Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 16.4s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l/ 99.3%

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

   2. *-lft-identity 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*l* 99.3%

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

   5. div-sub 99.3%

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

   6. sub-neg 99.3%

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

   7. distribute-frac-neg 99.3%

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

   8. metadata-eval 99.3%

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

   9. neg-mul-1 99.3%

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

   10. associate-/l* 99.3%

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

   11. associate-/r/ 99.3%

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

   12. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.55e-110)
   (/ (sqrt (* 2.0 (* PI n))) (sqrt k))
   (sqrt (/ (pow (* n (* PI 2.0)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.55e-110) {
		tmp = sqrt((2.0 * (((double) M_PI) * n))) / sqrt(k);
	} else {
		tmp = sqrt((pow((n * (((double) M_PI) * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.55e-110) {
		tmp = Math.sqrt((2.0 * (Math.PI * n))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow((n * (Math.PI * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.55e-110:
		tmp = math.sqrt((2.0 * (math.pi * n))) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow((n * (math.pi * 2.0)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.55e-110)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi * n))) / sqrt(k));
	else
		tmp = sqrt(Float64((Float64(n * Float64(pi * 2.0)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.55e-110)
		tmp = sqrt((2.0 * (pi * n))) / sqrt(k);
	else
		tmp = sqrt((((n * (pi * 2.0)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.55e-110], N[(N[Sqrt[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.55000000000000004e-110

   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. Step-by-step derivation

      1. associate-*l/ 99.4%

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

      2. *-lft-identity 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-*l* 99.4%

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

      5. div-sub 99.4%

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

      6. sub-neg 99.4%

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

      7. distribute-frac-neg 99.4%

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

      8. metadata-eval 99.4%

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

      9. neg-mul-1 99.4%

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

      10. associate-/l* 99.4%

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

      11. associate-/r/ 99.4%

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

      12. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. +-commutative 99.4%

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

      2. unpow-prod-up 99.4%

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

      3. associate-*r* 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-*l* 99.4%

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

      6. *-commutative 99.4%

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

      7. pow1/2 99.4%

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

      8. associate-*r* 99.4%

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

      9. *-commutative 99.4%

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

      10. associate-*l* 99.4%

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

   5. Applied egg-rr 99.4%

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

   6. Taylor expanded in k around 0 99.4%

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

   if 1.55000000000000004e-110 < k

   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. Step-by-step derivation

      1. add-sqr-sqrt 98.4%

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

      2. sqrt-unprod 98.6%

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

      3. associate-*l/ 98.6%

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

      4. *-un-lft-identity 98.6%

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

      5. associate-*l/ 98.6%

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

      6. *-un-lft-identity 98.6%

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

      7. frac-times 98.6%

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

   3. Applied egg-rr 98.6%

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

   5. Simplified 98.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 3: 49.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 98.6%

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

   2. sqrt-unprod 87.5%

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

   3. associate-*l/ 87.5%

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

   4. *-un-lft-identity 87.5%

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

   5. associate-*l/ 87.6%

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

   6. *-un-lft-identity 87.6%

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

   7. frac-times 87.5%

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

3. Applied egg-rr 87.5%

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

5. Simplified 87.7%

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

6. Taylor expanded in k around 0 42.3%

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

   1. associate-*r/ 42.3%

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

   2. *-commutative 42.3%

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

   3. associate-/l* 42.3%

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

8. Simplified 42.3%

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

   1. div-inv 42.3%

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

   2. clear-num 42.3%

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

   3. associate-*l* 42.3%

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

10. Applied egg-rr 42.3%

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

    1. sqrt-prod 53.6%

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

    2. associate-*r/ 53.6%

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

    3. *-commutative 53.6%

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

    4. *-un-lft-identity 53.6%

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

    5. times-frac 53.6%

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

    6. metadata-eval 53.6%

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

12. Applied egg-rr 53.6%

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

    1. associate-*r/ 53.6%

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

    2. associate-*l/ 53.6%

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

14. Simplified 53.6%

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

15. Final simplification 53.6%

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

Alternative 4: 49.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 98.6%

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

   2. sqrt-unprod 87.5%

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

   3. associate-*l/ 87.5%

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

   4. *-un-lft-identity 87.5%

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

   5. associate-*l/ 87.6%

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

   6. *-un-lft-identity 87.6%

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

   7. frac-times 87.5%

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

3. Applied egg-rr 87.5%

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

5. Simplified 87.7%

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

6. Taylor expanded in k around 0 42.3%

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

   1. associate-*r/ 42.3%

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

   2. *-commutative 42.3%

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

   3. associate-/l* 42.3%

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

8. Simplified 42.3%

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

   1. div-inv 42.3%

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

   2. sqrt-prod 53.6%

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

   3. *-commutative 53.6%

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

   4. clear-num 53.6%

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

10. Applied egg-rr 53.6%

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

11. Final simplification 53.6%

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

Alternative 5: 50.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{n}}{\sqrt{\frac{0.5 \cdot k}{\pi}}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 98.6%

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

   2. sqrt-unprod 87.5%

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

   3. associate-*l/ 87.5%

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

   4. *-un-lft-identity 87.5%

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

   5. associate-*l/ 87.6%

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

   6. *-un-lft-identity 87.6%

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

   7. frac-times 87.5%

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

3. Applied egg-rr 87.5%

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

5. Simplified 87.7%

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

6. Taylor expanded in k around 0 42.3%

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

   1. associate-*r/ 42.3%

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

   2. *-commutative 42.3%

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

   3. associate-/l* 42.3%

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

8. Simplified 42.3%

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

   1. associate-/l* 42.3%

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

   2. sqrt-div 53.7%

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

   3. div-inv 53.7%

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

   4. metadata-eval 53.7%

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

10. Applied egg-rr 53.7%

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

11. Final simplification 53.7%

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

Alternative 6: 38.7% accurate, 1.5× speedup

Math

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

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(k * Float64(Float64(0.5 / 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[(k * N[(N[(0.5 / Pi), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 98.6%

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

   2. sqrt-unprod 87.5%

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

   3. associate-*l/ 87.5%

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

   4. *-un-lft-identity 87.5%

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

   5. associate-*l/ 87.6%

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

   6. *-un-lft-identity 87.6%

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

   7. frac-times 87.5%

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

3. Applied egg-rr 87.5%

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

5. Simplified 87.7%

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

6. Taylor expanded in k around 0 42.3%

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

   1. associate-*r/ 42.3%

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

   2. *-commutative 42.3%

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

   3. associate-/l* 42.3%

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

8. Simplified 42.3%

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

   1. clear-num 42.3%

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

   2. sqrt-div 42.6%

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

   3. metadata-eval 42.6%

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

   4. div-inv 42.6%

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

   5. metadata-eval 42.6%

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

   6. *-un-lft-identity 42.6%

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

   7. times-frac 42.6%

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

   8. /-rgt-identity 42.6%

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

   9. *-commutative 42.6%

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

10. Applied egg-rr 42.6%

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

    1. pow1/2 42.6%

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

    2. pow-flip 42.6%

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

    3. associate-/r* 42.6%

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

    4. metadata-eval 42.6%

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

12. Applied egg-rr 42.6%

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

13. Final simplification 42.6%

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

Alternative 7: 38.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 98.6%

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

   2. sqrt-unprod 87.5%

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

   3. associate-*l/ 87.5%

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

   4. *-un-lft-identity 87.5%

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

   5. associate-*l/ 87.6%

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

   6. *-un-lft-identity 87.6%

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

   7. frac-times 87.5%

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

3. Applied egg-rr 87.5%

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

5. Simplified 87.7%

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

6. Taylor expanded in k around 0 42.3%

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

   1. associate-*r/ 42.3%

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

   2. *-commutative 42.3%

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

   3. associate-/l* 42.3%

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

8. Simplified 42.3%

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

   1. div-inv 42.3%

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

   2. clear-num 42.3%

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

   3. associate-*l* 42.3%

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

10. Applied egg-rr 42.3%

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

11. Final simplification 42.3%

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

Alternative 8: 38.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 98.6%

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

   2. sqrt-unprod 87.5%

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

   3. associate-*l/ 87.5%

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

   4. *-un-lft-identity 87.5%

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

   5. associate-*l/ 87.6%

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

   6. *-un-lft-identity 87.6%

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

   7. frac-times 87.5%

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

3. Applied egg-rr 87.5%

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

5. Simplified 87.7%

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

6. Taylor expanded in k around 0 42.3%

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

   1. associate-*r/ 42.3%

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

   2. *-commutative 42.3%

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

   3. associate-/l* 42.3%

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

8. Simplified 42.3%

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

   1. div-inv 42.3%

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

   2. *-commutative 42.3%

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

   3. clear-num 42.3%

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

   4. associate-*l* 42.3%

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

10. Applied egg-rr 42.3%

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

11. Final simplification 42.3%

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

Alternative 9: 38.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{n}{k} \cdot \frac{\pi}{0.5}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 98.6%

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

   2. sqrt-unprod 87.5%

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

   3. associate-*l/ 87.5%

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

   4. *-un-lft-identity 87.5%

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

   5. associate-*l/ 87.6%

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

   6. *-un-lft-identity 87.6%

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

   7. frac-times 87.5%

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

3. Applied egg-rr 87.5%

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

5. Simplified 87.7%

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

6. Taylor expanded in k around 0 42.3%

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

   1. associate-*r/ 42.3%

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

   2. *-commutative 42.3%

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

   3. associate-/l* 42.3%

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

8. Simplified 42.3%

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

   1. div-inv 42.3%

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

   2. metadata-eval 42.3%

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

   3. times-frac 42.3%

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

10. Applied egg-rr 42.3%

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

11. Final simplification 42.3%

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

Alternative 10: 38.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{\pi}{k} \cdot \frac{n}{0.5}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 98.6%

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

   2. sqrt-unprod 87.5%

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

   3. associate-*l/ 87.5%

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

   4. *-un-lft-identity 87.5%

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

   5. associate-*l/ 87.6%

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

   6. *-un-lft-identity 87.6%

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

   7. frac-times 87.5%

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

3. Applied egg-rr 87.5%

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

5. Simplified 87.7%

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

6. Taylor expanded in k around 0 42.3%

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

   1. associate-*r/ 42.3%

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

   2. *-commutative 42.3%

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

   3. associate-/l* 42.3%

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

8. Simplified 42.3%

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

   1. *-commutative 42.3%

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

   2. div-inv 42.3%

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

   3. metadata-eval 42.3%

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

   4. times-frac 42.3%

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

10. Applied egg-rr 42.3%

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

11. Final simplification 42.3%

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

Alternative 11: 38.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 98.6%

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

   2. sqrt-unprod 87.5%

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

   3. associate-*l/ 87.5%

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

   4. *-un-lft-identity 87.5%

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

   5. associate-*l/ 87.6%

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

   6. *-un-lft-identity 87.6%

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

   7. frac-times 87.5%

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

3. Applied egg-rr 87.5%

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

5. Simplified 87.7%

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

6. Taylor expanded in k around 0 42.3%

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

   1. associate-*r/ 42.3%

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

   2. *-commutative 42.3%

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

   3. associate-/l* 42.3%

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

8. Simplified 42.3%

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

9. Final simplification 42.3%

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

Reproduce

herbie shell --seed 2023322 
(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))))