Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 15.0s

Alternatives: 15

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. 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. *-un-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. associate-*r* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

   6. pow-div 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. *-rgt-identity 99.7%

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

   2. *-commutative 99.7%

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

   3. times-frac 99.7%

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

   4. associate-*l/ 99.7%

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

   5. *-lft-identity 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

6. Simplified 99.7%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.5%

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

   2. *-un-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. associate-*r* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

   6. pow-div 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-/l/ 99.7%

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

   2. unpow1/2 99.7%

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

   3. metadata-eval 99.7%

      \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{{k}^{\color{blue}{\left(2 \cdot 0.25\right)}}} \]

   4. pow-sqr 99.5%

      \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{{k}^{0.25} \cdot {k}^{0.25}}} \]

   5. fabs-sqr 99.5%

      \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{\left|{k}^{0.25} \cdot {k}^{0.25}\right|}} \]

   6. pow-sqr 99.7%

      \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\color{blue}{{k}^{\left(2 \cdot 0.25\right)}}\right|} \]

   7. metadata-eval 99.7%

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

   8. unpow1/2 99.7%

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

   9. fabs-neg 99.7%

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

   10. neg-mul-1 99.7%

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

   11. rem-square-sqrt 0.0%

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

   12. unpow1/2 0.0%

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

   13. metadata-eval 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot {k}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right|} \]

   14. pow-sqr 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \color{blue}{\left({k}^{0.25} \cdot {k}^{0.25}\right)}\right|} \]

   15. unswap-sqr 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\color{blue}{\left(\sqrt{-1} \cdot {k}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {k}^{0.25}\right)}\right|} \]

   16. fabs-sqr 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{\left(\sqrt{-1} \cdot {k}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {k}^{0.25}\right)}} \]

   17. unswap-sqr 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \left({k}^{0.25} \cdot {k}^{0.25}\right)}} \]

   18. rem-square-sqrt 26.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{-1} \cdot \left({k}^{0.25} \cdot {k}^{0.25}\right)} \]

   19. pow-sqr 26.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{-1 \cdot \color{blue}{{k}^{\left(2 \cdot 0.25\right)}}} \]

   20. metadata-eval 26.0%

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

   21. unpow1/2 26.0%

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

6. Simplified 99.7%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot n\right)\\ \mathbf{if}\;k \leq 10^{-65}:\\ \;\;\;\;\frac{\sqrt{t\_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{t\_0}^{\left(1 - k\right)} \cdot \frac{1}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* 2.0 n))))
   (if (<= k 1e-65)
     (/ (sqrt t_0) (sqrt k))
     (sqrt (* (pow t_0 (- 1.0 k)) (/ 1.0 k))))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (2.0 * n);
	double tmp;
	if (k <= 1e-65) {
		tmp = sqrt(t_0) / sqrt(k);
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) * (1.0 / k)));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (2.0 * n);
	double tmp;
	if (k <= 1e-65) {
		tmp = Math.sqrt(t_0) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) * (1.0 / k)));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.pi * (2.0 * n)
	tmp = 0
	if k <= 1e-65:
		tmp = math.sqrt(t_0) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) * (1.0 / k)))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(2.0 * n))
	tmp = 0.0
	if (k <= 1e-65)
		tmp = Float64(sqrt(t_0) / sqrt(k));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) * Float64(1.0 / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = pi * (2.0 * n);
	tmp = 0.0;
	if (k <= 1e-65)
		tmp = sqrt(t_0) / sqrt(k);
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) * (1.0 / k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1e-65], N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 9.99999999999999923e-66

   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. Add Preprocessing

   3. Taylor expanded in k around 0 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-/l* 69.9%

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

   5. Simplified 69.9%

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

      1. pow1 69.9%

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

      2. *-commutative 69.9%

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

      3. sqrt-unprod 70.2%

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

   7. Applied egg-rr 70.2%

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

      1. unpow1 70.2%

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

      2. *-commutative 70.2%

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

      3. associate-*r* 70.2%

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

      4. *-commutative 70.2%

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

   9. Simplified 70.2%

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

       1. associate-*r/ 70.1%

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

       2. *-commutative 70.1%

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

       3. associate-*r* 70.1%

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

       4. sqrt-div 99.5%

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

       5. associate-*r* 99.5%

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

       6. *-commutative 99.5%

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

       7. add-sqr-sqrt 99.0%

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

       8. *-un-lft-identity 99.0%

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

       9. times-frac 99.1%

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

   11. Applied egg-rr 99.0%

       \[\leadsto \color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.25}}{1} \cdot \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.25}}{\sqrt{k}}} \]
   12. Step-by-step derivation

       1. /-rgt-identity 99.0%

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

       2. associate-*r/ 99.1%

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

       3. pow-sqr 99.5%

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

       4. metadata-eval 99.5%

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

       5. unpow1/2 99.5%

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

       6. *-commutative 99.5%

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

   13. Simplified 99.5%

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

   if 9.99999999999999923e-66 < 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
   3. 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. *-un-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. associate-*r* 99.5%

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

      4. div-sub 99.5%

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

      5. metadata-eval 99.5%

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

      6. pow-div 99.9%

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

      7. pow1/2 99.9%

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

      8. associate-/l/ 99.9%

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

      9. div-inv 99.9%

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

      10. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. *-rgt-identity 99.9%

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

      2. *-commutative 99.9%

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

      3. times-frac 99.9%

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

      4. associate-*l/ 99.9%

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

      5. *-lft-identity 99.9%

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

      6. *-commutative 99.9%

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

      7. *-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. add-sqr-sqrt 99.8%

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

      3. sqrt-unprod 99.3%

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

      4. frac-times 99.3%

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

   8. Applied egg-rr 98.9%

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

      1. *-lft-identity 98.9%

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

      2. *-commutative 98.9%

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

   10. Simplified 98.9%

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

       1. div-inv 98.9%

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

       2. *-commutative 98.9%

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

       3. associate-*l* 98.9%

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

   12. Applied egg-rr 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-65}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)} \cdot \frac{1}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5.5e-66)
   (/ (sqrt (* PI (* 2.0 n))) (sqrt k))
   (sqrt (/ (pow (* 2.0 (* n PI)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5.5e-66) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) / sqrt(k);
	} else {
		tmp = sqrt((pow((2.0 * (n * ((double) M_PI))), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 5.5e-66) {
		tmp = Math.sqrt((Math.PI * (2.0 * n))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow((2.0 * (n * Math.PI)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 5.5e-66:
		tmp = math.sqrt((math.pi * (2.0 * n))) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow((2.0 * (n * math.pi)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5.5e-66)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) / sqrt(k));
	else
		tmp = sqrt(Float64((Float64(2.0 * Float64(n * pi)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 5.5e-66)
		tmp = sqrt((pi * (2.0 * n))) / sqrt(k);
	else
		tmp = sqrt((((2.0 * (n * pi)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5.5e-66], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(2.0 * N[(n * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.50000000000000053e-66

   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. Add Preprocessing

   3. Taylor expanded in k around 0 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-/l* 69.9%

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

   5. Simplified 69.9%

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

      1. pow1 69.9%

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

      2. *-commutative 69.9%

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

      3. sqrt-unprod 70.2%

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

   7. Applied egg-rr 70.2%

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

      1. unpow1 70.2%

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

      2. *-commutative 70.2%

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

      3. associate-*r* 70.2%

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

      4. *-commutative 70.2%

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

   9. Simplified 70.2%

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

       1. associate-*r/ 70.1%

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

       2. *-commutative 70.1%

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

       3. associate-*r* 70.1%

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

       4. sqrt-div 99.5%

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

       5. associate-*r* 99.5%

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

       6. *-commutative 99.5%

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

       7. add-sqr-sqrt 99.0%

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

       8. *-un-lft-identity 99.0%

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

       9. times-frac 99.1%

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

   11. Applied egg-rr 99.0%

       \[\leadsto \color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.25}}{1} \cdot \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.25}}{\sqrt{k}}} \]
   12. Step-by-step derivation

       1. /-rgt-identity 99.0%

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

       2. associate-*r/ 99.1%

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

       3. pow-sqr 99.5%

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

       4. metadata-eval 99.5%

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

       5. unpow1/2 99.5%

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

       6. *-commutative 99.5%

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

   13. Simplified 99.5%

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

   if 5.50000000000000053e-66 < 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 98.9%

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

      1. distribute-rgt-in 98.9%

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

      2. metadata-eval 98.9%

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

      3. associate-*l* 98.9%

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

      4. metadata-eval 98.9%

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

      5. *-commutative 98.9%

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

      6. neg-mul-1 98.9%

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

      7. sub-neg 98.9%

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

      8. *-commutative 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.22 \cdot 10^{+76}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot \left(-1 + \mathsf{fma}\left(n, \frac{2}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.22e+76)
   (/ (sqrt (* PI (* 2.0 n))) (sqrt k))
   (sqrt (* PI (+ -1.0 (fma n (/ 2.0 k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.22e+76) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) / sqrt(k);
	} else {
		tmp = sqrt((((double) M_PI) * (-1.0 + fma(n, (2.0 / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.22e+76)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) / sqrt(k));
	else
		tmp = sqrt(Float64(pi * Float64(-1.0 + fma(n, Float64(2.0 / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.22e+76], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(Pi * N[(-1.0 + N[(n * N[(2.0 / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.22000000000000002e76

   1. Initial program 99.1%

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

   3. Taylor expanded in k around 0 63.1%

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

      1. *-commutative 63.1%

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

      2. associate-/l* 63.2%

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

   5. Simplified 63.2%

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

      1. pow1 63.2%

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

      2. *-commutative 63.2%

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

      3. sqrt-unprod 63.4%

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

   7. Applied egg-rr 63.4%

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

      1. unpow1 63.4%

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

      2. *-commutative 63.4%

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

      3. associate-*r* 63.4%

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

      4. *-commutative 63.4%

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

   9. Simplified 63.4%

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

       1. associate-*r/ 63.4%

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

       2. *-commutative 63.4%

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

       3. associate-*r* 63.4%

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

       4. sqrt-div 83.3%

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

       5. associate-*r* 83.3%

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

       6. *-commutative 83.3%

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

       7. add-sqr-sqrt 82.9%

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

       8. *-un-lft-identity 82.9%

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

       9. times-frac 82.9%

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

   11. Applied egg-rr 82.9%

       \[\leadsto \color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.25}}{1} \cdot \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.25}}{\sqrt{k}}} \]
   12. Step-by-step derivation

       1. /-rgt-identity 82.9%

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

       2. associate-*r/ 83.0%

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

       3. pow-sqr 83.3%

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

       4. metadata-eval 83.3%

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

       5. unpow1/2 83.3%

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

       6. *-commutative 83.3%

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

   13. Simplified 83.3%

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

   if 1.22000000000000002e76 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.7%

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

      1. *-commutative 2.7%

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

      2. associate-/l* 2.7%

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

   5. Simplified 2.7%

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

      1. pow1 2.7%

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

      2. *-commutative 2.7%

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

      3. sqrt-unprod 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. *-commutative 2.7%

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

      3. associate-*r* 2.7%

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

      4. *-commutative 2.7%

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

   9. Simplified 2.7%

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

   10. Taylor expanded in n around 0 2.7%

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

       1. associate-*r/ 2.7%

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

       2. associate-*r* 2.7%

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

       3. associate-*l/ 2.7%

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

       4. *-rgt-identity 2.7%

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

       5. associate-*r/ 2.7%

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

       6. *-commutative 2.7%

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

       7. *-commutative 2.7%

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

       8. associate-*r* 2.7%

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

       9. associate-*l/ 2.7%

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

       10. metadata-eval 2.7%

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

   12. Simplified 2.7%

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

       1. expm1-log1p-u 2.7%

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

       2. expm1-undefine 39.8%

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

       3. associate-*l/ 39.8%

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

       4. *-un-lft-identity 39.8%

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

       5. times-frac 39.8%

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

       6. metadata-eval 39.8%

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

   14. Applied egg-rr 39.8%

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

       1. sub-neg 39.8%

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

       2. metadata-eval 39.8%

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

       3. +-commutative 39.8%

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

       4. log1p-undefine 39.8%

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

       5. rem-exp-log 39.8%

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

       6. +-commutative 39.8%

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

       7. associate-*r/ 39.8%

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

       8. *-commutative 39.8%

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

       9. associate-/l* 39.8%

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

       10. fma-define 39.8%

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

   16. Simplified 39.8%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.22 \cdot 10^{+76}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot \left(-1 + \mathsf{fma}\left(n, \frac{2}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{+246}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \frac{n \cdot \pi}{k}\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.2e+246)
   (/ (sqrt (* PI (* 2.0 n))) (sqrt k))
   (cbrt (pow (* 2.0 (/ (* n PI) k)) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.2e+246) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) / sqrt(k);
	} else {
		tmp = cbrt(pow((2.0 * ((n * ((double) M_PI)) / k)), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.2e+246) {
		tmp = Math.sqrt((Math.PI * (2.0 * n))) / Math.sqrt(k);
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * ((n * Math.PI) / k)), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.2e+246)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) / sqrt(k));
	else
		tmp = cbrt((Float64(2.0 * Float64(Float64(n * pi) / k)) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.2e+246], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(N[(n * Pi), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.2e246

   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. Add Preprocessing

   3. Taylor expanded in k around 0 44.5%

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

      1. *-commutative 44.5%

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

      2. associate-/l* 44.5%

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

   5. Simplified 44.5%

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

      1. pow1 44.5%

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

      2. *-commutative 44.5%

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

      3. sqrt-unprod 44.7%

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

   7. Applied egg-rr 44.7%

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

      1. unpow1 44.7%

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

      2. *-commutative 44.7%

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

      3. associate-*r* 44.7%

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

      4. *-commutative 44.7%

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

   9. Simplified 44.7%

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

       1. associate-*r/ 44.7%

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

       2. *-commutative 44.7%

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

       3. associate-*r* 44.7%

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

       4. sqrt-div 58.5%

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

       5. associate-*r* 58.5%

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

       6. *-commutative 58.5%

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

       7. add-sqr-sqrt 58.2%

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

       8. *-un-lft-identity 58.2%

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

       9. times-frac 58.2%

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

   11. Applied egg-rr 58.2%

       \[\leadsto \color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.25}}{1} \cdot \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.25}}{\sqrt{k}}} \]
   12. Step-by-step derivation

       1. /-rgt-identity 58.2%

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

       2. associate-*r/ 58.2%

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

       3. pow-sqr 58.5%

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

       4. metadata-eval 58.5%

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

       5. unpow1/2 58.5%

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

       6. *-commutative 58.5%

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

   13. Simplified 58.5%

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

   if 4.2e246 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.9%

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

      1. *-commutative 2.9%

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

      2. associate-/l* 2.9%

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

   5. Simplified 2.9%

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

      1. pow1 2.9%

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

      2. *-commutative 2.9%

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

      3. sqrt-unprod 2.9%

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

   7. Applied egg-rr 2.9%

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

      1. unpow1 2.9%

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

      2. *-commutative 2.9%

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

      3. associate-*r* 2.9%

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

      4. *-commutative 2.9%

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

   9. Simplified 2.9%

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

       1. add-cbrt-cube 17.9%

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

       2. add-sqr-sqrt 17.9%

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

       3. associate-*r/ 17.9%

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

       4. *-commutative 17.9%

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

       5. associate-*r* 17.9%

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

       6. pow1 17.9%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{1}} \cdot \sqrt{\left(n \cdot 2\right) \cdot \frac{\pi}{k}}} \]

       7. pow1/2 17.9%

          \[\leadsto \sqrt[3]{{\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{1} \cdot \color{blue}{{\left(\left(n \cdot 2\right) \cdot \frac{\pi}{k}\right)}^{0.5}}} \]

       8. associate-*r/ 17.9%

          \[\leadsto \sqrt[3]{{\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{1} \cdot {\color{blue}{\left(\frac{\left(n \cdot 2\right) \cdot \pi}{k}\right)}}^{0.5}} \]

       9. *-commutative 17.9%

          \[\leadsto \sqrt[3]{{\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{1} \cdot {\left(\frac{\color{blue}{\left(2 \cdot n\right)} \cdot \pi}{k}\right)}^{0.5}} \]

       10. associate-*r* 17.9%

           \[\leadsto \sqrt[3]{{\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{1} \cdot {\left(\frac{\color{blue}{2 \cdot \left(n \cdot \pi\right)}}{k}\right)}^{0.5}} \]

       11. pow-prod-up 17.9%

           \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{\left(1 + 0.5\right)}}} \]

       12. associate-/l* 17.9%

           \[\leadsto \sqrt[3]{{\color{blue}{\left(2 \cdot \frac{n \cdot \pi}{k}\right)}}^{\left(1 + 0.5\right)}} \]

       13. *-commutative 17.9%

           \[\leadsto \sqrt[3]{{\left(2 \cdot \frac{\color{blue}{\pi \cdot n}}{k}\right)}^{\left(1 + 0.5\right)}} \]

       14. metadata-eval 17.9%

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

   11. Applied egg-rr 17.9%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{+246}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \frac{n \cdot \pi}{k}\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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. associate-*l* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 8: 50.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.3%

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

   1. *-commutative 39.3%

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

   2. associate-/l* 39.3%

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

5. Simplified 39.3%

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

   1. pow1 39.3%

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

   2. *-commutative 39.3%

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

   3. sqrt-unprod 39.5%

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

7. Applied egg-rr 39.5%

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

   1. unpow1 39.5%

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

   2. *-commutative 39.5%

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

   3. associate-*r* 39.5%

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

   4. *-commutative 39.5%

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

9. Simplified 39.5%

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

    1. associate-*r/ 39.4%

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

    2. *-commutative 39.4%

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

    3. associate-*r* 39.4%

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

    4. sqrt-div 51.5%

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

    5. associate-*r* 51.5%

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

    6. *-commutative 51.5%

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

    7. add-sqr-sqrt 51.3%

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

    8. *-un-lft-identity 51.3%

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

    9. times-frac 51.3%

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

11. Applied egg-rr 51.3%

    \[\leadsto \color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.25}}{1} \cdot \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.25}}{\sqrt{k}}} \]
12. Step-by-step derivation

    1. /-rgt-identity 51.3%

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

    2. associate-*r/ 51.3%

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

    3. pow-sqr 51.5%

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

    4. metadata-eval 51.5%

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

    5. unpow1/2 51.5%

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

    6. *-commutative 51.5%

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

13. Simplified 51.5%

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

14. Final simplification 51.5%

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

Alternative 9: 50.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.3%

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

   1. *-commutative 39.3%

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

   2. associate-/l* 39.3%

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

5. Simplified 39.3%

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

   1. pow1 39.3%

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

   2. *-commutative 39.3%

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

   3. sqrt-unprod 39.5%

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

7. Applied egg-rr 39.5%

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

   1. unpow1 39.5%

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

   2. *-commutative 39.5%

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

   3. associate-*r* 39.5%

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

   4. *-commutative 39.5%

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

9. Simplified 39.5%

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

    1. *-commutative 39.5%

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

    2. *-commutative 39.5%

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

    3. sqrt-prod 51.5%

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

11. Applied egg-rr 51.5%

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

    1. *-commutative 51.5%

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

13. Simplified 51.5%

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

14. Final simplification 51.5%

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

Alternative 10: 50.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.3%

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

   1. *-commutative 39.3%

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

   2. associate-/l* 39.3%

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

5. Simplified 39.3%

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

   1. pow1 39.3%

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

   2. *-commutative 39.3%

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

   3. sqrt-unprod 39.5%

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

7. Applied egg-rr 39.5%

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

   1. unpow1 39.5%

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

   2. *-commutative 39.5%

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

   3. associate-*r* 39.5%

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

   4. *-commutative 39.5%

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

9. Simplified 39.5%

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

    1. sqrt-prod 51.5%

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

    2. *-commutative 51.5%

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

    3. sqrt-prod 51.4%

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

    4. associate-*r* 51.4%

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

    5. *-commutative 51.4%

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

    6. associate-*r* 51.5%

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

    7. sqrt-unprod 51.5%

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

11. Applied egg-rr 51.5%

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

Alternative 11: 50.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.3%

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

   1. *-commutative 39.3%

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

   2. associate-/l* 39.3%

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

5. Simplified 39.3%

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

   1. pow1 39.3%

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

   2. *-commutative 39.3%

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

   3. sqrt-unprod 39.5%

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

7. Applied egg-rr 39.5%

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

   1. unpow1 39.5%

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

   2. *-commutative 39.5%

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

   3. associate-*r* 39.5%

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

   4. *-commutative 39.5%

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

9. Simplified 39.5%

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

    1. associate-*l* 39.5%

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

    2. sqrt-prod 51.5%

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

11. Applied egg-rr 51.5%

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

    1. *-commutative 51.5%

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

    2. associate-*l/ 51.5%

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

    3. associate-/l* 51.5%

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

13. Simplified 51.5%

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

Alternative 12: 39.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.3%

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

   1. *-commutative 39.3%

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

   2. associate-/l* 39.3%

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

5. Simplified 39.3%

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

   1. pow1 39.3%

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

   2. *-commutative 39.3%

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

   3. sqrt-unprod 39.5%

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

7. Applied egg-rr 39.5%

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

   1. unpow1 39.5%

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

   2. *-commutative 39.5%

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

   3. associate-*r* 39.5%

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

   4. *-commutative 39.5%

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

9. Simplified 39.5%

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

    1. associate-*r/ 39.4%

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

    2. *-commutative 39.4%

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

    3. associate-*r* 39.4%

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

    4. clear-num 39.4%

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

    5. sqrt-div 39.9%

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

    6. metadata-eval 39.9%

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

    7. *-un-lft-identity 39.9%

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

    8. times-frac 39.9%

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

    9. metadata-eval 39.9%

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

    10. *-commutative 39.9%

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

11. Applied egg-rr 39.9%

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

    1. associate-*r/ 39.9%

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

    2. *-commutative 39.9%

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

    3. times-frac 39.8%

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

13. Simplified 39.8%

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

    1. *-un-lft-identity 39.8%

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

    2. inv-pow 39.8%

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

    3. sqrt-pow2 39.9%

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

    4. *-commutative 39.9%

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

    5. frac-times 39.9%

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

    6. *-un-lft-identity 39.9%

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

    7. times-frac 39.9%

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

    8. metadata-eval 39.9%

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

    9. *-commutative 39.9%

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

    10. metadata-eval 39.9%

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

15. Applied egg-rr 39.9%

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

    1. *-lft-identity 39.9%

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

    2. associate-/r* 40.0%

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

17. Simplified 40.0%

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

Alternative 13: 38.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.3%

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

   1. *-commutative 39.3%

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

   2. associate-/l* 39.3%

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

5. Simplified 39.3%

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

   1. pow1 39.3%

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

   2. *-commutative 39.3%

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

   3. sqrt-unprod 39.5%

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

7. Applied egg-rr 39.5%

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

   1. unpow1 39.5%

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

   2. *-commutative 39.5%

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

   3. associate-*r* 39.5%

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

   4. *-commutative 39.5%

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

9. Simplified 39.5%

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

10. Final simplification 39.5%

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

Alternative 14: 38.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(Pi * N[(N[(2.0 * 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

3. Taylor expanded in k around 0 39.3%

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

   1. *-commutative 39.3%

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

   2. associate-/l* 39.3%

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

5. Simplified 39.3%

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

   1. pow1 39.3%

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

   2. *-commutative 39.3%

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

   3. sqrt-unprod 39.5%

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

7. Applied egg-rr 39.5%

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

   1. unpow1 39.5%

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

   2. *-commutative 39.5%

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

   3. associate-*r* 39.5%

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

   4. *-commutative 39.5%

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

9. Simplified 39.5%

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

10. Taylor expanded in n around 0 39.4%

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

    1. associate-*r/ 39.4%

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

    2. *-commutative 39.4%

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

    3. associate-*l/ 39.4%

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

    4. metadata-eval 39.4%

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

    5. associate-*r/ 39.4%

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

    6. *-commutative 39.4%

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

    7. associate-*l* 39.5%

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

    8. associate-*r/ 39.5%

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

    9. metadata-eval 39.5%

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

    10. associate-*r/ 39.5%

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

12. Simplified 39.5%

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

13. Final simplification 39.5%

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

Alternative 15: 38.6% accurate, 2.0× speedup

Math

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

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.3%

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

   1. *-commutative 39.3%

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

   2. associate-/l* 39.3%

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

5. Simplified 39.3%

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

   1. pow1 39.3%

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

   2. *-commutative 39.3%

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

   3. sqrt-unprod 39.5%

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

7. Applied egg-rr 39.5%

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

   1. unpow1 39.5%

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

   2. *-commutative 39.5%

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

   3. associate-*r* 39.5%

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

   4. *-commutative 39.5%

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

9. Simplified 39.5%

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

10. Taylor expanded in n around 0 39.4%

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

    1. associate-*r/ 39.4%

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

    2. associate-*r* 39.4%

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

    3. associate-*l/ 39.5%

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

    4. *-rgt-identity 39.5%

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

    5. associate-*r/ 39.5%

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

    6. *-commutative 39.5%

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

    7. *-commutative 39.5%

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

    8. associate-*r* 39.5%

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

    9. associate-*l/ 39.5%

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

    10. metadata-eval 39.5%

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

12. Simplified 39.5%

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

13. Final simplification 39.5%

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

Reproduce

herbie shell --seed 2024166 
(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))))