(k) * (((2.0 * pi) * n) ^ -0.5));
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.8e-51], N[(1.0 / N[(N[Sqrt[k], $MachinePrecision] * N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.8e-51

   1. Initial program 99.2%

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

   3. Taylor expanded in k around 0 99.2%

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

      1. associate-*l/ 99.3%

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

      2. *-un-lft-identity 99.3%

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

      3. clear-num 99.2%

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

      4. *-commutative 99.2%

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

      5. sqrt-prod 99.3%

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

      6. *-commutative 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. div-inv 99.3%

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

      2. pow1/2 99.3%

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

      3. pow-flip 99.4%

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

      4. metadata-eval 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*r* 99.4%

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

      3. rem-exp-log 99.0%

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

      4. rem-exp-log 92.4%

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

      5. exp-sum 91.6%

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

      6. +-commutative 91.6%

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

      7. exp-sum 92.4%

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

      8. rem-exp-log 99.0%

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

      9. rem-exp-log 99.4%

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

   9. Simplified 99.4%

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

   if 2.8e-51 < 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. Taylor expanded in k around 0 99.8%

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

   5. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.2%

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

      3. distribute-rgt-in 99.2%

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

      4. metadata-eval 99.2%

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

      5. associate-*l* 99.2%

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

      6. metadata-eval 99.2%

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

      7. *-commutative 99.2%

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

      8. neg-mul-1 99.2%

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

      9. sub-neg 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{\sqrt{k} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{\sqrt{k} \cdot \sqrt{\frac{\frac{0.5}{n}}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.7e-57)
   (/ 1.0 (* (sqrt k) (sqrt (/ (/ 0.5 n) PI))))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.7e-57) {
		tmp = 1.0 / (sqrt(k) * sqrt(((0.5 / n) / ((double) M_PI))));
	} else {
		tmp = sqrt((pow((((double) M_PI) * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.7e-57) {
		tmp = 1.0 / (Math.sqrt(k) * Math.sqrt(((0.5 / n) / Math.PI)));
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 4.7e-57:
		tmp = 1.0 / (math.sqrt(k) * math.sqrt(((0.5 / n) / math.pi)))
	else:
		tmp = math.sqrt((math.pow((math.pi * (2.0 * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.7e-57)
		tmp = Float64(1.0 / Float64(sqrt(k) * sqrt(Float64(Float64(0.5 / n) / pi))));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(2.0 * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 4.7e-57)
		tmp = 1.0 / (sqrt(k) * sqrt(((0.5 / n) / pi)));
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.7e-57], N[(1.0 / N[(N[Sqrt[k], $MachinePrecision] * N[Sqrt[N[(N[(0.5 / n), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.6999999999999998e-57

   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 99.2%

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

      1. associate-*l/ 99.2%

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

      2. *-un-lft-identity 99.2%

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

      3. clear-num 99.2%

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

      4. *-commutative 99.2%

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

      5. sqrt-prod 99.3%

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

      6. *-commutative 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in k around 0 77.4%

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

      1. associate-*r/ 77.5%

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

      2. *-rgt-identity 77.5%

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

   8. Simplified 77.5%

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

      1. sqrt-undiv 77.5%

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

      2. clear-num 75.8%

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

      3. associate-*l/ 75.6%

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

      4. *-commutative 75.6%

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

      5. associate-/l/ 75.6%

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

      6. *-commutative 75.6%

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

      7. metadata-eval 75.6%

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

      8. associate-*l/ 75.8%

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

      9. times-frac 75.8%

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

      10. *-un-lft-identity 75.8%

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

      11. *-commutative 75.8%

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

      12. clear-num 77.5%

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

      13. associate-/l* 77.4%

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

      14. sqrt-prod 99.4%

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

      15. associate-/r* 99.4%

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

   10. Applied egg-rr 99.4%

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

   if 4.6999999999999998e-57 < 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. Taylor expanded in k around 0 99.8%

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

   5. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.2%

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

      3. distribute-rgt-in 99.2%

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

      4. metadata-eval 99.2%

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

      5. associate-*l* 99.2%

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

      6. metadata-eval 99.2%

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

      7. *-commutative 99.2%

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

      8. neg-mul-1 99.2%

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

      9. sub-neg 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{\sqrt{k} \cdot \sqrt{\frac{\frac{0.5}{n}}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{\sqrt{k} \cdot \sqrt{\frac{\frac{0.5}{n}}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.8e+80)
   (/ 1.0 (* (sqrt k) (sqrt (/ (/ 0.5 n) PI))))
   (sqrt (* 2.0 (+ -1.0 (fma n (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.8e+80) {
		tmp = 1.0 / (sqrt(k) * sqrt(((0.5 / n) / ((double) M_PI))));
	} 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 <= 4.8e+80)
		tmp = Float64(1.0 / Float64(sqrt(k) * sqrt(Float64(Float64(0.5 / n) / pi))));
	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, 4.8e+80], N[(1.0 / N[(N[Sqrt[k], $MachinePrecision] * N[Sqrt[N[(N[(0.5 / n), $MachinePrecision] / Pi), $MachinePrecision]], $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 < 4.79999999999999958e80

   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 79.2%

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

      1. associate-*l/ 79.3%

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

      2. *-un-lft-identity 79.3%

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

      3. clear-num 79.2%

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

      4. *-commutative 79.2%

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

      5. sqrt-prod 79.3%

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

      6. *-commutative 79.3%

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

   5. Applied egg-rr 79.3%

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

   6. Taylor expanded in k around 0 64.6%

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

      1. associate-*r/ 64.7%

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

      2. *-rgt-identity 64.7%

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

   8. Simplified 64.7%

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

      1. sqrt-undiv 64.7%

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

      2. clear-num 63.5%

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

      3. associate-*l/ 63.4%

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

      4. *-commutative 63.4%

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

      5. associate-/l/ 63.4%

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

      6. *-commutative 63.4%

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

      7. metadata-eval 63.4%

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

      8. associate-*l/ 63.5%

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

      9. times-frac 63.5%

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

      10. *-un-lft-identity 63.5%

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

      11. *-commutative 63.5%

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

      12. clear-num 64.7%

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

      13. associate-/l* 64.6%

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

      14. sqrt-prod 79.3%

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

      15. associate-/r* 79.3%

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

   10. Applied egg-rr 79.3%

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

   if 4.79999999999999958e80 < 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 2.4%

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

      1. associate-/l* 2.4%

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

   5. Simplified 2.4%

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

      1. *-commutative 2.4%

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

      2. sqrt-unprod 2.4%

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

      3. associate-*r/ 2.4%

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

      4. *-commutative 2.4%

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

   7. Applied egg-rr 2.4%

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

      1. expm1-log1p-u 2.4%

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

      2. expm1-undefine 22.7%

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

      3. associate-/l* 22.7%

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

   9. Applied egg-rr 22.7%

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

       1. sub-neg 22.7%

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

       2. metadata-eval 22.7%

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

       3. +-commutative 22.7%

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

       4. log1p-undefine 22.7%

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

       5. rem-exp-log 22.7%

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

       6. +-commutative 22.7%

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

       7. associate-*r/ 22.7%

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

       8. associate-*l/ 22.7%

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

       9. *-commutative 22.7%

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

       10. fma-define 22.7%

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

   11. Simplified 22.7%

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

Alternative 5: 59.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{\frac{1}{k}} \cdot \sqrt{\left(2 \cdot \pi\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5e+80)
   (* (sqrt (/ 1.0 k)) (sqrt (* (* 2.0 PI) n)))
   (sqrt (* 2.0 (+ -1.0 (fma n (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5e+80) {
		tmp = sqrt((1.0 / k)) * sqrt(((2.0 * ((double) M_PI)) * n));
	} 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 <= 5e+80)
		tmp = Float64(sqrt(Float64(1.0 / k)) * sqrt(Float64(Float64(2.0 * pi) * n)));
	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, 5e+80], N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * Pi), $MachinePrecision] * n), $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 < 4.99999999999999961e80

   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 63.4%

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

      1. associate-/l* 63.3%

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

   5. Simplified 63.3%

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

      1. *-commutative 63.3%

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

      2. sqrt-unprod 63.5%

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

      3. associate-*r/ 63.6%

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

      4. *-commutative 63.6%

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

   7. Applied egg-rr 63.6%

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

      1. pow1/2 63.6%

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

      2. associate-*r/ 63.6%

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

      3. div-inv 63.5%

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

      4. unpow-prod-down 79.3%

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

      5. pow1/2 79.3%

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

   9. Applied egg-rr 79.3%

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

       1. unpow1/2 79.3%

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

       2. *-commutative 79.3%

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

       3. associate-*r* 79.3%

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

       4. rem-exp-log 79.1%

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

       5. rem-exp-log 73.9%

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

       6. exp-sum 73.3%

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

       7. +-commutative 73.3%

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

       8. exp-sum 73.9%

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

       9. rem-exp-log 79.1%

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

       10. rem-exp-log 79.3%

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

   11. Simplified 79.3%

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

   if 4.99999999999999961e80 < 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 2.4%

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

      1. associate-/l* 2.4%

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

   5. Simplified 2.4%

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

      1. *-commutative 2.4%

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

      2. sqrt-unprod 2.4%

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

      3. associate-*r/ 2.4%

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

      4. *-commutative 2.4%

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

   7. Applied egg-rr 2.4%

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

      1. expm1-log1p-u 2.4%

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

      2. expm1-undefine 22.7%

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

      3. associate-/l* 22.7%

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

   9. Applied egg-rr 22.7%

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

       1. sub-neg 22.7%

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

       2. metadata-eval 22.7%

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

       3. +-commutative 22.7%

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

       4. log1p-undefine 22.7%

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

       5. rem-exp-log 22.7%

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

       6. +-commutative 22.7%

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

       7. associate-*r/ 22.7%

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

       8. associate-*l/ 22.7%

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

       9. *-commutative 22.7%

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

       10. fma-define 22.7%

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

   11. Simplified 22.7%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{\frac{1}{k}} \cdot \sqrt{\left(2 \cdot \pi\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5e+80)
   (/ (sqrt (* PI (* 2.0 n))) (sqrt k))
   (sqrt (* 2.0 (+ -1.0 (fma n (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5e+80) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) / sqrt(k);
	} else {
		tmp = sqrt((2.0 * (-1.0 + fma(n, (((double) M_PI) / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5e+80)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) / sqrt(k));
	else
		tmp = sqrt(Float64(2.0 * Float64(-1.0 + fma(n, Float64(pi / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5e+80], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(-1.0 + N[(n * N[(Pi / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.99999999999999961e80

   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 79.2%

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

      1. associate-*l/ 79.3%

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

      2. *-un-lft-identity 79.3%

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

      3. clear-num 79.2%

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

      4. *-commutative 79.2%

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

      5. sqrt-prod 79.3%

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

      6. *-commutative 79.3%

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

   5. Applied egg-rr 79.3%

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

      1. associate-/r/ 79.2%

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

      2. associate-*l/ 79.3%

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

      3. *-lft-identity 79.3%

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

      4. *-commutative 79.3%

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

      5. associate-*l* 79.3%

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

   7. Simplified 79.3%

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

   if 4.99999999999999961e80 < 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 2.4%

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

      1. associate-/l* 2.4%

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

   5. Simplified 2.4%

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

      1. *-commutative 2.4%

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

      2. sqrt-unprod 2.4%

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

      3. associate-*r/ 2.4%

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

      4. *-commutative 2.4%

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

   7. Applied egg-rr 2.4%

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

      1. expm1-log1p-u 2.4%

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

      2. expm1-undefine 22.7%

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

      3. associate-/l* 22.7%

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

   9. Applied egg-rr 22.7%

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

       1. sub-neg 22.7%

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

       2. metadata-eval 22.7%

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

       3. +-commutative 22.7%

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

       4. log1p-undefine 22.7%

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

       5. rem-exp-log 22.7%

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

       6. +-commutative 22.7%

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

       7. associate-*r/ 22.7%

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

       8. associate-*l/ 22.7%

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

       9. *-commutative 22.7%

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

       10. fma-define 22.7%

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

   11. Simplified 22.7%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 99.5%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. 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. Add Preprocessing

Alternative 8: 49.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 48.1%

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

   1. associate-*l/ 48.1%

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

   2. *-un-lft-identity 48.1%

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

   3. clear-num 48.1%

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

   4. *-commutative 48.1%

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

   5. sqrt-prod 48.1%

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

   6. *-commutative 48.1%

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

5. Applied egg-rr 48.1%

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

   1. associate-/r/ 48.1%

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

   2. associate-*l/ 48.1%

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

   3. *-lft-identity 48.1%

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

   4. *-commutative 48.1%

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

   5. associate-*l* 48.1%

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

7. Simplified 48.1%

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

8. Final simplification 48.1%

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

Alternative 9: 38.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 38.6%

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

   1. associate-/l* 38.6%

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

5. Simplified 38.6%

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

   1. *-commutative 38.6%

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

   2. sqrt-unprod 38.7%

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

   3. associate-*r/ 38.7%

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

   4. *-commutative 38.7%

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

7. Applied egg-rr 38.7%

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

   1. associate-*r/ 38.7%

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

   2. sqrt-div 48.1%

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

   3. clear-num 48.1%

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

   4. inv-pow 48.1%

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

   5. sqrt-undiv 39.4%

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

   6. sqrt-pow2 39.4%

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

   7. *-un-lft-identity 39.4%

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

   8. times-frac 39.4%

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

   9. metadata-eval 39.4%

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

   10. *-commutative 39.4%

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

   11. associate-/r* 39.4%

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

   12. metadata-eval 39.4%

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

9. Applied egg-rr 39.4%

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

    1. associate-/l/ 39.4%

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

    2. associate-*r/ 39.4%

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

    3. *-commutative 39.4%

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

    4. *-commutative 39.4%

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

11. Simplified 39.4%

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

12. Final simplification 39.4%

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

Alternative 10: 38.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 38.6%

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

   1. associate-/l* 38.6%

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

5. Simplified 38.6%

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

   1. *-commutative 38.6%

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

   2. sqrt-unprod 38.7%

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

   3. associate-*r/ 38.7%

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

   4. *-commutative 38.7%

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

7. Applied egg-rr 38.7%

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

   1. associate-*r/ 38.7%

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

   2. sqrt-div 48.1%

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

   3. clear-num 48.1%

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

   4. inv-pow 48.1%

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

   5. sqrt-undiv 39.4%

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

   6. sqrt-pow2 39.4%

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

   7. *-un-lft-identity 39.4%

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

   8. times-frac 39.4%

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

   9. metadata-eval 39.4%

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

   10. *-commutative 39.4%

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

   11. associate-/r* 39.4%

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

   12. metadata-eval 39.4%

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

9. Applied egg-rr 39.4%

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

Alternative 11: 38.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 38.6%

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

   1. associate-/l* 38.6%

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

5. Simplified 38.6%

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

   1. *-commutative 38.6%

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

   2. sqrt-unprod 38.7%

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

   3. associate-*r/ 38.7%

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

   4. *-commutative 38.7%

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

7. Applied egg-rr 38.7%

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

Alternative 12: 38.0% 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 99.5%

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

3. Taylor expanded in k around 0 38.6%

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

   1. associate-/l* 38.6%

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

5. Simplified 38.6%

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

   1. *-commutative 38.6%

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

   2. sqrt-unprod 38.7%

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

   3. associate-*r/ 38.7%

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

   4. *-commutative 38.7%

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

7. Applied egg-rr 38.7%

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

   1. *-commutative 38.7%

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

   2. associate-*l/ 38.7%

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

9. Applied egg-rr 38.7%

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

10. Final simplification 38.7%

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

Reproduce

herbie shell --seed 2024116 
(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))))