Migdal et al, Equation (51)

Percentage Accurate: 98.4% → 99.6%

Time: 14.7s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. k = 4.94066e-324
  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: 98.4% 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: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \cdot \left(\pi \cdot 2\right) \leq 10^{-278}:\\ \;\;\;\;\sqrt{\frac{{\left(\left|\left(\pi \cdot n\right) \cdot -2\right|\right)}^{\left(1 - k\right)}}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* n (* PI 2.0)) 1e-278)
   (sqrt (/ (pow (fabs (* (* PI n) -2.0)) (- 1.0 k)) k))
   (/ (pow (* 2.0 (* PI n)) (- 0.5 (/ k 2.0))) (sqrt k))))

C

double code(double k, double n) {
	double tmp;
	if ((n * (((double) M_PI) * 2.0)) <= 1e-278) {
		tmp = sqrt((pow(fabs(((((double) M_PI) * n) * -2.0)), (1.0 - k)) / k));
	} else {
		tmp = pow((2.0 * (((double) M_PI) * n)), (0.5 - (k / 2.0))) / sqrt(k);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if ((n * (Math.PI * 2.0)) <= 1e-278) {
		tmp = Math.sqrt((Math.pow(Math.abs(((Math.PI * n) * -2.0)), (1.0 - k)) / k));
	} else {
		tmp = Math.pow((2.0 * (Math.PI * n)), (0.5 - (k / 2.0))) / Math.sqrt(k);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if (n * (math.pi * 2.0)) <= 1e-278:
		tmp = math.sqrt((math.pow(math.fabs(((math.pi * n) * -2.0)), (1.0 - k)) / k))
	else:
		tmp = math.pow((2.0 * (math.pi * n)), (0.5 - (k / 2.0))) / math.sqrt(k)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (Float64(n * Float64(pi * 2.0)) <= 1e-278)
		tmp = sqrt(Float64((abs(Float64(Float64(pi * n) * -2.0)) ^ Float64(1.0 - k)) / k));
	else
		tmp = Float64((Float64(2.0 * Float64(pi * n)) ^ Float64(0.5 - Float64(k / 2.0))) / sqrt(k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if ((n * (pi * 2.0)) <= 1e-278)
		tmp = sqrt(((abs(((pi * n) * -2.0)) ^ (1.0 - k)) / k));
	else
		tmp = ((2.0 * (pi * n)) ^ (0.5 - (k / 2.0))) / sqrt(k);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision], 1e-278], N[Sqrt[N[(N[Power[N[Abs[N[(N[(Pi * n), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision], 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. Split input into 2 regimes

2. if (*.f64 (*.f64 2 (PI.f64)) n) < 9.99999999999999938e-279

   1. Initial program 94.0%

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

      1. add-sqr-sqrt 93.9%

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

      2. sqrt-unprod 94.0%

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

      3. *-commutative 94.0%

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

      4. associate-*r* 94.0%

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

      5. div-sub 94.0%

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

      6. metadata-eval 94.0%

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

      7. div-inv 94.0%

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

      8. *-commutative 94.0%

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

   4. Applied egg-rr 94.0%

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

   6. Simplified 94.0%

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

      1. add-sqr-sqrt 11.9%

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

      2. fabs-sqr 11.9%

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

      3. add-sqr-sqrt 100.0%

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

      4. neg-fabs 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. *-commutative 100.0%

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

      9. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

   if 9.99999999999999938e-279 < (*.f64 (*.f64 2 (PI.f64)) n)

   1. Initial program 99.3%

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

      1. 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. associate-*l* 99.4%

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

      4. div-sub 99.4%

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

      5. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \cdot \left(\pi \cdot 2\right) \leq 10^{-278}:\\ \;\;\;\;\sqrt{\frac{{\left(\left|\left(\pi \cdot n\right) \cdot -2\right|\right)}^{\left(1 - k\right)}}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

   1. associate-*r* 97.9%

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

   2. add-sqr-sqrt 76.4%

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

   3. sqrt-unprod 82.9%

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

   4. *-commutative 82.9%

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

   5. *-commutative 82.9%

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

   6. swap-sqr 82.9%

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

   7. pow2 82.9%

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

   8. metadata-eval 82.9%

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

4. Applied egg-rr 82.9%

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

   1. *-commutative 82.9%

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

   2. metadata-eval 82.9%

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

   3. unpow2 82.9%

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

   4. swap-sqr 82.9%

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

   5. rem-sqrt-square 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{n \cdot 2}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.35e-62)
   (/ (sqrt (* n 2.0)) (sqrt (/ k PI)))
   (/ 1.0 (sqrt (/ k (pow (* PI (* n 2.0)) (- 1.0 k)))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.35e-62) {
		tmp = sqrt((n * 2.0)) / sqrt((k / ((double) M_PI)));
	} else {
		tmp = 1.0 / sqrt((k / pow((((double) M_PI) * (n * 2.0)), (1.0 - k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.35e-62) {
		tmp = Math.sqrt((n * 2.0)) / Math.sqrt((k / Math.PI));
	} else {
		tmp = 1.0 / Math.sqrt((k / Math.pow((Math.PI * (n * 2.0)), (1.0 - k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.35e-62:
		tmp = math.sqrt((n * 2.0)) / math.sqrt((k / math.pi))
	else:
		tmp = 1.0 / math.sqrt((k / math.pow((math.pi * (n * 2.0)), (1.0 - k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.35e-62)
		tmp = Float64(sqrt(Float64(n * 2.0)) / sqrt(Float64(k / pi)));
	else
		tmp = Float64(1.0 / sqrt(Float64(k / (Float64(pi * Float64(n * 2.0)) ^ Float64(1.0 - k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.35e-62)
		tmp = sqrt((n * 2.0)) / sqrt((k / pi));
	else
		tmp = 1.0 / sqrt((k / ((pi * (n * 2.0)) ^ (1.0 - k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.35e-62], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(k / N[Power[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3500000000000001e-62

   1. Initial program 99.2%

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

   3. Taylor expanded in k around 0 71.8%

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

      1. *-commutative 71.8%

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

      2. associate-/l* 71.8%

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

   5. Simplified 71.8%

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

      1. pow1 71.8%

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

      2. sqrt-unprod 72.2%

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

      3. associate-*r/ 72.1%

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

      4. *-commutative 72.1%

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

   7. Applied egg-rr 72.1%

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

      1. unpow1 72.1%

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

      2. *-commutative 72.1%

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

      3. associate-/l* 72.2%

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

   9. Simplified 72.2%

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

       1. clear-num 72.1%

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

       2. un-div-inv 72.1%

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

   11. Applied egg-rr 72.1%

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

       1. associate-*r/ 72.1%

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

       2. *-commutative 72.1%

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

       3. sqrt-div 99.4%

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

   13. Applied egg-rr 99.4%

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

   if 1.3500000000000001e-62 < k

   1. Initial program 97.4%

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

      1. associate-*r* 97.4%

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

      2. add-sqr-sqrt 66.4%

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

      3. sqrt-unprod 93.9%

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

      4. *-commutative 93.9%

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

      5. *-commutative 93.9%

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

      6. swap-sqr 93.9%

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

      7. pow2 93.9%

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

      8. metadata-eval 93.9%

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

   4. Applied egg-rr 93.9%

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

      1. *-commutative 93.9%

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

      2. metadata-eval 93.9%

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

      3. unpow2 93.9%

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

      4. swap-sqr 93.9%

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

      5. rem-sqrt-square 99.6%

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

      6. *-commutative 99.6%

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

      7. associate-*l* 99.6%

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

   6. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. add-sqr-sqrt 66.4%

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

      3. fabs-sqr 66.4%

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

      4. add-sqr-sqrt 97.4%

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

      5. div-inv 97.4%

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

      6. sqrt-pow1 96.9%

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

      7. sqrt-div 96.4%

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

      8. clear-num 96.4%

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

      9. sqrt-div 96.9%

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

      10. metadata-eval 96.9%

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

   8. Applied egg-rr 96.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{n \cdot 2}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{n \cdot 2}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.35e-36)
   (/ (sqrt (* n 2.0)) (sqrt (/ k PI)))
   (sqrt (/ (pow (* PI (* n 2.0)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.35e-36) {
		tmp = sqrt((n * 2.0)) / sqrt((k / ((double) M_PI)));
	} else {
		tmp = sqrt((pow((((double) M_PI) * (n * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.35e-36) {
		tmp = Math.sqrt((n * 2.0)) / Math.sqrt((k / Math.PI));
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (n * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.35e-36:
		tmp = math.sqrt((n * 2.0)) / math.sqrt((k / math.pi))
	else:
		tmp = math.sqrt((math.pow((math.pi * (n * 2.0)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.35e-36)
		tmp = Float64(sqrt(Float64(n * 2.0)) / sqrt(Float64(k / pi)));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(n * 2.0)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.35e-36)
		tmp = sqrt((n * 2.0)) / sqrt((k / pi));
	else
		tmp = sqrt((((pi * (n * 2.0)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.35e-36], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.35000000000000004e-36

   1. Initial program 99.2%

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

   3. Taylor expanded in k around 0 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-/l* 74.2%

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

   5. Simplified 74.2%

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

      1. pow1 74.2%

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

      2. sqrt-unprod 74.5%

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

      3. associate-*r/ 74.5%

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

      4. *-commutative 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. unpow1 74.5%

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

      2. *-commutative 74.5%

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

      3. associate-/l* 74.5%

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

   9. Simplified 74.5%

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

       1. clear-num 74.5%

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

       2. un-div-inv 74.5%

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

   11. Applied egg-rr 74.5%

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

       1. associate-*r/ 74.5%

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

       2. *-commutative 74.5%

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

       3. sqrt-div 99.4%

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

   13. Applied egg-rr 99.4%

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

   if 1.35000000000000004e-36 < k

   1. Initial program 97.2%

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

      1. add-sqr-sqrt 97.2%

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

      2. sqrt-unprod 96.7%

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

      3. *-commutative 96.7%

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

      4. associate-*r* 96.7%

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

      5. div-sub 96.7%

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

      6. metadata-eval 96.7%

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

      7. div-inv 96.7%

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

      8. *-commutative 96.7%

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

   4. Applied egg-rr 96.7%

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

   6. Simplified 96.7%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{n \cdot 2}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.8e+29)
   (* (sqrt (* n 2.0)) (sqrt (/ PI k)))
   (sqrt (* 2.0 (+ -1.0 (fma n (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.8e+29) {
		tmp = sqrt((n * 2.0)) * sqrt((((double) M_PI) / k));
	} else {
		tmp = sqrt((2.0 * (-1.0 + fma(n, (((double) M_PI) / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.8e+29)
		tmp = Float64(sqrt(Float64(n * 2.0)) * sqrt(Float64(pi / k)));
	else
		tmp = sqrt(Float64(2.0 * Float64(-1.0 + fma(n, Float64(pi / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.8e+29], N[(N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(-1.0 + N[(n * N[(Pi / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.79999999999999971e29

   1. Initial program 95.5%

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

   3. Taylor expanded in k around 0 65.0%

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

      1. *-commutative 65.0%

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

      2. associate-/l* 65.0%

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

   5. Simplified 65.0%

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

      1. pow1 65.0%

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

      2. sqrt-unprod 65.3%

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

      3. associate-*r/ 65.3%

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

      4. *-commutative 65.3%

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

   7. Applied egg-rr 65.3%

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

      1. unpow1 65.3%

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

      2. *-commutative 65.3%

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

      3. associate-/l* 65.3%

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

   9. Simplified 65.3%

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

       1. associate-*r* 65.3%

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

       2. *-commutative 65.3%

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

       3. sqrt-prod 84.2%

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

       4. *-commutative 84.2%

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

   11. Applied egg-rr 84.2%

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

       1. *-commutative 84.2%

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

   13. Simplified 84.2%

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

   if 3.79999999999999971e29 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 1.6%

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

      1. *-commutative 1.6%

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

      2. associate-/l* 1.6%

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

   5. Simplified 1.6%

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

      1. pow1 1.6%

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

      2. sqrt-unprod 1.6%

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

      3. associate-*r/ 1.6%

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

      4. *-commutative 1.6%

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

   7. Applied egg-rr 1.6%

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

      1. unpow1 1.6%

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

      2. *-commutative 1.6%

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

      3. associate-/l* 1.6%

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

   9. Simplified 1.6%

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

       1. associate-*r/ 1.6%

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

       2. *-commutative 1.6%

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

       3. associate-*r/ 1.6%

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

       4. expm1-log1p-u 1.6%

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

       5. expm1-undefine 32.6%

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

   11. Applied egg-rr 32.6%

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

       1. sub-neg 32.6%

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

       2. metadata-eval 32.6%

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

       3. +-commutative 32.6%

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

       4. log1p-undefine 32.6%

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

       5. rem-exp-log 32.6%

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

       6. +-commutative 32.6%

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

       7. associate-*r/ 32.6%

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

       8. *-commutative 32.6%

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

       9. associate-/l* 32.6%

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

       10. fma-define 32.6%

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

   13. Simplified 32.6%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 98.0%

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

   2. *-lft-identity 98.0%

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

   3. associate-*l* 98.0%

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

   4. div-sub 98.0%

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

   5. metadata-eval 98.0%

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

3. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 7: 30.8% 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 97.9%

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

3. Taylor expanded in k around 0 30.5%

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

   1. *-commutative 30.5%

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

   2. associate-/l* 30.5%

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

5. Simplified 30.5%

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

   1. pow1 30.5%

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

   2. sqrt-unprod 30.7%

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

   3. associate-*r/ 30.7%

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

   4. *-commutative 30.7%

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

7. Applied egg-rr 30.7%

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

   1. unpow1 30.7%

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

   2. *-commutative 30.7%

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

   3. associate-/l* 30.7%

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

9. Simplified 30.7%

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

    1. clear-num 30.7%

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

    2. un-div-inv 30.7%

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

11. Applied egg-rr 30.7%

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

    1. pow1 30.7%

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

    2. associate-/r/ 30.7%

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

    3. *-commutative 30.7%

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

    4. metadata-eval 30.7%

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

    5. pow-prod-up 30.6%

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

    6. pow-prod-down 28.1%

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

    7. pow2 28.1%

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

13. Applied egg-rr 28.1%

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

    1. unpow1/2 28.1%

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

    2. unpow2 28.1%

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

    3. rem-sqrt-square 31.4%

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

15. Simplified 31.4%

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

16. Final simplification 31.4%

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

Alternative 8: 39.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in k around 0 30.5%

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

   1. *-commutative 30.5%

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

   2. associate-/l* 30.5%

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

5. Simplified 30.5%

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

   1. pow1 30.5%

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

   2. sqrt-unprod 30.7%

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

   3. associate-*r/ 30.7%

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

   4. *-commutative 30.7%

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

7. Applied egg-rr 30.7%

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

   1. unpow1 30.7%

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

   2. *-commutative 30.7%

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

   3. associate-/l* 30.7%

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

9. Simplified 30.7%

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

    1. associate-*r* 30.7%

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

    2. *-commutative 30.7%

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

    3. sqrt-prod 39.4%

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

    4. *-commutative 39.4%

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

11. Applied egg-rr 39.4%

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

    1. *-commutative 39.4%

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

13. Simplified 39.4%

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

14. Final simplification 39.4%

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

Alternative 9: 30.1% 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 97.9%

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

3. Taylor expanded in k around 0 30.5%

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

   1. *-commutative 30.5%

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

   2. associate-/l* 30.5%

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

5. Simplified 30.5%

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

   1. pow1 30.5%

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

   2. sqrt-unprod 30.7%

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

   3. associate-*r/ 30.7%

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

   4. *-commutative 30.7%

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

7. Applied egg-rr 30.7%

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

   1. unpow1 30.7%

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

   2. *-commutative 30.7%

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

   3. associate-/l* 30.7%

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

9. Simplified 30.7%

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

10. Final simplification 30.7%

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

Reproduce

herbie shell --seed 2024067 
(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))))