Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 24.6s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot n\right)\\ \frac{\sqrt{t\_0}}{\sqrt{k} \cdot {t\_0}^{\left(k \cdot 0.5\right)}} \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* 2.0 n))))
   (/ (sqrt t_0) (* (sqrt k) (pow t_0 (* k 0.5))))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (2.0 * n);
	return sqrt(t_0) / (sqrt(k) * pow(t_0, (k * 0.5)));
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (2.0 * n);
	return Math.sqrt(t_0) / (Math.sqrt(k) * Math.pow(t_0, (k * 0.5)));
}

Python

def code(k, n):
	t_0 = math.pi * (2.0 * n)
	return math.sqrt(t_0) / (math.sqrt(k) * math.pow(t_0, (k * 0.5)))

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(2.0 * n))
	return Float64(sqrt(t_0) / Float64(sqrt(k) * (t_0 ^ Float64(k * 0.5))))
end

MATLAB

function tmp = code(k, n)
	t_0 = pi * (2.0 * n);
	tmp = sqrt(t_0) / (sqrt(k) * (t_0 ^ (k * 0.5)));
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, N[(N[Sqrt[t$95$0], $MachinePrecision] / N[(N[Sqrt[k], $MachinePrecision] * N[Power[t$95$0, N[(k * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.6%

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

   2. *-un-lft-identity 99.6%

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

   3. unpow-prod-down 71.6%

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

   4. unpow-prod-down 99.6%

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

   5. div-sub 99.6%

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

   6. metadata-eval 99.6%

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

   7. pow-div 99.7%

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

   8. pow1/2 99.7%

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

   9. associate-/l/ 99.7%

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

   10. associate-*l* 99.7%

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

   11. associate-*l* 99.7%

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

   12. div-inv 99.7%

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

   13. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*r* 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-*l* 99.7%

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

   4. associate-*r* 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l* 99.7%

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

   7. *-commutative 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.6e-43)
   (/ (sqrt (* n (* PI 2.0))) (sqrt k))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.6e-43) {
		tmp = sqrt((n * (((double) M_PI) * 2.0))) / sqrt(k);
	} else {
		tmp = sqrt((pow((((double) M_PI) * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.6e-43) {
		tmp = Math.sqrt((n * (Math.PI * 2.0))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.6e-43:
		tmp = math.sqrt((n * (math.pi * 2.0))) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow((math.pi * (2.0 * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.6e-43)
		tmp = Float64(sqrt(Float64(n * Float64(pi * 2.0))) / sqrt(k));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(2.0 * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.6e-43)
		tmp = sqrt((n * (pi * 2.0))) / sqrt(k);
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.6e-43], N[(N[Sqrt[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.6e-43

   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. Add Preprocessing

   3. Taylor expanded in k around 0 99.3%

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

      1. associate-*r* 99.3%

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

      2. *-commutative 99.3%

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

   5. Simplified 99.3%

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

      1. associate-*l/ 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. associate-*r* 99.5%

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

   7. Applied egg-rr 99.5%

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

   if 2.6e-43 < k

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 99.6%

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

      2. sqrt-unprod 99.0%

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

      3. *-commutative 99.0%

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

      4. div-sub 99.0%

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

      5. metadata-eval 99.0%

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

      6. div-inv 99.0%

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

      7. *-commutative 99.0%

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

      8. div-sub 99.0%

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

      9. metadata-eval 99.0%

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

      10. div-inv 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. associate-*r* 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*l* 99.0%

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

      4. distribute-lft-in 99.0%

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

      5. metadata-eval 99.0%

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

      6. *-commutative 99.0%

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

      7. associate-*r* 99.0%

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

      8. metadata-eval 99.0%

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

      9. mul-1-neg 99.0%

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

      10. sub-neg 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 7.2e+76)
   (/ (sqrt (* n (* PI 2.0))) (sqrt k))
   (sqrt (* 2.0 (+ -1.0 (fma n (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 7.2e+76) {
		tmp = sqrt((n * (((double) M_PI) * 2.0))) / sqrt(k);
	} else {
		tmp = sqrt((2.0 * (-1.0 + fma(n, (((double) M_PI) / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 7.2e+76)
		tmp = Float64(sqrt(Float64(n * Float64(pi * 2.0))) / sqrt(k));
	else
		tmp = sqrt(Float64(2.0 * Float64(-1.0 + fma(n, Float64(pi / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 7.2e+76], N[(N[Sqrt[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(-1.0 + N[(n * N[(Pi / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.2000000000000006e76

   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 74.9%

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

      1. associate-*r* 74.9%

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

      2. *-commutative 74.9%

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

   5. Simplified 74.9%

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

      1. associate-*l/ 75.0%

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

      2. *-un-lft-identity 75.0%

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

      3. associate-*r* 75.0%

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

   7. Applied egg-rr 75.0%

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

   if 7.2000000000000006e76 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.7%

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

      1. associate-*r* 2.7%

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

      2. *-commutative 2.7%

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

   5. Simplified 2.7%

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

      1. pow1 2.7%

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

      2. associate-*l/ 2.7%

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

      3. *-un-lft-identity 2.7%

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

      4. associate-*r* 2.7%

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

      5. *-commutative 2.7%

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

      6. associate-*r* 2.7%

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

      7. sqrt-undiv 2.6%

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

      8. associate-*r* 2.6%

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

      9. *-commutative 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. unpow1 2.6%

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

      2. associate-/l* 2.6%

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

      3. *-commutative 2.6%

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

      4. associate-*l* 2.6%

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

   9. Simplified 2.6%

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

   10. Taylor expanded in n around 0 2.6%

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

       1. associate-*r/ 2.6%

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

   12. Simplified 2.6%

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

       1. expm1-log1p-u 2.6%

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

       2. expm1-undefine 29.6%

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

   14. Applied egg-rr 29.6%

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

       1. sub-neg 29.6%

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

       2. metadata-eval 29.6%

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

       3. +-commutative 29.6%

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

       4. log1p-undefine 29.6%

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

       5. rem-exp-log 29.6%

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

       6. +-commutative 29.6%

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

       7. fma-define 29.6%

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

   16. Simplified 29.6%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. div-sub 99.6%

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

   4. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 5: 48.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 47.0%

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

   1. associate-*r* 47.0%

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

   2. *-commutative 47.0%

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

5. Simplified 47.0%

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

   1. pow1 47.0%

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

   2. associate-*l/ 47.0%

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

   3. *-un-lft-identity 47.0%

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

   4. associate-*r* 47.0%

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

   5. *-commutative 47.0%

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

   6. associate-*r* 47.0%

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

   7. sqrt-undiv 37.5%

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

   8. associate-*r* 37.5%

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

   9. *-commutative 37.5%

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

7. Applied egg-rr 37.5%

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

   1. unpow1 37.5%

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

   2. associate-/l* 37.5%

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

   3. *-commutative 37.5%

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

   4. associate-*l* 37.5%

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

9. Simplified 37.5%

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

10. Taylor expanded in n around 0 37.5%

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

    1. associate-*r/ 37.5%

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

12. Simplified 37.5%

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

    1. pow1/2 37.5%

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

    2. associate-*r* 37.5%

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

    3. unpow-prod-down 47.0%

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

    4. *-commutative 47.0%

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

    5. pow1/2 47.0%

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

14. Applied egg-rr 47.0%

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

    1. unpow1/2 47.0%

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

16. Simplified 47.0%

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

17. Final simplification 47.0%

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

Alternative 6: 48.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 47.0%

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

   1. associate-*r* 47.0%

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

   2. *-commutative 47.0%

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

5. Simplified 47.0%

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

   1. pow1 47.0%

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

   2. associate-*l/ 47.0%

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

   3. *-un-lft-identity 47.0%

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

   4. associate-*r* 47.0%

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

   5. *-commutative 47.0%

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

   6. associate-*r* 47.0%

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

   7. sqrt-undiv 37.5%

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

   8. associate-*r* 37.5%

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

   9. *-commutative 37.5%

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

7. Applied egg-rr 37.5%

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

   1. unpow1 37.5%

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

   2. associate-/l* 37.5%

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

   3. *-commutative 37.5%

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

   4. associate-*l* 37.5%

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

9. Simplified 37.5%

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

10. Taylor expanded in n around 0 37.5%

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

    1. associate-*r/ 37.5%

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

12. Simplified 37.5%

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

    1. metadata-eval 37.5%

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

    2. associate-*r/ 37.5%

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

    3. times-frac 37.5%

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

    4. *-un-lft-identity 37.5%

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

    5. *-commutative 37.5%

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

    6. clear-num 37.5%

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

    7. metadata-eval 37.5%

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

    8. add-sqr-sqrt 37.4%

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

    9. frac-times 37.4%

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

    10. sqrt-unprod 38.0%

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

    11. add-sqr-sqrt 38.1%

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

    12. sqrt-div 46.9%

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

    13. associate-/r/ 46.9%

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

14. Applied egg-rr 46.9%

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

    1. associate-*l/ 47.0%

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

    2. *-lft-identity 47.0%

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

16. Simplified 47.0%

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

17. Final simplification 47.0%

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

Alternative 7: 48.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 47.0%

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

   1. associate-*r* 47.0%

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

   2. *-commutative 47.0%

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

5. Simplified 47.0%

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

   1. associate-*l/ 47.0%

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

   2. *-un-lft-identity 47.0%

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

   3. associate-*r* 47.0%

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

7. Applied egg-rr 47.0%

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

8. Final simplification 47.0%

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

Alternative 8: 38.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 47.0%

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

   1. associate-*r* 47.0%

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

   2. *-commutative 47.0%

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

5. Simplified 47.0%

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

   1. pow1 47.0%

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

   2. associate-*l/ 47.0%

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

   3. *-un-lft-identity 47.0%

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

   4. associate-*r* 47.0%

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

   5. *-commutative 47.0%

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

   6. associate-*r* 47.0%

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

   7. sqrt-undiv 37.5%

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

   8. associate-*r* 37.5%

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

   9. *-commutative 37.5%

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

7. Applied egg-rr 37.5%

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

   1. unpow1 37.5%

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

   2. associate-/l* 37.5%

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

   3. *-commutative 37.5%

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

   4. associate-*l* 37.5%

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

9. Simplified 37.5%

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

10. Taylor expanded in n around 0 37.5%

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

    1. associate-*r/ 37.5%

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

12. Simplified 37.5%

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

    1. metadata-eval 37.5%

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

    2. associate-*r/ 37.5%

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

    3. times-frac 37.5%

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

    4. *-un-lft-identity 37.5%

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

    5. *-commutative 37.5%

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

    6. clear-num 37.5%

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

    7. metadata-eval 37.5%

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

    8. add-sqr-sqrt 37.4%

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

    9. frac-times 37.4%

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

    10. sqrt-unprod 38.0%

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

    11. add-sqr-sqrt 38.1%

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

    12. pow1/2 38.1%

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

    13. pow-flip 38.2%

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

    14. associate-/l* 38.2%

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

    15. metadata-eval 38.2%

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

14. Applied egg-rr 38.2%

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

15. Final simplification 38.2%

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

Alternative 9: 37.5% 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 99.5%

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

3. Taylor expanded in k around 0 47.0%

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

   1. associate-*r* 47.0%

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

   2. *-commutative 47.0%

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

5. Simplified 47.0%

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

   1. pow1 47.0%

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

   2. associate-*l/ 47.0%

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

   3. *-un-lft-identity 47.0%

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

   4. associate-*r* 47.0%

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

   5. *-commutative 47.0%

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

   6. associate-*r* 47.0%

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

   7. sqrt-undiv 37.5%

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

   8. associate-*r* 37.5%

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

   9. *-commutative 37.5%

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

7. Applied egg-rr 37.5%

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

   1. unpow1 37.5%

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

   2. associate-/l* 37.5%

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

   3. *-commutative 37.5%

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

   4. associate-*l* 37.5%

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

9. Simplified 37.5%

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

10. Taylor expanded in n around 0 37.5%

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

    1. associate-*r/ 37.5%

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

12. Simplified 37.5%

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

13. Final simplification 37.5%

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

Alternative 10: 37.5% 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 99.5%

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

3. Taylor expanded in k around 0 47.0%

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

   1. associate-*r* 47.0%

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

   2. *-commutative 47.0%

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

5. Simplified 47.0%

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

   1. pow1 47.0%

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

   2. associate-*l/ 47.0%

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

   3. *-un-lft-identity 47.0%

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

   4. associate-*r* 47.0%

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

   5. *-commutative 47.0%

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

   6. associate-*r* 47.0%

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

   7. sqrt-undiv 37.5%

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

   8. associate-*r* 37.5%

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

   9. *-commutative 37.5%

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

7. Applied egg-rr 37.5%

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

   1. unpow1 37.5%

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

   2. associate-/l* 37.5%

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

   3. *-commutative 37.5%

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

   4. associate-*l* 37.5%

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

9. Simplified 37.5%

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

10. Final simplification 37.5%

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

Reproduce

herbie shell --seed 2024044 
(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))))