Migdal et al, Equation (51)

Percentage Accurate: 99.6% → 99.5%

Time: 12.7s

Alternatives: 14

Speedup: 1.0×

• could not determine a ground truth

  1. k = 1.7361819508803435e+232
  2. n = -3.0721595746876004e-5

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: 99.6% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-22}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.35e-22)
   (/ (sqrt (* PI (* 2.0 n))) (sqrt k))
   (sqrt (/ (pow (* 2.0 (* PI n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.35e-22) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) / sqrt(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 <= 1.35e-22) {
		tmp = Math.sqrt((Math.PI * (2.0 * n))) / Math.sqrt(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 <= 1.35e-22:
		tmp = math.sqrt((math.pi * (2.0 * n))) / math.sqrt(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 <= 1.35e-22)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) / sqrt(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 <= 1.35e-22)
		tmp = sqrt((pi * (2.0 * n))) / sqrt(k);
	else
		tmp = sqrt((((2.0 * (pi * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.35e-22], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $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 < 1.3500000000000001e-22

   1. Initial program 99.5%

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

   4. Applied egg-rr 82.9%

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

      1. distribute-lft-in 82.9%

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

      2. metadata-eval 82.9%

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

      3. *-commutative 82.9%

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

      4. associate-*r* 82.9%

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

      5. metadata-eval 82.9%

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

      6. neg-mul-1 82.9%

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

      7. sub-neg 82.9%

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

      8. *-commutative 82.9%

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

   6. Simplified 82.9%

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

   7. Taylor expanded in k around 0 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-/l* 82.9%

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

   9. Simplified 82.9%

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

       1. *-commutative 82.9%

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

       2. *-commutative 82.9%

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

       3. associate-*r/ 82.9%

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

       4. *-commutative 82.9%

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

       5. associate-*r/ 82.9%

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

       6. sqrt-div 99.6%

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

       7. *-commutative 99.6%

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

   11. Applied egg-rr 99.6%

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

       1. *-commutative 99.6%

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

       2. associate-*l* 99.6%

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

   13. Simplified 99.6%

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

   if 1.3500000000000001e-22 < k

   1. Initial program 99.9%

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

   4. Applied egg-rr 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. *-commutative 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-22}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(-1 + \mathsf{fma}\left(\pi, \frac{2}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.4e-56)
   (/ (sqrt (* PI (* 2.0 n))) (sqrt k))
   (sqrt (* n (+ -1.0 (fma PI (/ 2.0 k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.4e-56) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) / sqrt(k);
	} else {
		tmp = sqrt((n * (-1.0 + fma(((double) M_PI), (2.0 / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.4e-56)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) / sqrt(k));
	else
		tmp = sqrt(Float64(n * Float64(-1.0 + fma(pi, Float64(2.0 / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.4e-56], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(n * N[(-1.0 + N[(Pi * N[(2.0 / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.40000000000000001e-56

   1. Initial program 99.5%

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

   4. Applied egg-rr 81.4%

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

      1. distribute-lft-in 81.4%

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

      2. metadata-eval 81.4%

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

      3. *-commutative 81.4%

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

      4. associate-*r* 81.4%

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

      5. metadata-eval 81.4%

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

      6. neg-mul-1 81.4%

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

      7. sub-neg 81.4%

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

      8. *-commutative 81.4%

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

   6. Simplified 81.4%

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

   7. Taylor expanded in k around 0 81.4%

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

      1. *-commutative 81.4%

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

      2. associate-/l* 81.4%

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

   9. Simplified 81.4%

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

       1. *-commutative 81.4%

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

       2. *-commutative 81.4%

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

       3. associate-*r/ 81.4%

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

       4. *-commutative 81.4%

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

       5. associate-*r/ 81.4%

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

       6. sqrt-div 99.6%

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

       7. *-commutative 99.6%

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

   11. Applied egg-rr 99.6%

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

       1. *-commutative 99.6%

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

       2. associate-*l* 99.6%

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

   13. Simplified 99.6%

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

   if 2.40000000000000001e-56 < k

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. distribute-lft-in 99.9%

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

      2. metadata-eval 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r* 99.9%

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

      5. metadata-eval 99.9%

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

      6. neg-mul-1 99.9%

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

      7. sub-neg 99.9%

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

      8. *-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. pow-sub 99.4%

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

      2. pow1 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-*r* 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.4%

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

   8. Applied egg-rr 99.4%

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

      1. associate-/l* 99.4%

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

      2. associate-*l* 99.4%

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

      3. *-commutative 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in k around 0 9.2%

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

       1. associate-*l/ 9.2%

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

       2. associate-/r/ 9.2%

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

       3. associate-*r/ 9.2%

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

       4. associate-*l/ 9.2%

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

       5. *-commutative 9.2%

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

   13. Simplified 9.2%

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

       1. div-inv 9.2%

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

       2. clear-num 9.2%

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

       3. expm1-log1p-u 8.8%

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

       4. expm1-undefine 67.5%

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

       5. *-commutative 67.5%

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

       6. associate-*l/ 67.5%

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

   15. Applied egg-rr 67.5%

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

       1. sub-neg 67.5%

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

       2. metadata-eval 67.5%

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

       3. +-commutative 67.5%

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

       4. log1p-undefine 67.5%

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

       5. rem-exp-log 67.9%

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

       6. +-commutative 67.9%

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

       7. associate-*r/ 67.9%

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

       8. fma-define 67.9%

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

   17. Simplified 67.9%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(-1 + \mathsf{fma}\left(\pi, \frac{2}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 41.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{+226}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.45e+226)
   (/ (sqrt (* PI (* 2.0 n))) (sqrt k))
   (cbrt (pow (* 2.0 (* n (/ PI k))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.45e+226) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) / sqrt(k);
	} else {
		tmp = cbrt(pow((2.0 * (n * (((double) M_PI) / k))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.45e+226) {
		tmp = Math.sqrt((Math.PI * (2.0 * n))) / Math.sqrt(k);
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * (n * (Math.PI / k))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.45e+226)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) / sqrt(k));
	else
		tmp = cbrt((Float64(2.0 * Float64(n * Float64(pi / k))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.45e+226], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.44999999999999987e226

   1. Initial program 99.7%

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

   4. Applied egg-rr 92.0%

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

      1. distribute-lft-in 92.0%

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

      2. metadata-eval 92.0%

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

      3. *-commutative 92.0%

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

      4. associate-*r* 92.0%

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

      5. metadata-eval 92.0%

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

      6. neg-mul-1 92.0%

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

      7. sub-neg 92.0%

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

      8. *-commutative 92.0%

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

   6. Simplified 92.0%

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

   7. Taylor expanded in k around 0 41.8%

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

      1. *-commutative 41.8%

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

      2. associate-/l* 41.8%

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

   9. Simplified 41.8%

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

       1. *-commutative 41.8%

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

       2. *-commutative 41.8%

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

       3. associate-*r/ 41.8%

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

       4. *-commutative 41.8%

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

       5. associate-*r/ 41.8%

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

       6. sqrt-div 49.6%

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

       7. *-commutative 49.6%

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

   11. Applied egg-rr 49.6%

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

       1. *-commutative 49.6%

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

       2. associate-*l* 49.6%

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

   13. Simplified 49.6%

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

   if 1.44999999999999987e226 < k

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in k around 0 2.0%

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

      1. *-commutative 2.0%

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

      2. associate-/l* 2.0%

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

   9. Simplified 2.0%

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

       1. *-commutative 2.0%

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

       2. associate-*r/ 2.0%

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

       3. *-commutative 2.0%

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

       4. associate-*r/ 2.0%

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

       5. add-cbrt-cube 11.1%

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

       6. pow1/3 11.1%

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

   11. Applied egg-rr 11.1%

       \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   12. Step-by-step derivation

       1. unpow1/3 11.1%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}}} \]

       2. associate-*r/ 11.1%

          \[\leadsto \sqrt[3]{{\left(2 \cdot \color{blue}{\frac{\pi \cdot n}{k}}\right)}^{1.5}} \]

       3. associate-*l/ 11.1%

          \[\leadsto \sqrt[3]{{\left(2 \cdot \color{blue}{\left(\frac{\pi}{k} \cdot n\right)}\right)}^{1.5}} \]

       4. *-commutative 11.1%

          \[\leadsto \sqrt[3]{{\left(2 \cdot \color{blue}{\left(n \cdot \frac{\pi}{k}\right)}\right)}^{1.5}} \]

   13. Simplified 11.1%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{+226}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.8%

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

   2. *-lft-identity 99.8%

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

   3. associate-*l* 99.8%

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

   4. div-sub 99.8%

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

   5. metadata-eval 99.8%

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

3. Simplified 99.8%

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

Alternative 5: 40.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 93.6%

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

   1. distribute-lft-in 93.6%

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

   2. metadata-eval 93.6%

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

   3. *-commutative 93.6%

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

   4. associate-*r* 93.6%

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

   5. metadata-eval 93.6%

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

   6. neg-mul-1 93.6%

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

   7. sub-neg 93.6%

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

   8. *-commutative 93.6%

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

6. Simplified 93.6%

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

7. Taylor expanded in k around 0 33.7%

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

   1. *-commutative 33.7%

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

   2. associate-/l* 33.7%

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

9. Simplified 33.7%

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

    1. *-commutative 33.7%

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

    2. *-commutative 33.7%

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

    3. associate-*r/ 33.7%

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

    4. *-commutative 33.7%

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

    5. associate-*r/ 33.7%

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

    6. sqrt-div 40.0%

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

    7. *-commutative 40.0%

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

11. Applied egg-rr 40.0%

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

    1. *-commutative 40.0%

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

    2. associate-*l* 40.0%

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

13. Simplified 40.0%

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

14. Final simplification 40.0%

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

Alternative 6: 40.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 93.6%

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

   1. distribute-lft-in 93.6%

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

   2. metadata-eval 93.6%

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

   3. *-commutative 93.6%

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

   4. associate-*r* 93.6%

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

   5. metadata-eval 93.6%

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

   6. neg-mul-1 93.6%

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

   7. sub-neg 93.6%

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

   8. *-commutative 93.6%

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

6. Simplified 93.6%

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

7. Taylor expanded in k around 0 33.7%

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

   1. *-commutative 33.7%

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

   2. associate-/l* 33.7%

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

9. Simplified 33.7%

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

    1. pow1/2 33.7%

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

    2. *-commutative 33.7%

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

    3. associate-*r/ 33.7%

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

    4. *-commutative 33.7%

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

    5. associate-*r/ 33.7%

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

    6. associate-*l* 33.7%

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

    7. unpow-prod-down 40.0%

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

    8. pow1/2 40.0%

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

    9. *-commutative 40.0%

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

11. Applied egg-rr 40.0%

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

    1. unpow1/2 40.0%

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

13. Simplified 40.0%

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

Alternative 7: 31.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in k around 0 33.6%

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

   1. associate-/l* 33.6%

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

5. Simplified 33.6%

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

   1. sqrt-unprod 33.7%

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

7. Applied egg-rr 33.7%

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

   1. add-sqr-sqrt 33.6%

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

   2. pow1/2 33.6%

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

   3. pow1/2 33.6%

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

   4. pow-prod-down 30.0%

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

   5. pow2 30.0%

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

9. Applied egg-rr 30.0%

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

    1. unpow1/2 30.0%

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

    2. unpow2 30.0%

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

    3. rem-sqrt-square 34.4%

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

    4. associate-*r/ 34.4%

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

    5. *-commutative 34.4%

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

    6. associate-/l* 34.4%

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

11. Simplified 34.4%

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

12. Final simplification 34.4%

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

Alternative 8: 31.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 93.6%

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

   1. distribute-lft-in 93.6%

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

   2. metadata-eval 93.6%

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

   3. *-commutative 93.6%

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

   4. associate-*r* 93.6%

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

   5. metadata-eval 93.6%

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

   6. neg-mul-1 93.6%

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

   7. sub-neg 93.6%

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

   8. *-commutative 93.6%

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

6. Simplified 93.6%

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

7. Taylor expanded in k around 0 33.7%

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

   1. *-commutative 33.7%

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

   2. add-sqr-sqrt 33.6%

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

   3. sqrt-unprod 21.1%

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

   4. pow2 21.1%

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

9. Applied egg-rr 21.1%

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

    1. unpow2 21.1%

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

    2. rem-sqrt-square 34.4%

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

    3. *-commutative 34.4%

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

11. Simplified 34.4%

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

12. Final simplification 34.4%

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

Alternative 9: 31.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 93.6%

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

   1. distribute-lft-in 93.6%

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

   2. metadata-eval 93.6%

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

   3. *-commutative 93.6%

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

   4. associate-*r* 93.6%

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

   5. metadata-eval 93.6%

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

   6. neg-mul-1 93.6%

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

   7. sub-neg 93.6%

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

   8. *-commutative 93.6%

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

6. Simplified 93.6%

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

7. Taylor expanded in k around 0 33.7%

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

   1. *-commutative 33.7%

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

   2. associate-/l* 33.7%

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

9. Simplified 33.7%

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

    1. pow1 33.7%

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

    2. metadata-eval 33.7%

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

    3. pow-prod-up 33.7%

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

    4. pow-prod-down 29.9%

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

    5. pow2 29.9%

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

11. Applied egg-rr 29.9%

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

    1. unpow1/2 29.9%

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

    2. unpow2 29.9%

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

    3. rem-sqrt-square 34.4%

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

    4. associate-*r* 34.4%

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

    5. *-commutative 34.4%

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

    6. *-commutative 34.4%

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

    7. associate-*l* 34.4%

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

    8. associate-/r/ 34.4%

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

    9. associate-*l/ 34.4%

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

    10. *-commutative 34.4%

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

    11. associate-*l/ 34.4%

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

    12. *-commutative 34.4%

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

13. Simplified 34.4%

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

    1. associate-*r/ 34.4%

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

15. Applied egg-rr 34.4%

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

16. Final simplification 34.4%

    \[\leadsto \sqrt{\left|\frac{2 \cdot n}{\frac{k}{\pi}}\right|} \]
17. Add Preprocessing

Alternative 10: 31.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 93.6%

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

   1. distribute-lft-in 93.6%

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

   2. metadata-eval 93.6%

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

   3. *-commutative 93.6%

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

   4. associate-*r* 93.6%

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

   5. metadata-eval 93.6%

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

   6. neg-mul-1 93.6%

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

   7. sub-neg 93.6%

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

   8. *-commutative 93.6%

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

6. Simplified 93.6%

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

7. Taylor expanded in k around 0 33.7%

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

   1. *-commutative 33.7%

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

   2. associate-/l* 33.7%

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

9. Simplified 33.7%

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

    1. pow1 33.7%

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

    2. metadata-eval 33.7%

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

    3. pow-prod-up 33.7%

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

    4. pow-prod-down 29.9%

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

    5. pow2 29.9%

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

11. Applied egg-rr 29.9%

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

    1. unpow1/2 29.9%

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

    2. unpow2 29.9%

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

    3. rem-sqrt-square 34.4%

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

    4. associate-*r* 34.4%

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

    5. *-commutative 34.4%

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

    6. *-commutative 34.4%

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

    7. associate-*l* 34.4%

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

    8. associate-/r/ 34.4%

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

    9. associate-*l/ 34.4%

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

    10. *-commutative 34.4%

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

    11. associate-*l/ 34.4%

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

    12. *-commutative 34.4%

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

13. Simplified 34.4%

    \[\leadsto \sqrt{\color{blue}{\left|n \cdot \frac{2}{\frac{k}{\pi}}\right|}} \]
14. Add Preprocessing

Alternative 11: 31.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 93.6%

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

   1. distribute-lft-in 93.6%

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

   2. metadata-eval 93.6%

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

   3. *-commutative 93.6%

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

   4. associate-*r* 93.6%

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

   5. metadata-eval 93.6%

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

   6. neg-mul-1 93.6%

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

   7. sub-neg 93.6%

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

   8. *-commutative 93.6%

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

6. Simplified 93.6%

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

7. Taylor expanded in k around 0 33.7%

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

   1. *-commutative 33.7%

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

   2. associate-/l* 33.7%

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

9. Simplified 33.7%

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

    1. *-commutative 33.7%

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

    2. *-commutative 33.7%

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

    3. associate-*r/ 33.7%

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

    4. *-commutative 33.7%

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

    5. associate-*r/ 33.7%

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

    6. sqrt-div 40.0%

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

    7. *-commutative 40.0%

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

11. Applied egg-rr 40.0%

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

    1. *-commutative 40.0%

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

    2. associate-*l* 40.0%

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

13. Simplified 40.0%

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

    1. clear-num 39.9%

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

    2. inv-pow 39.9%

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

    3. sqrt-undiv 34.2%

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

15. Applied egg-rr 34.2%

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

    1. unpow-1 34.2%

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

17. Simplified 34.2%

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

18. Final simplification 34.2%

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

Alternative 12: 31.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{0.5 \cdot \frac{\frac{k}{n}}{\pi}}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (/ 1.0 (sqrt (* 0.5 (/ (/ k n) PI)))))

C

double code(double k, double n) {
	return 1.0 / sqrt((0.5 * ((k / n) / ((double) M_PI))));
}

Java

public static double code(double k, double n) {
	return 1.0 / Math.sqrt((0.5 * ((k / n) / Math.PI)));
}

Python

def code(k, n):
	return 1.0 / math.sqrt((0.5 * ((k / n) / math.pi)))

Julia

function code(k, n)
	return Float64(1.0 / sqrt(Float64(0.5 * Float64(Float64(k / n) / pi))))
end

MATLAB

function tmp = code(k, n)
	tmp = 1.0 / sqrt((0.5 * ((k / n) / pi)));
end

Wolfram

code[k_, n_] := N[(1.0 / N[Sqrt[N[(0.5 * N[(N[(k / n), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 93.6%

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

   1. distribute-lft-in 93.6%

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

   2. metadata-eval 93.6%

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

   3. *-commutative 93.6%

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

   4. associate-*r* 93.6%

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

   5. metadata-eval 93.6%

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

   6. neg-mul-1 93.6%

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

   7. sub-neg 93.6%

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

   8. *-commutative 93.6%

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

6. Simplified 93.6%

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

7. Taylor expanded in k around 0 33.7%

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

   1. *-commutative 33.7%

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

   2. associate-/l* 33.7%

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

9. Simplified 33.7%

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

    1. *-commutative 33.7%

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

    2. *-commutative 33.7%

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

    3. associate-*r/ 33.7%

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

    4. *-commutative 33.7%

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

    5. associate-*r/ 33.7%

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

    6. sqrt-div 40.0%

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

    7. *-commutative 40.0%

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

11. Applied egg-rr 40.0%

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

    1. *-commutative 40.0%

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

    2. associate-*l* 40.0%

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

13. Simplified 40.0%

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

    1. clear-num 39.9%

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

    2. inv-pow 39.9%

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

    3. sqrt-undiv 34.2%

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

15. Applied egg-rr 34.2%

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

    1. unpow-1 34.2%

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

    2. associate-/r* 34.2%

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

    3. *-lft-identity 34.2%

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

    4. *-commutative 34.2%

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

    5. times-frac 34.2%

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

    6. metadata-eval 34.2%

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

    7. associate-/l/ 34.2%

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

    8. associate-/r* 34.2%

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

17. Simplified 34.2%

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

Alternative 13: 30.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in k around 0 33.6%

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

   1. associate-/l* 33.6%

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

5. Simplified 33.6%

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

   1. sqrt-unprod 33.7%

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

7. Applied egg-rr 33.7%

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

8. Final simplification 33.7%

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

Alternative 14: 30.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 93.6%

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

   1. distribute-lft-in 93.6%

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

   2. metadata-eval 93.6%

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

   3. *-commutative 93.6%

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

   4. associate-*r* 93.6%

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

   5. metadata-eval 93.6%

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

   6. neg-mul-1 93.6%

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

   7. sub-neg 93.6%

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

   8. *-commutative 93.6%

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

6. Simplified 93.6%

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

7. Taylor expanded in k around 0 33.7%

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

   1. *-commutative 33.7%

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

   2. associate-/l* 33.7%

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

9. Simplified 33.7%

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

Reproduce

herbie shell --seed 2024177 
(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))))