Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 13.9s

Alternatives: 8

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, 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 2: 59.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5.2e+39)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (sqrt (* 2.0 (+ -1.0 (fma n (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5.2e+39) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = sqrt((2.0 * (-1.0 + fma(n, (((double) M_PI) / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5.2e+39)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = sqrt(Float64(2.0 * Float64(-1.0 + fma(n, Float64(pi / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5.2e+39], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(-1.0 + N[(n * N[(Pi / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.2e39

   1. Initial program 99.0%

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

   3. Taylor expanded in k around 0 66.5%

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

      1. associate-/l* 66.5%

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

   5. Simplified 66.5%

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

      1. sqrt-unprod 66.7%

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

      2. *-commutative 66.7%

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

      3. associate-*l* 66.7%

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

   7. Applied egg-rr 66.7%

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

      1. sqrt-prod 85.8%

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

   9. Applied egg-rr 85.8%

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

   if 5.2e39 < 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%

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

      2. *-commutative 2.5%

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

      3. associate-*l* 2.5%

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

   7. Applied egg-rr 2.5%

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

   8. Taylor expanded in k around 0 2.5%

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

      1. *-commutative 2.5%

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

      2. associate-/l* 2.5%

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

      3. associate-*r* 2.5%

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

      4. *-commutative 2.5%

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

      5. *-commutative 2.5%

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

      6. associate-*l* 2.5%

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

   10. Simplified 2.5%

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

       1. expm1-log1p-u 2.5%

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

       2. expm1-undefine 25.9%

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

       3. *-commutative 25.9%

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

   12. Applied egg-rr 25.9%

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

       1. sub-neg 25.9%

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

       2. metadata-eval 25.9%

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

       3. +-commutative 25.9%

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

       4. log1p-undefine 25.9%

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

       5. rem-exp-log 25.9%

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

       6. +-commutative 25.9%

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

       7. fma-define 25.9%

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

   14. Simplified 25.9%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $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 36.8%

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

   1. associate-/l* 36.8%

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

5. Simplified 36.8%

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

   1. sqrt-unprod 36.9%

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

   2. *-commutative 36.9%

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

   3. associate-*l* 36.9%

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

7. Applied egg-rr 36.9%

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

   1. sqrt-prod 47.2%

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

9. Applied egg-rr 47.2%

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

10. Final simplification 47.2%

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

Alternative 4: 48.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(2.0 / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * n), $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 36.8%

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

   1. associate-/l* 36.8%

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

5. Simplified 36.8%

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

   1. sqrt-unprod 36.9%

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

   2. *-commutative 36.9%

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

   3. associate-*l* 36.9%

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

7. Applied egg-rr 36.9%

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

8. Taylor expanded in k around 0 36.9%

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

   1. *-commutative 36.9%

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

   2. associate-/l* 36.9%

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

   3. associate-*r* 36.5%

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

   4. *-commutative 36.5%

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

   5. *-commutative 36.5%

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

   6. associate-*l* 36.9%

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

10. Simplified 36.9%

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

    1. sqrt-prod 36.8%

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

    2. associate-*l/ 36.8%

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

    3. clear-num 36.8%

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

    4. sqrt-prod 36.8%

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

    5. div-inv 36.8%

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

    6. associate-/r/ 36.8%

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

    7. sqrt-prod 47.0%

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

12. Applied egg-rr 47.0%

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

Alternative 5: 37.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Power[N[(N[(k / n), $MachinePrecision] * N[(0.5 / Pi), $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 36.8%

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

   1. associate-/l* 36.8%

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

5. Simplified 36.8%

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

   1. sqrt-unprod 36.9%

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

   2. *-commutative 36.9%

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

   3. associate-*l* 36.9%

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

7. Applied egg-rr 36.9%

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

8. Taylor expanded in k around 0 36.9%

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

   1. *-commutative 36.9%

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

   2. associate-/l* 36.9%

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

   3. associate-*r* 36.5%

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

   4. *-commutative 36.5%

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

   5. *-commutative 36.5%

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

   6. associate-*l* 36.9%

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

10. Simplified 36.9%

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

    1. metadata-eval 36.9%

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

    2. associate-*l/ 36.9%

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

    3. times-frac 36.8%

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

    4. *-commutative 36.8%

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

    5. *-un-lft-identity 36.8%

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

    6. clear-num 36.8%

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

    7. metadata-eval 36.8%

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

    8. *-commutative 36.8%

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

    9. add-sqr-sqrt 36.8%

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

    10. frac-times 36.8%

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

    11. sqrt-unprod 37.2%

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

    12. add-sqr-sqrt 37.3%

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

    13. pow1/2 37.3%

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

    14. pow-flip 37.4%

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

    15. times-frac 37.4%

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

    16. metadata-eval 37.4%

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

12. Applied egg-rr 37.4%

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

Alternative 6: 37.8% 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 36.8%

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

   1. associate-/l* 36.8%

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

5. Simplified 36.8%

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

   1. sqrt-unprod 36.9%

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

   2. *-commutative 36.9%

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

   3. associate-*l* 36.9%

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

7. Applied egg-rr 36.9%

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

8. Taylor expanded in k around 0 36.9%

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

   1. *-commutative 36.9%

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

   2. associate-/l* 36.9%

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

   3. associate-*r* 36.5%

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

   4. *-commutative 36.5%

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

   5. *-commutative 36.5%

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

   6. associate-*l* 36.9%

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

10. Simplified 36.9%

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

    1. metadata-eval 36.9%

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

    2. associate-*l/ 36.9%

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

    3. times-frac 36.8%

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

    4. *-commutative 36.8%

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

    5. *-un-lft-identity 36.8%

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

    6. clear-num 36.8%

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

    7. metadata-eval 36.8%

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

    8. *-commutative 36.8%

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

    9. add-sqr-sqrt 36.8%

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

    10. frac-times 36.8%

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

    11. sqrt-unprod 37.2%

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

    12. add-sqr-sqrt 37.3%

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

    13. pow1/2 37.3%

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

    14. pow-flip 37.4%

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

    15. times-frac 37.4%

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

    16. metadata-eval 37.4%

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

12. Applied egg-rr 37.4%

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

    1. associate-*l/ 37.4%

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

    2. associate-*r/ 37.4%

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

    3. associate-/l/ 37.4%

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

14. Simplified 37.4%

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

15. Final simplification 37.4%

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

Alternative 7: 37.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 36.8%

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

   1. associate-/l* 36.8%

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

5. Simplified 36.8%

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

   1. sqrt-unprod 36.9%

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

   2. *-commutative 36.9%

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

   3. associate-*l* 36.9%

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

7. Applied egg-rr 36.9%

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

8. Taylor expanded in k around 0 36.9%

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

   1. *-commutative 36.9%

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

   2. associate-/l* 36.9%

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

   3. associate-*r* 36.5%

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

   4. *-commutative 36.5%

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

   5. *-commutative 36.5%

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

   6. associate-*l* 36.9%

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

10. Simplified 36.9%

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

11. Final simplification 36.9%

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

Alternative 8: 37.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 36.8%

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

   1. associate-/l* 36.8%

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

5. Simplified 36.8%

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

   1. sqrt-unprod 36.9%

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

   2. *-commutative 36.9%

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

   3. associate-*l* 36.9%

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

7. Applied egg-rr 36.9%

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

8. Taylor expanded in k around 0 36.9%

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

   1. *-commutative 36.9%

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

   2. associate-/l* 36.9%

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

   3. associate-*r* 36.5%

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

   4. *-commutative 36.5%

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

   5. *-commutative 36.5%

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

   6. associate-*l* 36.9%

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

10. Simplified 36.9%

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

11. Taylor expanded in k around 0 36.9%

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

    1. *-commutative 36.9%

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

    2. associate-/l* 36.8%

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

13. Simplified 36.8%

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

Reproduce

herbie shell --seed 2024137 
(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))))