Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.5%

Time: 16.9s

Alternatives: 7

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

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

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(Float64(2.0 * 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[(N[(2.0 * Pi), $MachinePrecision] * n), $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/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. sqr-pow 99.3%

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

   4. pow-sqr 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-*l/ 99.5%

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

   7. associate-/l* 99.5%

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

   8. metadata-eval 99.5%

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

   9. /-rgt-identity 99.5%

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

   10. div-sub 99.5%

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

   11. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.3 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)} \cdot \frac{1}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6.3e-21)
   (* (sqrt (* 2.0 n)) (sqrt (/ PI k)))
   (sqrt (* (pow (* PI (* 2.0 n)) (- 1.0 k)) (/ 1.0 k)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6.3e-21) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	} else {
		tmp = sqrt((pow((((double) M_PI) * (2.0 * n)), (1.0 - k)) * (1.0 / k)));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 6.3e-21) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((Math.PI / k));
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (2.0 * n)), (1.0 - k)) * (1.0 / k)));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 6.3e-21:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((math.pi / k))
	else:
		tmp = math.sqrt((math.pow((math.pi * (2.0 * n)), (1.0 - k)) * (1.0 / k)))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 6.3e-21)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(2.0 * n)) ^ Float64(1.0 - k)) * Float64(1.0 / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 6.3e-21)
		tmp = sqrt((2.0 * n)) * sqrt((pi / k));
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) * (1.0 / k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6.3e-21], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.3e-21

   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. Step-by-step derivation

      1. *-commutative 99.3%

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

      2. div-sub 99.3%

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

      3. metadata-eval 99.3%

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

      4. div-inv 99.4%

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

      5. add-sqr-sqrt 99.0%

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

      6. sqrt-unprod 76.1%

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

      7. frac-times 76.0%

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

   3. Applied egg-rr 76.2%

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

   4. Taylor expanded in k around 0 76.2%

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

      1. associate-/l* 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in n around 0 76.2%

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

      1. associate-/l* 76.3%

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

      2. *-rgt-identity 76.3%

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

      3. associate-*r/ 76.2%

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

      4. associate-/r/ 76.2%

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

      5. associate-*l/ 76.3%

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

      6. *-lft-identity 76.3%

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

   9. Simplified 76.3%

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

       1. associate-*r* 76.3%

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

       2. sqrt-prod 99.4%

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

   11. Applied egg-rr 99.4%

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

       1. *-commutative 99.4%

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

   13. Simplified 99.4%

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

   if 6.3e-21 < 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. Step-by-step derivation

      1. *-commutative 99.6%

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

      2. div-sub 99.6%

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

      3. metadata-eval 99.6%

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

      4. div-inv 99.6%

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

      5. add-sqr-sqrt 99.6%

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

      6. sqrt-unprod 99.6%

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

      7. frac-times 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. div-inv 99.6%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*l* 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.3 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)} \cdot \frac{1}{k}}\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6.6e-21)
   (* (sqrt (* 2.0 n)) (sqrt (/ PI k)))
   (sqrt (/ (pow (* 2.0 (* PI n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6.6e-21) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	} 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 <= 6.6e-21) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((Math.PI / k));
	} 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 <= 6.6e-21:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((math.pi / k))
	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 <= 6.6e-21)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	else
		tmp = sqrt(Float64((Float64(2.0 * Float64(pi * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 6.6e-21)
		tmp = sqrt((2.0 * n)) * sqrt((pi / k));
	else
		tmp = sqrt((((2.0 * (pi * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6.6e-21], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.60000000000000018e-21

   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. Step-by-step derivation

      1. *-commutative 99.3%

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

      2. div-sub 99.3%

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

      3. metadata-eval 99.3%

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

      4. div-inv 99.4%

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

      5. add-sqr-sqrt 99.0%

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

      6. sqrt-unprod 76.1%

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

      7. frac-times 76.0%

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

   3. Applied egg-rr 76.2%

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

   4. Taylor expanded in k around 0 76.2%

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

      1. associate-/l* 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in n around 0 76.2%

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

      1. associate-/l* 76.3%

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

      2. *-rgt-identity 76.3%

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

      3. associate-*r/ 76.2%

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

      4. associate-/r/ 76.2%

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

      5. associate-*l/ 76.3%

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

      6. *-lft-identity 76.3%

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

   9. Simplified 76.3%

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

       1. associate-*r* 76.3%

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

       2. sqrt-prod 99.4%

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

   11. Applied egg-rr 99.4%

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

       1. *-commutative 99.4%

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

   13. Simplified 99.4%

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

   if 6.60000000000000018e-21 < 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. Step-by-step derivation

      1. *-commutative 99.6%

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

      2. div-sub 99.6%

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

      3. metadata-eval 99.6%

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

      4. div-inv 99.6%

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

      5. add-sqr-sqrt 99.6%

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

      6. sqrt-unprod 99.6%

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

      7. frac-times 99.6%

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

   3. Applied egg-rr 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 4: 54.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\pi \cdot \left(2 \cdot \frac{n}{k}\right)\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.2e+143)
   (* (sqrt (* 2.0 n)) (sqrt (/ PI k)))
   (pow (pow (* PI (* 2.0 (/ n k))) 3.0) 0.16666666666666666)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.2e+143) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	} else {
		tmp = pow(pow((((double) M_PI) * (2.0 * (n / k))), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.2e+143) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((Math.PI / k));
	} else {
		tmp = Math.pow(Math.pow((Math.PI * (2.0 * (n / k))), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.2e+143:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((math.pi / k))
	else:
		tmp = math.pow(math.pow((math.pi * (2.0 * (n / k))), 3.0), 0.16666666666666666)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.2e+143)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	else
		tmp = (Float64(pi * Float64(2.0 * Float64(n / k))) ^ 3.0) ^ 0.16666666666666666;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.2e+143)
		tmp = sqrt((2.0 * n)) * sqrt((pi / k));
	else
		tmp = ((pi * (2.0 * (n / k))) ^ 3.0) ^ 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.2e+143], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(Pi * N[(2.0 * N[(n / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1999999999999999e143

   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. Step-by-step derivation

      1. *-commutative 99.3%

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

      2. div-sub 99.3%

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

      3. metadata-eval 99.3%

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

      4. div-inv 99.4%

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

      5. add-sqr-sqrt 99.1%

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

      6. sqrt-unprod 84.5%

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

      7. frac-times 84.5%

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

   3. Applied egg-rr 84.6%

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

   4. Taylor expanded in k around 0 52.8%

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

      1. associate-/l* 52.8%

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

   6. Simplified 52.8%

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

   7. Taylor expanded in n around 0 52.8%

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

      1. associate-/l* 52.8%

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

      2. *-rgt-identity 52.8%

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

      3. associate-*r/ 52.8%

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

      4. associate-/r/ 52.8%

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

      5. associate-*l/ 52.9%

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

      6. *-lft-identity 52.9%

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

   9. Simplified 52.9%

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

       1. associate-*r* 52.9%

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

       2. sqrt-prod 67.6%

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

   11. Applied egg-rr 67.6%

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

       1. *-commutative 67.6%

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

   13. Simplified 67.6%

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

   if 1.1999999999999999e143 < 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. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. div-sub 100.0%

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

      3. metadata-eval 100.0%

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

      4. div-inv 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. frac-times 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in k around 0 2.6%

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

      1. associate-/l* 2.6%

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

   6. Simplified 2.6%

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

   7. Taylor expanded in n around 0 2.6%

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

      1. associate-/l* 2.6%

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

      2. *-rgt-identity 2.6%

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

      3. associate-*r/ 2.6%

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

      4. associate-/r/ 2.6%

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

      5. associate-*l/ 2.6%

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

      6. *-lft-identity 2.6%

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

   9. Simplified 2.6%

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

       1. pow1/2 2.6%

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

       2. associate-*r/ 2.6%

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

       3. associate-*l/ 2.6%

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

       4. metadata-eval 2.6%

          \[\leadsto {\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       5. pow-pow 4.1%

          \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       6. sqr-pow 4.1%

          \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{1.5}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left({\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{1.5}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

       7. pow-prod-down 16.7%

          \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{1.5} \cdot {\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{1.5}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

       8. pow-prod-up 16.7%

          \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{\left(1.5 + 1.5\right)}\right)}}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       9. associate-*r* 16.7%

          \[\leadsto {\left({\color{blue}{\left(\left(2 \cdot \frac{n}{k}\right) \cdot \pi\right)}}^{\left(1.5 + 1.5\right)}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       10. metadata-eval 16.7%

           \[\leadsto {\left({\left(\left(2 \cdot \frac{n}{k}\right) \cdot \pi\right)}^{\color{blue}{3}}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       11. metadata-eval 16.7%

           \[\leadsto {\left({\left(\left(2 \cdot \frac{n}{k}\right) \cdot \pi\right)}^{3}\right)}^{\color{blue}{0.16666666666666666}} \]

   11. Applied egg-rr 16.7%

       \[\leadsto \color{blue}{{\left({\left(\left(2 \cdot \frac{n}{k}\right) \cdot \pi\right)}^{3}\right)}^{0.16666666666666666}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\pi \cdot \left(2 \cdot \frac{n}{k}\right)\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \]

Alternative 5: 49.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $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. Step-by-step derivation

   1. *-commutative 99.4%

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

   2. div-sub 99.4%

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

   3. metadata-eval 99.4%

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

   4. div-inv 99.5%

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

   5. add-sqr-sqrt 99.3%

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

   6. sqrt-unprod 88.2%

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

   7. frac-times 88.2%

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

3. Applied egg-rr 88.3%

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

4. Taylor expanded in k around 0 40.9%

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

   1. associate-/l* 40.9%

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

6. Simplified 40.9%

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

7. Taylor expanded in n around 0 40.9%

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

   1. associate-/l* 40.9%

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

   2. *-rgt-identity 40.9%

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

   3. associate-*r/ 40.9%

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

   4. associate-/r/ 40.9%

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

   5. associate-*l/ 40.9%

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

   6. *-lft-identity 40.9%

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

9. Simplified 40.9%

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

    1. associate-*r* 40.9%

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

    2. sqrt-prod 52.1%

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

11. Applied egg-rr 52.1%

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

    1. *-commutative 52.1%

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

13. Simplified 52.1%

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

14. Final simplification 52.1%

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

Alternative 6: 39.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(1.0 / N[Sqrt[N[(k / N[(2.0 * N[(Pi * n), $MachinePrecision]), $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. Step-by-step derivation

   1. expm1-log1p-u 96.0%

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

   2. expm1-udef 78.0%

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

   3. inv-pow 78.0%

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

   4. sqrt-pow2 78.0%

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

   5. metadata-eval 78.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left({k}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

3. Applied egg-rr 78.0%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-0.5}\right)} - 1\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
4. Step-by-step derivation

   1. expm1-def 96.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. expm1-log1p 99.5%

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

5. Simplified 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 88.2%

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

   3. swap-sqr 88.2%

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

   4. pow-prod-up 88.2%

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

   5. metadata-eval 88.2%

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

   6. inv-pow 88.2%

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

   7. associate-*r* 88.2%

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

   8. associate-*r* 88.2%

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

   9. sqr-pow 88.3%

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

   10. associate-/r/ 88.3%

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

   11. sqrt-div 89.5%

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

7. Applied egg-rr 89.5%

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

8. Taylor expanded in k around 0 42.1%

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

9. Final simplification 42.1%

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

Alternative 7: 39.0% accurate, 1.5× 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. Step-by-step derivation

   1. *-commutative 99.4%

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

   2. div-sub 99.4%

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

   3. metadata-eval 99.4%

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

   4. div-inv 99.5%

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

   5. add-sqr-sqrt 99.3%

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

   6. sqrt-unprod 88.2%

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

   7. frac-times 88.2%

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

3. Applied egg-rr 88.3%

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

4. Taylor expanded in k around 0 40.9%

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

   1. associate-/l* 40.9%

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

6. Simplified 40.9%

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

7. Taylor expanded in n around 0 40.9%

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

   1. associate-/l* 40.9%

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

   2. *-rgt-identity 40.9%

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

   3. associate-*r/ 40.9%

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

   4. associate-/r/ 40.9%

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

   5. associate-*l/ 40.9%

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

   6. *-lft-identity 40.9%

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

9. Simplified 40.9%

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

10. Final simplification 40.9%

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

Reproduce

herbie shell --seed 2023283 
(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))))