Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.6%

Time: 12.3s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. k = 1.62701749440182e+222
  2. n = -7.00941495342652e-39

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. associate-*r* 99.6%

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

   2. div-sub 99.6%

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

   3. metadata-eval 99.6%

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

   4. pow-div 81.7%

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

   5. pow1/2 81.7%

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

   6. associate-*r/ 81.7%

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

   7. pow1/2 81.7%

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

   8. pow-flip 81.8%

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

   9. metadata-eval 81.8%

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

   10. div-inv 81.8%

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

   11. metadata-eval 81.8%

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

4. Applied egg-rr 81.8%

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

   1. associate-/l* 81.8%

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

   2. *-rgt-identity 81.8%

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

   3. associate-*l* 81.8%

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

   4. *-lft-identity 81.8%

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

   5. remove-double-neg 81.8%

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

   6. distribute-neg-frac 81.8%

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

   7. distribute-neg-frac 81.8%

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

   8. remove-double-neg 81.8%

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

   9. *-commutative 81.8%

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

   10. *-commutative 81.8%

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

6. Simplified 81.8%

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

   1. pow1/2 81.8%

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

   2. *-commutative 81.8%

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

   3. metadata-eval 81.8%

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

   4. div-inv 81.8%

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

   5. *-commutative 81.8%

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

   6. pow-sub 99.6%

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

   7. metadata-eval 99.6%

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

   8. div-sub 99.6%

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

   9. div-inv 99.6%

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

   10. metadata-eval 99.6%

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

   11. pow-unpow 99.6%

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

   12. associate-*r* 99.6%

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

   13. *-commutative 99.6%

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

8. Applied egg-rr 99.6%

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

   1. pow-sub 99.7%

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

   2. pow1 99.7%

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

10. Applied egg-rr 99.7%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6.4e-49)
   (/ (sqrt (* n (* 2.0 PI))) (sqrt k))
   (sqrt (/ (pow (* 2.0 (* n PI)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6.4e-49) {
		tmp = sqrt((n * (2.0 * ((double) M_PI)))) / sqrt(k);
	} else {
		tmp = sqrt((pow((2.0 * (n * ((double) M_PI))), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 6.4e-49) {
		tmp = Math.sqrt((n * (2.0 * Math.PI))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow((2.0 * (n * Math.PI)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 6.4e-49:
		tmp = math.sqrt((n * (2.0 * math.pi))) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow((2.0 * (n * math.pi)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 6.4e-49)
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * pi))) / sqrt(k));
	else
		tmp = sqrt(Float64((Float64(2.0 * Float64(n * pi)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 6.4e-49)
		tmp = sqrt((n * (2.0 * pi))) / sqrt(k);
	else
		tmp = sqrt((((2.0 * (n * pi)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6.4e-49], N[(N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(2.0 * N[(n * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.40000000000000005e-49

   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 66.4%

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

      1. associate-/l* 66.4%

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

   5. Simplified 66.4%

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

      1. sqrt-unprod 66.7%

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

   7. Applied egg-rr 66.7%

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

      1. associate-*r/ 66.6%

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

      2. *-commutative 66.6%

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

   9. Applied egg-rr 66.6%

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

       1. associate-*l/ 66.6%

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

       2. sqrt-div 99.5%

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

       3. *-commutative 99.5%

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

       4. associate-*r* 99.5%

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

       5. *-commutative 99.5%

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

   11. Applied egg-rr 99.5%

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

       1. *-commutative 99.5%

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

   13. Simplified 99.5%

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

   if 6.40000000000000005e-49 < k

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. distribute-lft-in 99.8%

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

      2. metadata-eval 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r* 99.8%

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

      5. metadata-eval 99.8%

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

      6. neg-mul-1 99.8%

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

      7. sub-neg 99.8%

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

      8. *-commutative 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \mathsf{fma}\left(2 \cdot \pi, \frac{n}{k}, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 8e+86)
   (/ (sqrt (* n (* 2.0 PI))) (sqrt k))
   (sqrt (+ 1.0 (fma (* 2.0 PI) (/ n k) -1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 8e+86) {
		tmp = sqrt((n * (2.0 * ((double) M_PI)))) / sqrt(k);
	} else {
		tmp = sqrt((1.0 + fma((2.0 * ((double) M_PI)), (n / k), -1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 8e+86)
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * pi))) / sqrt(k));
	else
		tmp = sqrt(Float64(1.0 + fma(Float64(2.0 * pi), Float64(n / k), -1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 8e+86], N[(N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(1.0 + N[(N[(2.0 * Pi), $MachinePrecision] * N[(n / k), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.0000000000000001e86

   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 52.0%

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

      1. associate-/l* 52.0%

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

   5. Simplified 52.0%

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

      1. sqrt-unprod 52.2%

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

   7. Applied egg-rr 52.2%

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

      1. associate-*r/ 52.2%

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

      2. *-commutative 52.2%

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

   9. Applied egg-rr 52.2%

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

       1. associate-*l/ 52.2%

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

       2. sqrt-div 74.7%

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

       3. *-commutative 74.7%

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

       4. associate-*r* 74.7%

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

       5. *-commutative 74.7%

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

   11. Applied egg-rr 74.7%

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

       1. *-commutative 74.7%

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

   13. Simplified 74.7%

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

   if 8.0000000000000001e86 < 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 1.9%

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

      1. associate-/l* 1.9%

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

   5. Simplified 1.9%

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

      1. sqrt-unprod 1.9%

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

   7. Applied egg-rr 1.9%

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

      1. associate-*r/ 1.9%

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

      2. *-commutative 1.9%

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

   9. Applied egg-rr 1.9%

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

       1. expm1-log1p-u 1.9%

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

       2. expm1-undefine 40.6%

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

       3. *-commutative 40.6%

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

       4. associate-*r/ 40.6%

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

       5. associate-*l* 40.6%

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

       6. *-commutative 40.6%

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

       7. associate-*r/ 40.6%

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

   11. Applied egg-rr 40.6%

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

       1. log1p-undefine 40.6%

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

       2. rem-exp-log 40.6%

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

       3. associate-+r- 40.6%

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

       4. *-commutative 40.6%

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

       5. associate-*l/ 40.6%

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

       6. associate-/l* 40.6%

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

       7. fmm-def 40.6%

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

       8. *-commutative 40.6%

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

       9. metadata-eval 40.6%

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

   13. Simplified 40.6%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \mathsf{fma}\left(2 \cdot \pi, \frac{n}{k}, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.7%

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

   2. *-lft-identity 99.7%

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

   3. associate-*l* 99.7%

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

   4. div-sub 99.7%

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

   5. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 5: 40.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in k around 0 25.4%

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

   1. associate-/l* 25.4%

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

5. Simplified 25.4%

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

   1. sqrt-unprod 25.5%

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

7. Applied egg-rr 25.5%

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

   1. associate-*r/ 25.5%

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

   2. *-commutative 25.5%

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

9. Applied egg-rr 25.5%

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

    1. associate-*l/ 25.5%

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

    2. sqrt-div 36.0%

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

    3. *-commutative 36.0%

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

    4. associate-*r* 36.0%

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

    5. *-commutative 36.0%

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

11. Applied egg-rr 36.0%

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

    1. *-commutative 36.0%

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

13. Simplified 36.0%

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

14. Final simplification 36.0%

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

Alternative 6: 40.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in k around 0 25.4%

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

   1. associate-/l* 25.4%

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

5. Simplified 25.4%

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

   1. sqrt-unprod 25.5%

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

7. Applied egg-rr 25.5%

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

   1. pow1/2 25.5%

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

   2. associate-*l* 25.5%

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

   3. unpow-prod-down 36.0%

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

   4. pow1/2 36.0%

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

9. Applied egg-rr 36.0%

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

    1. unpow1/2 36.0%

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

    2. associate-*l/ 36.0%

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

    3. *-commutative 36.0%

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

11. Simplified 36.0%

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

Alternative 7: 40.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 25.4%

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

   1. associate-/l* 25.4%

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

5. Simplified 25.4%

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

   1. sqrt-unprod 25.5%

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

7. Applied egg-rr 25.5%

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

   1. associate-*r/ 25.5%

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

   2. *-commutative 25.5%

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

9. Applied egg-rr 25.5%

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

    1. sqrt-prod 25.4%

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

    2. *-commutative 25.4%

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

    3. associate-*r/ 25.4%

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

    4. sqrt-prod 25.5%

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

    5. associate-*r* 25.5%

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

    6. *-commutative 25.5%

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

    7. sqrt-prod 36.0%

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

    8. associate-*l/ 36.0%

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

    9. *-commutative 36.0%

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

11. Applied egg-rr 36.0%

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

    1. *-commutative 36.0%

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

    2. *-commutative 36.0%

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

    3. associate-/l* 36.0%

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

13. Simplified 36.0%

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

Alternative 8: 32.0% accurate, 1.0× 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 (fabs (* n (/ PI k))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in k around 0 25.4%

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

   1. associate-/l* 25.4%

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

5. Simplified 25.4%

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

   1. sqrt-unprod 25.5%

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

7. Applied egg-rr 25.5%

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

   1. add-sqr-sqrt 25.4%

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

   2. pow1/2 25.4%

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

   3. pow1/2 25.4%

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

   4. pow-prod-down 25.7%

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

   5. pow2 25.7%

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

9. Applied egg-rr 25.7%

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

    1. unpow1/2 25.7%

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

    2. unpow2 25.7%

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

    3. rem-sqrt-square 26.1%

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

11. Simplified 26.1%

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

12. Final simplification 26.1%

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

Alternative 9: 32.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in k around 0 25.4%

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

   1. associate-/l* 25.4%

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

5. Simplified 25.4%

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

   1. sqrt-unprod 25.5%

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

7. Applied egg-rr 25.5%

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

   1. associate-*r/ 25.5%

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

   2. *-commutative 25.5%

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

9. Applied egg-rr 25.5%

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

    1. *-commutative 25.5%

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

    2. associate-*r/ 25.5%

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

    3. pow1 25.5%

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

    4. associate-*r* 25.5%

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

    5. metadata-eval 25.5%

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

    6. pow-sqr 25.5%

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

    7. pow-prod-down 25.7%

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

11. Applied egg-rr 25.7%

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

    1. unpow1/2 25.7%

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

    2. *-commutative 25.7%

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

    3. metadata-eval 25.7%

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

    4. unpow2 25.7%

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

    5. swap-sqr 25.7%

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

    6. associate-*r/ 25.7%

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

    7. associate-*r/ 25.7%

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

    8. associate-*r* 25.7%

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

    9. *-commutative 25.7%

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

    10. associate-*r* 25.7%

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

    11. associate-*r/ 25.7%

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

    12. associate-*r/ 25.7%

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

    13. associate-*r/ 25.7%

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

    14. associate-*r* 25.7%

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

    15. *-commutative 25.7%

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

    16. associate-*r* 25.7%

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

    17. associate-*r/ 25.7%

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

13. Simplified 26.0%

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

Alternative 10: 31.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in k around 0 25.4%

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

   1. associate-/l* 25.4%

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

5. Simplified 25.4%

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

   1. sqrt-unprod 25.5%

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

7. Applied egg-rr 25.5%

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

   1. div-inv 25.5%

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

9. Applied egg-rr 25.5%

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

10. Final simplification 25.5%

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

Alternative 11: 31.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 25.4%

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

   1. associate-/l* 25.4%

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

5. Simplified 25.4%

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

   1. sqrt-unprod 25.5%

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

7. Applied egg-rr 25.5%

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

8. Final simplification 25.5%

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

Alternative 12: 31.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in k around 0 25.4%

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

   1. associate-/l* 25.4%

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

5. Simplified 25.4%

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

   1. sqrt-unprod 25.5%

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

7. Applied egg-rr 25.5%

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

   1. associate-*r/ 25.5%

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

   2. *-commutative 25.5%

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

9. Applied egg-rr 25.5%

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

10. Taylor expanded in n around 0 25.5%

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

    1. associate-*r/ 25.5%

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

    2. *-commutative 25.5%

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

    3. associate-/l* 25.5%

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

12. Simplified 25.5%

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

Reproduce

herbie shell --seed 2024180 
(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))))