Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 15.4s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 * N[(Pi * 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. associate-*r* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

   6. pow-div 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. 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. Add Preprocessing

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5.2e-74)
   (/ (sqrt (* PI (* 2.0 n))) (sqrt k))
   (sqrt (/ (pow (* 2.0 (* PI n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5.2e-74) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) / sqrt(k);
	} else {
		tmp = sqrt((pow((2.0 * (((double) M_PI) * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 5.2e-74) {
		tmp = Math.sqrt((Math.PI * (2.0 * n))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow((2.0 * (Math.PI * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 5.2e-74:
		tmp = math.sqrt((math.pi * (2.0 * n))) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow((2.0 * (math.pi * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5.2e-74)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) / sqrt(k));
	else
		tmp = sqrt(Float64((Float64(2.0 * Float64(pi * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 5.2e-74)
		tmp = sqrt((pi * (2.0 * n))) / sqrt(k);
	else
		tmp = sqrt((((2.0 * (pi * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5.2e-74], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.2000000000000002e-74

   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 72.7%

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

      1. associate-/l* 72.7%

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

   5. Simplified 72.7%

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

      1. pow1 72.7%

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

      2. sqrt-unprod 72.8%

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

   7. Applied egg-rr 72.8%

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

      1. unpow1 72.8%

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

      2. *-commutative 72.8%

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

      3. associate-*r/ 72.9%

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

   9. Simplified 72.9%

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

       1. associate-*r/ 72.9%

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

       2. sqrt-div 99.5%

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

       3. *-commutative 99.5%

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

       4. associate-*r* 99.5%

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

       5. *-commutative 99.5%

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

       6. *-commutative 99.5%

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

   11. Applied egg-rr 99.5%

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

       1. associate-*r* 99.5%

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

       2. *-commutative 99.5%

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

       3. associate-*r* 99.5%

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

   13. Simplified 99.5%

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

   if 5.2000000000000002e-74 < k

   1. Initial program 99.7%

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

   4. Applied egg-rr 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{+191}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{n}{\frac{k}{\pi}}\right)}^{2} \cdot 4\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3e+191)
   (/ (sqrt (* PI (* 2.0 n))) (sqrt k))
   (pow (* (pow (/ n (/ k PI)) 2.0) 4.0) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3e+191) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) / sqrt(k);
	} else {
		tmp = pow((pow((n / (k / ((double) M_PI))), 2.0) * 4.0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3e+191) {
		tmp = Math.sqrt((Math.PI * (2.0 * n))) / Math.sqrt(k);
	} else {
		tmp = Math.pow((Math.pow((n / (k / Math.PI)), 2.0) * 4.0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3e+191:
		tmp = math.sqrt((math.pi * (2.0 * n))) / math.sqrt(k)
	else:
		tmp = math.pow((math.pow((n / (k / math.pi)), 2.0) * 4.0), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3e+191)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) / sqrt(k));
	else
		tmp = Float64((Float64(n / Float64(k / pi)) ^ 2.0) * 4.0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3e+191)
		tmp = sqrt((pi * (2.0 * n))) / sqrt(k);
	else
		tmp = (((n / (k / pi)) ^ 2.0) * 4.0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3e+191], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(n / N[(k / Pi), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 4.0), $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.9999999999999997e191

   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 45.6%

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

      1. associate-/l* 45.6%

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

   5. Simplified 45.6%

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

      1. pow1 45.6%

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

      2. sqrt-unprod 45.7%

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

   7. Applied egg-rr 45.7%

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

      1. unpow1 45.7%

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

      2. *-commutative 45.7%

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

      3. associate-*r/ 45.8%

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

   9. Simplified 45.8%

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

       1. associate-*r/ 45.8%

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

       2. sqrt-div 57.0%

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

       3. *-commutative 57.0%

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

       4. associate-*r* 57.0%

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

       5. *-commutative 57.0%

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

       6. *-commutative 57.0%

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

   11. Applied egg-rr 57.0%

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

       1. associate-*r* 57.0%

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

       2. *-commutative 57.0%

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

       3. associate-*r* 57.0%

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

   13. Simplified 57.0%

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

   if 2.9999999999999997e191 < 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 2.5%

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

      1. associate-/l* 2.5%

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

   5. Simplified 2.5%

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

      1. sqrt-unprod 2.5%

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

   7. Applied egg-rr 2.5%

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

      1. *-commutative 2.5%

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

      2. metadata-eval 2.5%

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

      3. associate-*r/ 2.5%

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

      4. times-frac 2.5%

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

      5. associate-*r* 2.5%

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

      6. *-un-lft-identity 2.5%

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

      7. pow1/2 2.5%

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

      8. *-commutative 2.5%

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

      9. associate-*r* 2.5%

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

      10. associate-*r/ 2.5%

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

      11. metadata-eval 2.5%

          \[\leadsto {\left(\left(\pi \cdot 2\right) \cdot \frac{n}{k}\right)}^{\color{blue}{\left(2 \cdot 0.25\right)}} \]

      12. pow-sqr 2.5%

          \[\leadsto \color{blue}{{\left(\left(\pi \cdot 2\right) \cdot \frac{n}{k}\right)}^{0.25} \cdot {\left(\left(\pi \cdot 2\right) \cdot \frac{n}{k}\right)}^{0.25}} \]

      13. pow-prod-down 14.1%

          \[\leadsto \color{blue}{{\left(\left(\left(\pi \cdot 2\right) \cdot \frac{n}{k}\right) \cdot \left(\left(\pi \cdot 2\right) \cdot \frac{n}{k}\right)\right)}^{0.25}} \]

   9. Applied egg-rr 14.1%

      \[\leadsto \color{blue}{{\left({\left(\frac{n}{\frac{k}{\pi}}\right)}^{2} \cdot 4\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{+191}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{n}{\frac{k}{\pi}}\right)}^{2} \cdot 4\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. associate-*l* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

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

Alternative 5: 50.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 38.7%

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

   1. associate-/l* 38.7%

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

5. Simplified 38.7%

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

   1. pow1 38.7%

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

   2. sqrt-unprod 38.8%

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

7. Applied egg-rr 38.8%

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

   1. unpow1 38.8%

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

   2. *-commutative 38.8%

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

   3. associate-*r/ 38.8%

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

9. Simplified 38.8%

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

    1. associate-*r/ 38.8%

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

    2. sqrt-div 48.3%

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

    3. *-commutative 48.3%

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

    4. associate-*r* 48.3%

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

    5. *-commutative 48.3%

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

    6. *-commutative 48.3%

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

11. Applied egg-rr 48.3%

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

    1. associate-*r* 48.3%

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

    2. *-commutative 48.3%

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

    3. associate-*r* 48.3%

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

13. Simplified 48.3%

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

14. Final simplification 48.3%

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

Alternative 6: 50.3% 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 38.7%

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

   1. associate-/l* 38.7%

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

5. Simplified 38.7%

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

   1. sqrt-unprod 38.8%

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

7. Applied egg-rr 38.8%

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

   1. *-commutative 38.8%

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

   2. metadata-eval 38.8%

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

   3. associate-*r/ 38.8%

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

   4. times-frac 38.8%

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

   5. associate-*r* 38.8%

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

   6. *-un-lft-identity 38.8%

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

   7. sqrt-undiv 48.3%

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

   8. sqrt-prod 48.2%

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

   9. *-un-lft-identity 48.2%

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

   10. times-frac 48.1%

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

   11. sqrt-div 48.3%

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

9. Applied egg-rr 48.3%

   \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot n}}{1} \cdot \sqrt{\frac{\pi}{k}}} \]
10. Step-by-step derivation

    1. /-rgt-identity 48.3%

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

    2. *-commutative 48.3%

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

11. Simplified 48.3%

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

12. Final simplification 48.3%

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

Alternative 7: 50.3% 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 99.5%

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

3. Taylor expanded in k around 0 38.7%

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

   1. associate-/l* 38.7%

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

5. Simplified 38.7%

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

   1. sqrt-unprod 38.8%

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

7. Applied egg-rr 38.8%

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

   1. pow1/2 38.8%

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

   2. associate-*l* 38.8%

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

   3. unpow-prod-down 48.2%

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

   4. pow1/2 48.2%

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

9. Applied egg-rr 48.2%

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

    1. unpow1/2 48.2%

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

    2. associate-*l/ 48.2%

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

    3. associate-/l* 48.2%

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

11. Simplified 48.2%

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

Alternative 8: 39.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Power[N[(N[(0.5 * N[(k / n), $MachinePrecision]), $MachinePrecision] / Pi), $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 38.7%

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

   1. associate-/l* 38.7%

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

5. Simplified 38.7%

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

   1. pow1 38.7%

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

   2. sqrt-unprod 38.8%

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

7. Applied egg-rr 38.8%

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

   1. unpow1 38.8%

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

   2. *-commutative 38.8%

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

   3. associate-*r/ 38.8%

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

9. Simplified 38.8%

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

    1. clear-num 38.8%

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

    2. un-div-inv 38.8%

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

    3. sqrt-undiv 38.9%

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

    4. clear-num 38.9%

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

    5. inv-pow 38.9%

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

    6. sqrt-undiv 38.9%

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

    7. sqrt-pow2 39.0%

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

    8. div-inv 39.0%

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

    9. metadata-eval 39.0%

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

    10. metadata-eval 39.0%

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

11. Applied egg-rr 39.0%

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

    1. *-commutative 39.0%

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

    2. associate-/r* 39.0%

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

    3. associate-*r/ 39.0%

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

13. Simplified 39.0%

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

Alternative 9: 39.6% 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 38.7%

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

   1. associate-/l* 38.7%

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

5. Simplified 38.7%

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

   1. pow1 38.7%

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

   2. sqrt-unprod 38.8%

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

7. Applied egg-rr 38.8%

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

   1. unpow1 38.8%

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

   2. *-commutative 38.8%

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

   3. associate-*r/ 38.8%

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

9. Simplified 38.8%

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

    1. clear-num 38.8%

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

    2. un-div-inv 38.8%

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

    3. sqrt-undiv 38.9%

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

    4. clear-num 38.9%

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

    5. frac-2neg 38.9%

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

    6. metadata-eval 38.9%

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

    7. div-inv 38.9%

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

    8. sqrt-undiv 38.9%

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

    9. div-inv 38.9%

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

    10. metadata-eval 38.9%

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

11. Applied egg-rr 38.9%

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

    1. mul-1-neg 38.9%

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

    2. distribute-frac-neg2 38.9%

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

    3. remove-double-neg 38.9%

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

    4. *-commutative 38.9%

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

    5. associate-/r* 38.9%

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

    6. associate-*r/ 38.9%

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

13. Simplified 38.9%

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

    1. inv-pow 38.9%

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

    2. sqrt-pow2 39.0%

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

    3. clear-num 38.8%

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

    4. un-div-inv 38.8%

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

    5. metadata-eval 38.8%

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

15. Applied egg-rr 38.8%

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

    1. associate-/r/ 39.0%

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

    2. associate-*l/ 39.0%

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

    3. associate-/r* 39.0%

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

    4. *-commutative 39.0%

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

    5. associate-/l* 39.0%

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

17. Simplified 39.0%

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

18. Final simplification 39.0%

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

Alternative 10: 38.8% accurate, 2.0× speedup

Math

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

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(Float64(2.0 * 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[(N[(2.0 * Pi), $MachinePrecision] * N[(n / 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 38.7%

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

   1. associate-/l* 38.7%

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

5. Simplified 38.7%

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

   1. associate-*r/ 38.7%

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

   2. *-commutative 38.7%

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

7. Applied egg-rr 38.7%

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

   1. sqrt-unprod 38.8%

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

   2. *-commutative 38.8%

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

   3. *-commutative 38.8%

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

   4. *-commutative 38.8%

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

   5. associate-/l* 38.9%

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

   6. associate-*r* 38.9%

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

   7. *-commutative 38.9%

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

9. Applied egg-rr 38.9%

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

10. Final simplification 38.9%

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

Alternative 11: 38.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(2.0 * N[(n / N[(k / Pi), $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 38.7%

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

   1. associate-/l* 38.7%

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

5. Simplified 38.7%

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

   1. pow1 38.7%

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

   2. sqrt-unprod 38.8%

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

7. Applied egg-rr 38.8%

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

   1. unpow1 38.8%

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

   2. *-commutative 38.8%

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

   3. associate-*r/ 38.8%

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

9. Simplified 38.8%

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

    1. associate-*r/ 38.8%

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

    2. sqrt-div 48.3%

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

    3. *-commutative 48.3%

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

    4. associate-*r* 48.3%

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

    5. *-commutative 48.3%

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

    6. *-commutative 48.3%

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

11. Applied egg-rr 48.3%

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

    1. associate-*r* 48.3%

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

    2. *-commutative 48.3%

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

    3. associate-*r* 48.3%

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

13. Simplified 48.3%

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

    1. sqrt-undiv 38.8%

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

    2. associate-*r* 38.8%

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

    3. *-un-lft-identity 38.8%

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

    4. times-frac 38.8%

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

    5. metadata-eval 38.8%

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

    6. associate-*r/ 38.8%

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

    7. clear-num 38.8%

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

    8. un-div-inv 38.8%

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

15. Applied egg-rr 38.8%

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

Reproduce

herbie shell --seed 2024152 
(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))))