Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 19.1s

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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $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. add-sqr-sqrt 99.4%

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

   2. sqrt-unprod 99.5%

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

   3. frac-times 99.5%

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

   4. metadata-eval 99.5%

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

   5. add-sqr-sqrt 99.5%

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.6e-67)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (/ 1.0 (sqrt (/ k (pow (* (* 2.0 PI) n) (- 1.0 k)))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.6e-67) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = 1.0 / sqrt((k / pow(((2.0 * ((double) M_PI)) * n), (1.0 - k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.6e-67) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((2.0 * n));
	} else {
		tmp = 1.0 / Math.sqrt((k / Math.pow(((2.0 * Math.PI) * n), (1.0 - k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 4.6e-67:
		tmp = math.sqrt((math.pi / k)) * math.sqrt((2.0 * n))
	else:
		tmp = 1.0 / math.sqrt((k / math.pow(((2.0 * math.pi) * n), (1.0 - k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.6e-67)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = Float64(1.0 / sqrt(Float64(k / (Float64(Float64(2.0 * pi) * n) ^ Float64(1.0 - k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 4.6e-67)
		tmp = sqrt((pi / k)) * sqrt((2.0 * n));
	else
		tmp = 1.0 / sqrt((k / (((2.0 * pi) * n) ^ (1.0 - k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.6e-67], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(k / N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.6000000000000001e-67

   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

   4. Applied egg-rr 74.9%

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

   6. Simplified 75.0%

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

   7. Taylor expanded in k around 0 75.0%

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

      1. associate-*r* 75.0%

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

      2. *-commutative 75.0%

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

      3. associate-*l* 75.0%

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

   9. Simplified 75.0%

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

       1. associate-*r* 75.0%

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

       2. associate-*r/ 75.0%

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

       3. sqrt-prod 99.5%

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

       4. *-commutative 99.5%

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

       5. *-commutative 99.5%

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

   11. Applied egg-rr 99.5%

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

   if 4.6000000000000001e-67 < k

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. 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. sqr-pow 98.9%

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

      4. pow-sqr 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*l* 99.6%

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

      7. associate-*r/ 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/l* 99.6%

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

      10. metadata-eval 99.6%

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

      11. /-rgt-identity 99.6%

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

      12. div-sub 99.6%

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

      13. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. div-inv 99.6%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*l* 99.6%

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

      5. sub-neg 99.6%

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

      6. div-inv 99.6%

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

      7. metadata-eval 99.6%

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

      8. distribute-rgt-neg-in 99.6%

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

      9. metadata-eval 99.6%

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

      10. inv-pow 99.6%

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

      11. sqrt-pow2 99.6%

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

      12. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. sqr-pow 98.9%

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

      2. pow-prod-down 91.3%

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

      3. sqrt-pow1 91.3%

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

      4. pow-prod-down 99.0%

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

      5. pow2 99.0%

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

      6. pow-unpow 99.0%

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

      7. *-commutative 99.0%

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

      8. associate-*r* 99.0%

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

      9. +-commutative 99.0%

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

      10. fma-udef 99.0%

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

      11. *-commutative 99.0%

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

      12. associate-*r* 99.0%

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

   8. Applied egg-rr 99.0%

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

      1. *-commutative 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*l* 99.0%

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

      4. fma-udef 99.0%

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

      5. distribute-lft-in 99.0%

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

      6. *-commutative 99.0%

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

      7. associate-*r* 99.0%

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

      8. metadata-eval 99.0%

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

      9. neg-mul-1 99.0%

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

      10. metadata-eval 99.0%

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

      11. +-commutative 99.0%

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

      12. sub-neg 99.0%

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

   10. Simplified 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.35e-96)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (sqrt (/ (pow (* (* 2.0 PI) n) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.35e-96) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} 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 <= 1.35e-96) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((2.0 * n));
	} 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 <= 1.35e-96:
		tmp = math.sqrt((math.pi / k)) * math.sqrt((2.0 * n))
	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 <= 1.35e-96)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = sqrt(Float64((Float64(Float64(2.0 * pi) * n) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.35e-96)
		tmp = sqrt((pi / k)) * sqrt((2.0 * n));
	else
		tmp = sqrt(((((2.0 * pi) * n) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.35e-96], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.35e-96

   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

   4. Applied egg-rr 72.5%

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

   6. Simplified 72.5%

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

   7. Taylor expanded in k around 0 72.5%

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

      1. associate-*r* 72.5%

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

      2. *-commutative 72.5%

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

      3. associate-*l* 72.5%

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

   9. Simplified 72.5%

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

       1. associate-*r* 72.5%

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

       2. associate-*r/ 72.5%

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

       3. sqrt-prod 99.5%

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

       4. *-commutative 99.5%

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

       5. *-commutative 99.5%

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

   11. Applied egg-rr 99.5%

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

   if 1.35e-96 < k

   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

   4. Applied egg-rr 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in n around 0 97.9%

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

   9. Simplified 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(Pi * N[(2.0 * 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/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. sqr-pow 98.9%

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

   4. pow-sqr 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-*l* 99.5%

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

   7. associate-*r/ 99.5%

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

   8. *-commutative 99.5%

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

   9. associate-/l* 99.5%

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

   10. metadata-eval 99.5%

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

   11. /-rgt-identity 99.5%

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

   12. div-sub 99.5%

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

   13. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 5: 49.6% 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

4. Applied egg-rr 90.7%

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

6. Simplified 90.7%

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

7. Taylor expanded in k around 0 35.1%

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

   1. associate-*r* 35.1%

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

   2. *-commutative 35.1%

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

   3. associate-*l* 35.1%

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

9. Simplified 35.1%

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

    1. pow1/2 35.1%

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

    2. associate-*r* 35.1%

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

    3. associate-*r/ 35.1%

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

    4. associate-*l* 35.1%

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

    5. unpow-prod-down 43.5%

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

    6. pow1/2 43.5%

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

    7. clear-num 43.5%

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

    8. un-div-inv 43.5%

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

11. Applied egg-rr 43.5%

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

    1. unpow1/2 43.5%

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

    2. associate-/r/ 43.5%

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

13. Simplified 43.5%

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

14. Final simplification 43.5%

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

Alternative 6: 49.6% 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

4. Applied egg-rr 90.7%

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

6. Simplified 90.7%

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

7. Taylor expanded in k around 0 35.1%

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

   1. associate-*r* 35.1%

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

   2. *-commutative 35.1%

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

   3. associate-*l* 35.1%

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

9. Simplified 35.1%

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

    1. associate-*r* 35.1%

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

    2. associate-*r/ 35.1%

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

    3. sqrt-prod 43.5%

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

    4. *-commutative 43.5%

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

    5. *-commutative 43.5%

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

11. Applied egg-rr 43.5%

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

12. Final simplification 43.5%

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

Alternative 7: 38.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{0.5 \cdot \frac{k}{\pi \cdot n}}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (/ 1.0 (sqrt (* 0.5 (/ k (* PI n))))))

C

double code(double k, double n) {
	return 1.0 / sqrt((0.5 * (k / (((double) M_PI) * n))));
}

Java

public static double code(double k, double n) {
	return 1.0 / Math.sqrt((0.5 * (k / (Math.PI * n))));
}

Python

def code(k, n):
	return 1.0 / math.sqrt((0.5 * (k / (math.pi * n))))

Julia

function code(k, n)
	return Float64(1.0 / sqrt(Float64(0.5 * Float64(k / Float64(pi * n)))))
end

MATLAB

function tmp = code(k, n)
	tmp = 1.0 / sqrt((0.5 * (k / (pi * n))));
end

Wolfram

code[k_, n_] := N[(1.0 / N[Sqrt[N[(0.5 * N[(k / N[(Pi * n), $MachinePrecision]), $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

4. Applied egg-rr 90.7%

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

6. Simplified 90.7%

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

7. Taylor expanded in k around 0 35.1%

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

   1. associate-*r* 35.1%

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

   2. *-commutative 35.1%

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

   3. associate-*l* 35.1%

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

9. Simplified 35.1%

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

    1. clear-num 35.0%

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

    2. sqrt-div 36.2%

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

    3. metadata-eval 36.2%

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

    4. *-un-lft-identity 36.2%

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

    5. *-commutative 36.2%

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

    6. associate-*r* 36.2%

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

    7. *-commutative 36.2%

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

    8. times-frac 36.2%

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

    9. metadata-eval 36.2%

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

    10. *-commutative 36.2%

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

    11. associate-/r* 36.2%

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

11. Applied egg-rr 36.2%

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

    1. associate-/l/ 36.2%

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

13. Simplified 36.2%

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

14. Final simplification 36.2%

    \[\leadsto \frac{1}{\sqrt{0.5 \cdot \frac{k}{\pi \cdot n}}} \]
15. Add Preprocessing

Alternative 8: 38.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{0.5 \cdot \frac{\frac{k}{\pi}}{n}}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (/ 1.0 (sqrt (* 0.5 (/ (/ k PI) n)))))

C

double code(double k, double n) {
	return 1.0 / sqrt((0.5 * ((k / ((double) M_PI)) / n)));
}

Java

public static double code(double k, double n) {
	return 1.0 / Math.sqrt((0.5 * ((k / Math.PI) / n)));
}

Python

def code(k, n):
	return 1.0 / math.sqrt((0.5 * ((k / math.pi) / n)))

Julia

function code(k, n)
	return Float64(1.0 / sqrt(Float64(0.5 * Float64(Float64(k / pi) / n))))
end

MATLAB

function tmp = code(k, n)
	tmp = 1.0 / sqrt((0.5 * ((k / pi) / n)));
end

Wolfram

code[k_, n_] := N[(1.0 / N[Sqrt[N[(0.5 * N[(N[(k / Pi), $MachinePrecision] / n), $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

4. Applied egg-rr 90.7%

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

6. Simplified 90.7%

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

7. Taylor expanded in k around 0 35.1%

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

   1. associate-*r* 35.1%

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

   2. *-commutative 35.1%

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

   3. associate-*l* 35.1%

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

9. Simplified 35.1%

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

    1. clear-num 35.0%

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

    2. sqrt-div 36.2%

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

    3. metadata-eval 36.2%

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

    4. *-un-lft-identity 36.2%

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

    5. *-commutative 36.2%

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

    6. associate-*r* 36.2%

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

    7. *-commutative 36.2%

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

    8. times-frac 36.2%

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

    9. metadata-eval 36.2%

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

    10. *-commutative 36.2%

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

    11. associate-/r* 36.2%

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

11. Applied egg-rr 36.2%

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

12. Final simplification 36.2%

    \[\leadsto \frac{1}{\sqrt{0.5 \cdot \frac{\frac{k}{\pi}}{n}}} \]
13. Add Preprocessing

Alternative 9: 37.6% 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

4. Applied egg-rr 90.7%

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

6. Simplified 90.7%

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

7. Taylor expanded in k around 0 35.1%

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

   1. associate-*r* 35.1%

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

   2. *-commutative 35.1%

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

   3. associate-*l* 35.1%

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

9. Simplified 35.1%

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

    1. associate-/l* 35.1%

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

    2. sqrt-div 43.5%

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

    3. *-un-lft-identity 43.5%

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

    4. times-frac 43.5%

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

    5. metadata-eval 43.5%

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

11. Applied egg-rr 43.5%

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

    1. sqrt-undiv 35.1%

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

    2. *-un-lft-identity 35.1%

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

    3. times-frac 35.1%

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

    4. metadata-eval 35.1%

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

13. Applied egg-rr 35.1%

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

14. Final simplification 35.1%

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

Alternative 10: 37.6% accurate, 2.0× speedup

Math

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

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

4. Applied egg-rr 90.7%

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

6. Simplified 90.7%

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

7. Taylor expanded in k around 0 35.1%

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

   1. associate-*r* 35.1%

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

   2. *-commutative 35.1%

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

   3. associate-*l* 35.1%

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

9. Simplified 35.1%

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

10. Taylor expanded in n around 0 35.1%

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

11. Final simplification 35.1%

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

Alternative 11: 37.5% accurate, 2.0× speedup

Math

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

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(n * Float64(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 * 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

4. Applied egg-rr 90.7%

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

6. Simplified 90.7%

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

7. Taylor expanded in k around 0 35.1%

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

   1. associate-*r* 35.1%

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

   2. *-commutative 35.1%

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

   3. associate-*l* 35.1%

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

9. Simplified 35.1%

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

    1. associate-*r* 35.1%

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

    2. associate-*r/ 35.1%

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

    3. expm1-log1p-u 33.5%

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

    4. expm1-udef 33.8%

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

11. Applied egg-rr 33.8%

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

    1. expm1-def 33.5%

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

    2. expm1-log1p 35.1%

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

    3. associate-/r/ 35.1%

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

13. Simplified 35.1%

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

14. Final simplification 35.1%

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

Reproduce

herbie shell --seed 2024017 
(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))))