Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 17.0s

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, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(N[(Pi / N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], k], $MachinePrecision]), $MachinePrecision] * 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. Step-by-step derivation

   1. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*r* 99.5%

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

   4. div-inv 99.5%

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

   5. expm1-log1p-u 97.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

   6. expm1-udef 85.6%

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

3. Applied egg-rr 76.0%

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

   1. expm1-def 87.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

   2. expm1-log1p 89.0%

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

   3. *-commutative 89.0%

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

   4. associate-*r* 89.0%

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

   5. *-commutative 89.0%

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

5. Simplified 89.0%

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

   1. pow-sub 89.2%

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

   2. pow1 89.2%

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

   3. *-commutative 89.2%

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

   4. *-commutative 89.2%

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

   5. *-commutative 89.2%

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

   6. *-commutative 89.2%

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

   7. *-commutative 89.2%

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

   8. *-commutative 89.2%

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

7. Applied egg-rr 89.2%

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

   1. associate-/l* 89.2%

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

   2. associate-/r/ 89.3%

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

   3. *-commutative 89.3%

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

   4. associate-*l* 89.3%

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

   5. *-commutative 89.3%

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

9. Simplified 89.3%

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

    1. sqrt-div 99.7%

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

    2. *-commutative 99.7%

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

    3. associate-*r* 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{{k}^{-0.5}}{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}\right)}^{-0.5}} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (/ (pow k -0.5) (pow (pow (* n (* PI 2.0)) (- 1.0 k)) -0.5)))

C

double code(double k, double n) {
	return pow(k, -0.5) / pow(pow((n * (((double) M_PI) * 2.0)), (1.0 - k)), -0.5);
}

Java

public static double code(double k, double n) {
	return Math.pow(k, -0.5) / Math.pow(Math.pow((n * (Math.PI * 2.0)), (1.0 - k)), -0.5);
}

Python

def code(k, n):
	return math.pow(k, -0.5) / math.pow(math.pow((n * (math.pi * 2.0)), (1.0 - k)), -0.5)

Julia

function code(k, n)
	return Float64((k ^ -0.5) / ((Float64(n * Float64(pi * 2.0)) ^ Float64(1.0 - k)) ^ -0.5))
end

MATLAB

function tmp = code(k, n)
	tmp = (k ^ -0.5) / (((n * (pi * 2.0)) ^ (1.0 - k)) ^ -0.5);
end

Wolfram

code[k_, n_] := N[(N[Power[k, -0.5], $MachinePrecision] / N[Power[N[Power[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision], -0.5], $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. *-commutative 99.5%

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

   2. associate-*r* 99.5%

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

   3. associate-/r/ 99.4%

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

   4. add-sqr-sqrt 99.3%

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

   5. sqrt-unprod 99.4%

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

   6. associate-*r* 99.4%

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

   7. *-commutative 99.4%

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

   8. associate-*r* 99.4%

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

   9. *-commutative 99.4%

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

   10. pow-prod-up 99.4%

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

3. Applied egg-rr 99.4%

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

   1. pow-sub 99.6%

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

   2. pow1 99.6%

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

5. Applied egg-rr 99.6%

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

   1. inv-pow 99.6%

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

   2. div-inv 99.6%

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

   3. unpow-prod-down 99.6%

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

   4. inv-pow 99.6%

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

   5. pow1/2 99.6%

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

   6. pow-flip 99.7%

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

   7. metadata-eval 99.7%

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

7. Applied egg-rr 99.5%

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

   1. unpow-1 99.5%

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

   2. associate-*r/ 99.5%

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

   3. *-rgt-identity 99.5%

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

   4. *-commutative 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-*l* 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 10^{-42}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\pi \cdot \frac{2}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1e-42)
   (* (sqrt n) (sqrt (* PI (/ 2.0 k))))
   (pow (/ k (pow (* n (* PI 2.0)) (- 1.0 k))) -0.5)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1e-42) {
		tmp = sqrt(n) * sqrt((((double) M_PI) * (2.0 / k)));
	} else {
		tmp = pow((k / pow((n * (((double) M_PI) * 2.0)), (1.0 - k))), -0.5);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1e-42) {
		tmp = Math.sqrt(n) * Math.sqrt((Math.PI * (2.0 / k)));
	} else {
		tmp = Math.pow((k / Math.pow((n * (Math.PI * 2.0)), (1.0 - k))), -0.5);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1e-42:
		tmp = math.sqrt(n) * math.sqrt((math.pi * (2.0 / k)))
	else:
		tmp = math.pow((k / math.pow((n * (math.pi * 2.0)), (1.0 - k))), -0.5)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1e-42)
		tmp = Float64(sqrt(n) * sqrt(Float64(pi * Float64(2.0 / k))));
	else
		tmp = Float64(k / (Float64(n * Float64(pi * 2.0)) ^ Float64(1.0 - k))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1e-42)
		tmp = sqrt(n) * sqrt((pi * (2.0 / k)));
	else
		tmp = (k / ((n * (pi * 2.0)) ^ (1.0 - k))) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1e-42], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(k / N[Power[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000004e-42

   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. Taylor expanded in n around 0 92.1%

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

   3. Taylor expanded in k around 0 92.1%

      \[\leadsto \sqrt{\frac{1}{k}} \cdot \color{blue}{e^{0.5 \cdot \left(\log \left(2 \cdot \pi\right) + \log n\right)}} \]
   4. Step-by-step derivation

      1. *-commutative 92.1%

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

      2. exp-prod 92.1%

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

      3. log-prod 92.8%

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

      4. associate-*r* 92.8%

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

      5. log-prod 92.3%

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

      6. exp-sum 92.5%

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

      7. rem-exp-log 92.5%

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

      8. rem-exp-log 99.4%

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

      9. *-commutative 99.4%

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

      10. associate-*r* 99.4%

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

      11. unpow1/2 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-*l* 99.4%

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

   5. Simplified 99.4%

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

      1. sqrt-unprod 72.5%

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

      2. pow1/2 72.5%

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

      3. *-commutative 72.5%

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

      4. associate-*r* 72.5%

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

   7. Applied egg-rr 72.5%

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

      1. *-commutative 72.5%

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

      2. unpow1/2 72.5%

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

      3. *-commutative 72.5%

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

      4. associate-*l* 72.5%

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

      5. *-commutative 72.5%

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

      6. *-commutative 72.5%

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

      7. associate-*l/ 72.5%

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

      8. metadata-eval 72.5%

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

   9. Simplified 72.5%

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

       1. associate-*l* 72.5%

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

       2. sqrt-prod 99.4%

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

   11. Applied egg-rr 99.4%

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

   if 1.00000000000000004e-42 < 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. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*r* 99.6%

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

      4. div-inv 99.5%

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

      5. expm1-log1p-u 99.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

      6. expm1-udef 94.6%

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

   3. Applied egg-rr 94.6%

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

      1. expm1-def 99.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

      2. expm1-log1p 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*r* 99.6%

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

      5. *-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. pow-sub 100.0%

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

      2. pow1 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. associate-/l* 99.9%

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

      2. associate-/r/ 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

   9. Simplified 100.0%

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

       1. sqrt-div 99.9%

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

       2. *-commutative 99.9%

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

       3. associate-*r* 99.9%

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

   11. Applied egg-rr 99.9%

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

       1. clear-num 99.9%

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

       2. associate-*l/ 99.9%

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

       3. *-commutative 99.9%

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

       4. pow1 99.9%

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

       5. *-commutative 99.9%

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

       6. associate-*r* 99.9%

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

       7. pow-sub 99.6%

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

       8. sqrt-div 99.6%

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

       9. pow1/2 99.6%

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

       10. pow-flip 99.6%

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

   13. Applied egg-rr 99.6%

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

   15. Simplified 99.6%

       \[\leadsto \color{blue}{{\left(\frac{k}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-42}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\pi \cdot \frac{2}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \]

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {k}^{-0.5} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (pow k -0.5) (pow (* n (* PI 2.0)) (/ (- 1.0 k) 2.0))))

C

double code(double k, double n) {
	return pow(k, -0.5) * pow((n * (((double) M_PI) * 2.0)), ((1.0 - k) / 2.0));
}

Java

public static double code(double k, double n) {
	return Math.pow(k, -0.5) * Math.pow((n * (Math.PI * 2.0)), ((1.0 - k) / 2.0));
}

Python

def code(k, n):
	return math.pow(k, -0.5) * math.pow((n * (math.pi * 2.0)), ((1.0 - k) / 2.0))

Julia

function code(k, n)
	return Float64((k ^ -0.5) * (Float64(n * Float64(pi * 2.0)) ^ Float64(Float64(1.0 - k) / 2.0)))
end

MATLAB

function tmp = code(k, n)
	tmp = (k ^ -0.5) * ((n * (pi * 2.0)) ^ ((1.0 - k) / 2.0));
end

Wolfram

code[k_, n_] := N[(N[Power[k, -0.5], $MachinePrecision] * N[Power[N[(n * N[(Pi * 2.0), $MachinePrecision]), $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. Step-by-step derivation

   1. expm1-log1p-u 96.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{k}}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. expm1-udef 74.1%

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

   3. inv-pow 74.1%

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

   4. sqrt-pow2 74.1%

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

   5. metadata-eval 74.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left({k}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

3. Applied egg-rr 74.1%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-0.5}\right)} - 1\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
4. Step-by-step derivation

   1. expm1-def 96.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. expm1-log1p 99.5%

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\pi \cdot \frac{2}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3e-34)
   (* (sqrt n) (sqrt (* PI (/ 2.0 k))))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3e-34) {
		tmp = sqrt(n) * sqrt((((double) M_PI) * (2.0 / k)));
	} else {
		tmp = sqrt((pow((((double) M_PI) * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3e-34) {
		tmp = Math.sqrt(n) * Math.sqrt((Math.PI * (2.0 / k)));
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3e-34:
		tmp = math.sqrt(n) * math.sqrt((math.pi * (2.0 / k)))
	else:
		tmp = math.sqrt((math.pow((math.pi * (2.0 * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3e-34)
		tmp = Float64(sqrt(n) * sqrt(Float64(pi * Float64(2.0 / k))));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(2.0 * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3e-34)
		tmp = sqrt(n) * sqrt((pi * (2.0 / k)));
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3e-34], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3e-34

   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. Taylor expanded in n around 0 92.2%

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

   3. Taylor expanded in k around 0 92.2%

      \[\leadsto \sqrt{\frac{1}{k}} \cdot \color{blue}{e^{0.5 \cdot \left(\log \left(2 \cdot \pi\right) + \log n\right)}} \]
   4. Step-by-step derivation

      1. *-commutative 92.2%

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

      2. exp-prod 92.2%

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

      3. log-prod 92.8%

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

      4. associate-*r* 92.8%

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

      5. log-prod 92.3%

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

      6. exp-sum 92.6%

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

      7. rem-exp-log 92.6%

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

      8. rem-exp-log 99.4%

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

      9. *-commutative 99.4%

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

      10. associate-*r* 99.4%

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

      11. unpow1/2 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-*l* 99.4%

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

   5. Simplified 99.4%

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

      1. sqrt-unprod 73.6%

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

      2. pow1/2 73.6%

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

      3. *-commutative 73.6%

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

      4. associate-*r* 73.6%

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

   7. Applied egg-rr 73.6%

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

      1. *-commutative 73.6%

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

      2. unpow1/2 73.6%

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

      3. *-commutative 73.6%

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

      4. associate-*l* 73.6%

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

      5. *-commutative 73.6%

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

      6. *-commutative 73.6%

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

      7. associate-*l/ 73.6%

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

      8. metadata-eval 73.6%

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

   9. Simplified 73.6%

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

       1. associate-*l* 73.5%

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

       2. sqrt-prod 99.4%

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

   11. Applied egg-rr 99.4%

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

   if 3e-34 < 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. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*r* 99.6%

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

      4. div-inv 99.6%

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

      5. expm1-log1p-u 99.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

      6. expm1-udef 95.8%

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

   3. Applied egg-rr 95.8%

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

      1. expm1-def 99.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

      2. expm1-log1p 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*r* 99.6%

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

      5. *-commutative 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\pi \cdot \frac{2}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 6: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $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. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 7: 49.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+236}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\pi \cdot \frac{2}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\pi \cdot \frac{n}{\frac{k}{2}}\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5e+236)
   (* (sqrt n) (sqrt (* PI (/ 2.0 k))))
   (cbrt (pow (* PI (/ n (/ k 2.0))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5e+236) {
		tmp = sqrt(n) * sqrt((((double) M_PI) * (2.0 / k)));
	} else {
		tmp = cbrt(pow((((double) M_PI) * (n / (k / 2.0))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 5e+236) {
		tmp = Math.sqrt(n) * Math.sqrt((Math.PI * (2.0 / k)));
	} else {
		tmp = Math.cbrt(Math.pow((Math.PI * (n / (k / 2.0))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5e+236)
		tmp = Float64(sqrt(n) * sqrt(Float64(pi * Float64(2.0 / k))));
	else
		tmp = cbrt((Float64(pi * Float64(n / Float64(k / 2.0))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5e+236], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(Pi * N[(n / N[(k / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.9999999999999997e236

   1. Initial program 99.4%

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

   2. Taylor expanded in n around 0 95.7%

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

   3. Taylor expanded in k around 0 50.4%

      \[\leadsto \sqrt{\frac{1}{k}} \cdot \color{blue}{e^{0.5 \cdot \left(\log \left(2 \cdot \pi\right) + \log n\right)}} \]
   4. Step-by-step derivation

      1. *-commutative 50.4%

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

      2. exp-prod 50.4%

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

      3. log-prod 50.7%

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

      4. associate-*r* 50.7%

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

      5. log-prod 50.5%

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

      6. exp-sum 50.6%

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

      7. rem-exp-log 50.6%

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

      8. rem-exp-log 54.1%

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

      9. *-commutative 54.1%

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

      10. associate-*r* 54.1%

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

      11. unpow1/2 54.1%

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

      12. *-commutative 54.1%

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

      13. associate-*l* 54.1%

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

   5. Simplified 54.1%

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

      1. sqrt-unprod 41.7%

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

      2. pow1/2 41.7%

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

      3. *-commutative 41.7%

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

      4. associate-*r* 41.7%

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

   7. Applied egg-rr 41.7%

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

      1. *-commutative 41.7%

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

      2. unpow1/2 41.7%

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

      3. *-commutative 41.7%

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

      4. associate-*l* 41.7%

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

      5. *-commutative 41.7%

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

      6. *-commutative 41.7%

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

      7. associate-*l/ 41.7%

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

      8. metadata-eval 41.7%

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

   9. Simplified 41.7%

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

       1. associate-*l* 41.7%

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

       2. sqrt-prod 54.1%

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

   11. Applied egg-rr 54.1%

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

   if 4.9999999999999997e236 < 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. Taylor expanded in n around 0 100.0%

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

   3. Taylor expanded in k around 0 2.9%

      \[\leadsto \sqrt{\frac{1}{k}} \cdot \color{blue}{e^{0.5 \cdot \left(\log \left(2 \cdot \pi\right) + \log n\right)}} \]
   4. Step-by-step derivation

      1. *-commutative 2.9%

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

      2. exp-prod 2.9%

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

      3. log-prod 2.9%

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

      4. associate-*r* 2.9%

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

      5. log-prod 2.9%

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

      6. exp-sum 2.9%

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

      7. rem-exp-log 2.9%

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

      8. rem-exp-log 2.9%

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

      9. *-commutative 2.9%

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

      10. associate-*r* 2.9%

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

      11. unpow1/2 2.9%

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

      12. *-commutative 2.9%

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

      13. associate-*l* 2.9%

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

   5. Simplified 2.9%

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

      1. sqrt-unprod 2.8%

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

      2. pow1/2 2.8%

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

      3. *-commutative 2.8%

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

      4. associate-*r* 2.8%

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

   7. Applied egg-rr 2.8%

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

      1. *-commutative 2.8%

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

      2. unpow1/2 2.8%

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

      3. *-commutative 2.8%

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

      4. associate-*l* 2.8%

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

      5. *-commutative 2.8%

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

      6. *-commutative 2.8%

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

      7. associate-*l/ 2.8%

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

      8. metadata-eval 2.8%

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

   9. Simplified 2.8%

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

       1. add-cbrt-cube 15.3%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\left(n \cdot \pi\right) \cdot \frac{2}{k}} \cdot \sqrt{\left(n \cdot \pi\right) \cdot \frac{2}{k}}\right) \cdot \sqrt{\left(n \cdot \pi\right) \cdot \frac{2}{k}}}} \]

       2. pow1/3 15.3%

          \[\leadsto \color{blue}{{\left(\left(\sqrt{\left(n \cdot \pi\right) \cdot \frac{2}{k}} \cdot \sqrt{\left(n \cdot \pi\right) \cdot \frac{2}{k}}\right) \cdot \sqrt{\left(n \cdot \pi\right) \cdot \frac{2}{k}}\right)}^{0.3333333333333333}} \]

       3. add-sqr-sqrt 15.3%

          \[\leadsto {\left(\color{blue}{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)} \cdot \sqrt{\left(n \cdot \pi\right) \cdot \frac{2}{k}}\right)}^{0.3333333333333333} \]

       4. pow1 15.3%

          \[\leadsto {\left(\color{blue}{{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{1}} \cdot \sqrt{\left(n \cdot \pi\right) \cdot \frac{2}{k}}\right)}^{0.3333333333333333} \]

       5. pow1/2 15.3%

          \[\leadsto {\left({\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{1} \cdot \color{blue}{{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

       6. pow-prod-up 15.3%

          \[\leadsto {\color{blue}{\left({\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

       7. *-commutative 15.3%

          \[\leadsto {\left({\color{blue}{\left(\frac{2}{k} \cdot \left(n \cdot \pi\right)\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

       8. associate-*r* 15.3%

          \[\leadsto {\left({\color{blue}{\left(\left(\frac{2}{k} \cdot n\right) \cdot \pi\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

       9. metadata-eval 15.3%

          \[\leadsto {\left({\left(\left(\frac{2}{k} \cdot n\right) \cdot \pi\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   11. Applied egg-rr 15.3%

       \[\leadsto \color{blue}{{\left({\left(\left(\frac{2}{k} \cdot n\right) \cdot \pi\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   12. Step-by-step derivation

       1. unpow1/3 15.3%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(\frac{2}{k} \cdot n\right) \cdot \pi\right)}^{1.5}}} \]

       2. *-commutative 15.3%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\pi \cdot \left(\frac{2}{k} \cdot n\right)\right)}}^{1.5}} \]

       3. associate-*l/ 15.3%

          \[\leadsto \sqrt[3]{{\left(\pi \cdot \color{blue}{\frac{2 \cdot n}{k}}\right)}^{1.5}} \]

       4. *-commutative 15.3%

          \[\leadsto \sqrt[3]{{\left(\pi \cdot \frac{\color{blue}{n \cdot 2}}{k}\right)}^{1.5}} \]

       5. associate-/l* 15.3%

          \[\leadsto \sqrt[3]{{\left(\pi \cdot \color{blue}{\frac{n}{\frac{k}{2}}}\right)}^{1.5}} \]

   13. Simplified 15.3%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\pi \cdot \frac{n}{\frac{k}{2}}\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+236}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\pi \cdot \frac{2}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\pi \cdot \frac{n}{\frac{k}{2}}\right)}^{1.5}}\\ \end{array} \]

Alternative 8: 48.5% 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. Taylor expanded in n around 0 96.3%

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

3. Taylor expanded in k around 0 43.3%

   \[\leadsto \sqrt{\frac{1}{k}} \cdot \color{blue}{e^{0.5 \cdot \left(\log \left(2 \cdot \pi\right) + \log n\right)}} \]
4. Step-by-step derivation

   1. *-commutative 43.3%

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

   2. exp-prod 43.3%

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

   3. log-prod 43.6%

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

   4. associate-*r* 43.6%

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

   5. log-prod 43.4%

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

   6. exp-sum 43.5%

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

   7. rem-exp-log 43.5%

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

   8. rem-exp-log 46.5%

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

   9. *-commutative 46.5%

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

   10. associate-*r* 46.5%

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

   11. unpow1/2 46.5%

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

   12. *-commutative 46.5%

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

   13. associate-*l* 46.5%

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

5. Simplified 46.5%

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

   1. sqrt-unprod 36.0%

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

   2. pow1/2 36.0%

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

   3. *-commutative 36.0%

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

   4. associate-*r* 36.0%

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

7. Applied egg-rr 36.0%

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

   1. *-commutative 36.0%

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

   2. unpow1/2 36.0%

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

   3. *-commutative 36.0%

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

   4. associate-*l* 36.0%

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

   5. *-commutative 36.0%

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

   6. *-commutative 36.0%

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

   7. associate-*l/ 36.0%

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

   8. metadata-eval 36.0%

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

9. Simplified 36.0%

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

    1. associate-*l* 35.9%

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

    2. sqrt-prod 46.5%

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

11. Applied egg-rr 46.5%

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

12. Final simplification 46.5%

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

Alternative 9: 37.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(Pi * N[(n / N[(k / 2.0), $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. Taylor expanded in n around 0 96.3%

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

3. Taylor expanded in k around 0 43.3%

   \[\leadsto \sqrt{\frac{1}{k}} \cdot \color{blue}{e^{0.5 \cdot \left(\log \left(2 \cdot \pi\right) + \log n\right)}} \]
4. Step-by-step derivation

   1. *-commutative 43.3%

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

   2. exp-prod 43.3%

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

   3. log-prod 43.6%

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

   4. associate-*r* 43.6%

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

   5. log-prod 43.4%

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

   6. exp-sum 43.5%

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

   7. rem-exp-log 43.5%

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

   8. rem-exp-log 46.5%

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

   9. *-commutative 46.5%

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

   10. associate-*r* 46.5%

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

   11. unpow1/2 46.5%

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

   12. *-commutative 46.5%

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

   13. associate-*l* 46.5%

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

5. Simplified 46.5%

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

   1. sqrt-unprod 36.0%

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

   2. pow1/2 36.0%

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

   3. *-commutative 36.0%

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

   4. associate-*r* 36.0%

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

7. Applied egg-rr 36.0%

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

   1. *-commutative 36.0%

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

   2. unpow1/2 36.0%

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

   3. *-commutative 36.0%

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

   4. associate-*l* 36.0%

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

   5. *-commutative 36.0%

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

   6. *-commutative 36.0%

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

   7. associate-*l/ 36.0%

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

   8. metadata-eval 36.0%

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

9. Simplified 36.0%

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

10. Taylor expanded in n around 0 36.0%

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

    1. associate-*r/ 36.0%

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

    2. associate-*r* 36.0%

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

    3. *-commutative 36.0%

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

    4. associate-*r/ 35.9%

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

    5. *-commutative 35.9%

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

    6. associate-/l* 35.9%

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

12. Simplified 35.9%

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

13. Final simplification 35.9%

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

Alternative 10: 37.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(Pi * n), $MachinePrecision] * N[(2.0 / 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. Taylor expanded in n around 0 96.3%

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

3. Taylor expanded in k around 0 43.3%

   \[\leadsto \sqrt{\frac{1}{k}} \cdot \color{blue}{e^{0.5 \cdot \left(\log \left(2 \cdot \pi\right) + \log n\right)}} \]
4. Step-by-step derivation

   1. *-commutative 43.3%

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

   2. exp-prod 43.3%

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

   3. log-prod 43.6%

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

   4. associate-*r* 43.6%

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

   5. log-prod 43.4%

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

   6. exp-sum 43.5%

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

   7. rem-exp-log 43.5%

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

   8. rem-exp-log 46.5%

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

   9. *-commutative 46.5%

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

   10. associate-*r* 46.5%

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

   11. unpow1/2 46.5%

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

   12. *-commutative 46.5%

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

   13. associate-*l* 46.5%

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

5. Simplified 46.5%

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

   1. sqrt-unprod 36.0%

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

   2. pow1/2 36.0%

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

   3. *-commutative 36.0%

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

   4. associate-*r* 36.0%

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

7. Applied egg-rr 36.0%

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

   1. *-commutative 36.0%

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

   2. unpow1/2 36.0%

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

   3. *-commutative 36.0%

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

   4. associate-*l* 36.0%

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

   5. *-commutative 36.0%

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

   6. *-commutative 36.0%

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

   7. associate-*l/ 36.0%

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

   8. metadata-eval 36.0%

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

9. Simplified 36.0%

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

10. Final simplification 36.0%

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

Alternative 11: 37.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(k, n)
	return sqrt(Float64(Float64(n * Float64(pi * 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 * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision] / 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. Taylor expanded in n around 0 96.3%

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

3. Taylor expanded in k around 0 43.3%

   \[\leadsto \sqrt{\frac{1}{k}} \cdot \color{blue}{e^{0.5 \cdot \left(\log \left(2 \cdot \pi\right) + \log n\right)}} \]
4. Step-by-step derivation

   1. *-commutative 43.3%

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

   2. exp-prod 43.3%

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

   3. log-prod 43.6%

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

   4. associate-*r* 43.6%

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

   5. log-prod 43.4%

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

   6. exp-sum 43.5%

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

   7. rem-exp-log 43.5%

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

   8. rem-exp-log 46.5%

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

   9. *-commutative 46.5%

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

   10. associate-*r* 46.5%

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

   11. unpow1/2 46.5%

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

   12. *-commutative 46.5%

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

   13. associate-*l* 46.5%

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

5. Simplified 46.5%

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

   1. sqrt-unprod 36.0%

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

   2. pow1/2 36.0%

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

   3. *-commutative 36.0%

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

   4. associate-*r* 36.0%

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

7. Applied egg-rr 36.0%

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

   1. *-commutative 36.0%

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

   2. unpow1/2 36.0%

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

   3. *-commutative 36.0%

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

   4. associate-*l* 36.0%

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

   5. *-commutative 36.0%

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

   6. *-commutative 36.0%

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

   7. associate-*l/ 36.0%

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

   8. metadata-eval 36.0%

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

9. Simplified 36.0%

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

10. Taylor expanded in n around 0 36.0%

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

    1. associate-*r/ 36.0%

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

    2. associate-*r* 36.0%

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

    3. *-commutative 36.0%

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

    4. associate-*r* 36.0%

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

    5. *-commutative 36.0%

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

12. Simplified 36.0%

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

13. Final simplification 36.0%

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

Reproduce

herbie shell --seed 2023200 
(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))))