Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 16.9s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / N[Power[t$95$0, N[(k * 0.5), $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-*r* 99.5%

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

   2. div-sub 99.5%

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

   3. metadata-eval 99.5%

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

   4. pow-div 99.6%

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

   5. pow1/2 99.6%

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

   6. associate-*r/ 99.6%

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

   7. pow1/2 99.6%

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

   8. pow-flip 99.7%

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

   9. metadata-eval 99.7%

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

   10. div-inv 99.7%

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

   11. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(2.0 * Pi), $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/ 99.6%

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

   9. div-inv 99.6%

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

   10. metadata-eval 99.6%

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

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k} \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}} \]
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 24.8%

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

   19. pow-sqr 24.8%

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

   20. metadata-eval 24.8%

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

   21. unpow1/2 24.8%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 58.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.6 \cdot 10^{+58}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(2 \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 9.6e+58)
   (* (pow k -0.5) (sqrt (* n (* 2.0 PI))))
   (sqrt (* 2.0 (+ -1.0 (fma n (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 9.6e+58) {
		tmp = pow(k, -0.5) * sqrt((n * (2.0 * ((double) M_PI))));
	} else {
		tmp = sqrt((2.0 * (-1.0 + fma(n, (((double) M_PI) / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 9.6e+58)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(n * Float64(2.0 * pi))));
	else
		tmp = sqrt(Float64(2.0 * Float64(-1.0 + fma(n, Float64(pi / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 9.6e+58], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(-1.0 + N[(n * N[(Pi / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.5999999999999999e58

   1. Initial program 99.2%

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

   3. Taylor expanded in k around 0 60.5%

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

      1. associate-/l* 60.6%

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

   5. Simplified 60.6%

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

      1. pow1 60.6%

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

      2. sqrt-unprod 60.6%

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

      3. clear-num 60.6%

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

      4. un-div-inv 60.7%

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

   7. Applied egg-rr 60.7%

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

      1. unpow1 60.7%

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

      2. *-commutative 60.7%

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

      3. associate-/r/ 60.7%

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

   9. Simplified 60.7%

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

       1. *-commutative 60.7%

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

       2. sqrt-prod 60.5%

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

       3. associate-*l/ 60.5%

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

       4. *-commutative 60.5%

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

       5. sqrt-div 82.2%

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

       6. *-commutative 82.2%

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

       7. *-un-lft-identity 82.2%

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

       8. associate-*l/ 82.2%

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

       9. associate-*r* 82.1%

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

       10. pow1/2 82.1%

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

       11. pow-flip 82.2%

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

       12. metadata-eval 82.2%

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

       13. sqrt-unprod 82.4%

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

       14. *-commutative 82.4%

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

       15. associate-*r* 82.4%

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

       16. *-commutative 82.4%

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

       17. associate-*l* 82.4%

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

   11. Applied egg-rr 82.4%

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

   if 9.5999999999999999e58 < 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. associate-/l* 2.7%

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

   5. Simplified 2.7%

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

      1. pow1 2.7%

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

      2. sqrt-unprod 2.7%

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

      3. clear-num 2.7%

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

      4. un-div-inv 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. *-commutative 2.7%

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

      3. associate-/r/ 2.7%

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

   9. Simplified 2.7%

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

       1. expm1-log1p-u 2.7%

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

       2. expm1-undefine 30.5%

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

       3. *-commutative 30.5%

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

   11. Applied egg-rr 30.5%

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

       1. sub-neg 30.5%

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

       2. metadata-eval 30.5%

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

       3. +-commutative 30.5%

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

       4. log1p-undefine 30.5%

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

       5. rem-exp-log 30.5%

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

       6. +-commutative 30.5%

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

       7. *-commutative 30.5%

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

       8. associate-*l/ 30.5%

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

       9. associate-*r/ 30.5%

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

       10. fma-define 30.5%

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

   13. Simplified 30.5%

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

Alternative 4: 49.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{+262}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(2 \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\pi \cdot \left(n \cdot \frac{2}{k}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.7e+262)
   (* (pow k -0.5) (sqrt (* n (* 2.0 PI))))
   (cbrt (pow (* PI (* n (/ 2.0 k))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.7e+262) {
		tmp = pow(k, -0.5) * sqrt((n * (2.0 * ((double) M_PI))));
	} else {
		tmp = cbrt(pow((((double) M_PI) * (n * (2.0 / k))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.7e+262) {
		tmp = Math.pow(k, -0.5) * Math.sqrt((n * (2.0 * Math.PI)));
	} else {
		tmp = Math.cbrt(Math.pow((Math.PI * (n * (2.0 / k))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.7e+262)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(n * Float64(2.0 * pi))));
	else
		tmp = cbrt((Float64(pi * Float64(n * Float64(2.0 / k))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.7e+262], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(Pi * N[(n * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.7000000000000001e262

   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 42.6%

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

      1. associate-/l* 42.6%

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

   5. Simplified 42.6%

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

      1. pow1 42.6%

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

      2. sqrt-unprod 42.7%

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

      3. clear-num 42.7%

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

      4. un-div-inv 42.7%

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

   7. Applied egg-rr 42.7%

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

      1. unpow1 42.7%

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

      2. *-commutative 42.7%

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

      3. associate-/r/ 42.7%

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

   9. Simplified 42.7%

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

       1. *-commutative 42.7%

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

       2. sqrt-prod 42.6%

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

       3. associate-*l/ 42.6%

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

       4. *-commutative 42.6%

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

       5. sqrt-div 57.6%

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

       6. *-commutative 57.6%

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

       7. *-un-lft-identity 57.6%

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

       8. associate-*l/ 57.6%

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

       9. associate-*r* 57.6%

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

       10. pow1/2 57.6%

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

       11. pow-flip 57.6%

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

       12. metadata-eval 57.6%

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

       13. sqrt-unprod 57.8%

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

       14. *-commutative 57.8%

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

       15. associate-*r* 57.8%

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

       16. *-commutative 57.8%

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

       17. associate-*l* 57.8%

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

   11. Applied egg-rr 57.8%

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

   if 1.7000000000000001e262 < 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 3.2%

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

      1. associate-/l* 3.2%

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

   5. Simplified 3.2%

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

      1. pow1 3.2%

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

      2. sqrt-unprod 3.2%

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

      3. clear-num 3.2%

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

      4. un-div-inv 3.2%

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

   7. Applied egg-rr 3.2%

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

      1. unpow1 3.2%

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

      2. *-commutative 3.2%

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

      3. associate-/r/ 3.2%

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

   9. Simplified 3.2%

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

       1. associate-*l/ 3.2%

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

       2. *-commutative 3.2%

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

       3. associate-*r/ 3.2%

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

       4. *-commutative 3.2%

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

       5. associate-*r* 3.2%

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

       6. add-cbrt-cube 31.8%

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

       7. pow1/3 31.8%

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

   11. Applied egg-rr 31.8%

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

       1. unpow1/3 31.8%

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

       2. associate-/r* 31.8%

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

       3. associate-/r/ 31.8%

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

       4. metadata-eval 31.8%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{2 \cdot 1}}{k} \cdot \left(\pi \cdot n\right)\right)}^{1.5}} \]

       5. associate-*r/ 31.8%

          \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(2 \cdot \frac{1}{k}\right)} \cdot \left(\pi \cdot n\right)\right)}^{1.5}} \]

       6. *-commutative 31.8%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(\pi \cdot n\right) \cdot \left(2 \cdot \frac{1}{k}\right)\right)}}^{1.5}} \]

       7. associate-*l* 31.8%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\pi \cdot \left(n \cdot \left(2 \cdot \frac{1}{k}\right)\right)\right)}}^{1.5}} \]

       8. associate-*r/ 31.8%

          \[\leadsto \sqrt[3]{{\left(\pi \cdot \left(n \cdot \color{blue}{\frac{2 \cdot 1}{k}}\right)\right)}^{1.5}} \]

       9. metadata-eval 31.8%

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

   13. Simplified 31.8%

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

Alternative 5: 49.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{+260}:\\ \;\;\;\;\sqrt{\frac{2}{k}} \cdot \sqrt{\pi \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\pi \cdot \left(n \cdot \frac{2}{k}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.4e+260)
   (* (sqrt (/ 2.0 k)) (sqrt (* PI n)))
   (cbrt (pow (* PI (* n (/ 2.0 k))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.4e+260) {
		tmp = sqrt((2.0 / k)) * sqrt((((double) M_PI) * n));
	} else {
		tmp = cbrt(pow((((double) M_PI) * (n * (2.0 / k))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.4e+260) {
		tmp = Math.sqrt((2.0 / k)) * Math.sqrt((Math.PI * n));
	} else {
		tmp = Math.cbrt(Math.pow((Math.PI * (n * (2.0 / k))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.4e+260)
		tmp = Float64(sqrt(Float64(2.0 / k)) * sqrt(Float64(pi * n)));
	else
		tmp = cbrt((Float64(pi * Float64(n * Float64(2.0 / k))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.4e+260], N[(N[Sqrt[N[(2.0 / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(Pi * N[(n * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.40000000000000023e260

   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 42.6%

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

      1. associate-/l* 42.6%

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

   5. Simplified 42.6%

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

      1. pow1 42.6%

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

      2. sqrt-unprod 42.7%

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

      3. clear-num 42.7%

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

      4. un-div-inv 42.7%

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

   7. Applied egg-rr 42.7%

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

      1. unpow1 42.7%

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

      2. *-commutative 42.7%

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

      3. associate-/r/ 42.7%

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

   9. Simplified 42.7%

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

       1. *-commutative 42.7%

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

       2. clear-num 42.7%

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

       3. un-div-inv 42.7%

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

   11. Applied egg-rr 42.7%

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

       1. clear-num 42.7%

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

       2. associate-/r* 42.7%

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

       3. div-inv 42.7%

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

       4. associate-/r/ 42.6%

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

       5. *-commutative 42.6%

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

       6. sqrt-prod 57.7%

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

       7. *-commutative 57.7%

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

   13. Applied egg-rr 57.7%

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

   if 4.40000000000000023e260 < 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 3.2%

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

      1. associate-/l* 3.2%

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

   5. Simplified 3.2%

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

      1. pow1 3.2%

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

      2. sqrt-unprod 3.2%

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

      3. clear-num 3.2%

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

      4. un-div-inv 3.2%

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

   7. Applied egg-rr 3.2%

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

      1. unpow1 3.2%

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

      2. *-commutative 3.2%

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

      3. associate-/r/ 3.2%

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

   9. Simplified 3.2%

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

       1. associate-*l/ 3.2%

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

       2. *-commutative 3.2%

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

       3. associate-*r/ 3.2%

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

       4. *-commutative 3.2%

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

       5. associate-*r* 3.2%

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

       6. add-cbrt-cube 31.8%

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

       7. pow1/3 31.8%

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

   11. Applied egg-rr 31.8%

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

       1. unpow1/3 31.8%

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

       2. associate-/r* 31.8%

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

       3. associate-/r/ 31.8%

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

       4. metadata-eval 31.8%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{2 \cdot 1}}{k} \cdot \left(\pi \cdot n\right)\right)}^{1.5}} \]

       5. associate-*r/ 31.8%

          \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(2 \cdot \frac{1}{k}\right)} \cdot \left(\pi \cdot n\right)\right)}^{1.5}} \]

       6. *-commutative 31.8%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(\pi \cdot n\right) \cdot \left(2 \cdot \frac{1}{k}\right)\right)}}^{1.5}} \]

       7. associate-*l* 31.8%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\pi \cdot \left(n \cdot \left(2 \cdot \frac{1}{k}\right)\right)\right)}}^{1.5}} \]

       8. associate-*r/ 31.8%

          \[\leadsto \sqrt[3]{{\left(\pi \cdot \left(n \cdot \color{blue}{\frac{2 \cdot 1}{k}}\right)\right)}^{1.5}} \]

       9. metadata-eval 31.8%

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

   13. Simplified 31.8%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{+260}:\\ \;\;\;\;\sqrt{\frac{2}{k}} \cdot \sqrt{\pi \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\pi \cdot \left(n \cdot \frac{2}{k}\right)\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. div-inv 99.5%

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

   2. div-inv 99.5%

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

   3. metadata-eval 99.5%

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

   4. pow1/2 99.5%

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

   5. pow-flip 99.6%

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

   6. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 7: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.5%

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

   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. Add Preprocessing

Alternative 8: 48.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.5%

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

   1. associate-/l* 39.6%

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

5. Simplified 39.6%

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

   1. pow1 39.6%

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

   2. sqrt-unprod 39.6%

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

   3. clear-num 39.6%

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

   4. un-div-inv 39.6%

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

7. Applied egg-rr 39.6%

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

   1. unpow1 39.6%

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

   2. *-commutative 39.6%

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

   3. associate-/r/ 39.6%

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

9. Simplified 39.6%

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

    1. *-commutative 39.6%

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

    2. clear-num 39.6%

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

    3. un-div-inv 39.6%

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

11. Applied egg-rr 39.6%

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

    1. clear-num 39.6%

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

    2. associate-/r* 39.6%

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

    3. div-inv 39.6%

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

    4. associate-/r/ 39.6%

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

    5. *-commutative 39.6%

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

    6. sqrt-prod 53.4%

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

    7. *-commutative 53.4%

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

13. Applied egg-rr 53.4%

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

14. Final simplification 53.4%

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

Alternative 9: 37.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.5%

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

   1. associate-/l* 39.6%

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

5. Simplified 39.6%

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

   1. pow1 39.6%

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

   2. sqrt-unprod 39.6%

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

   3. clear-num 39.6%

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

   4. un-div-inv 39.6%

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

7. Applied egg-rr 39.6%

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

   1. unpow1 39.6%

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

   2. *-commutative 39.6%

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

   3. associate-/r/ 39.6%

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

9. Simplified 39.6%

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

    1. *-commutative 39.6%

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

    2. sqrt-prod 39.5%

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

    3. associate-*l/ 39.5%

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

    4. *-commutative 39.5%

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

    5. sqrt-div 53.4%

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

    6. *-commutative 53.4%

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

    7. *-un-lft-identity 53.4%

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

    8. associate-*l/ 53.3%

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

    9. associate-*r* 53.3%

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

    10. associate-*l/ 53.3%

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

    11. *-un-lft-identity 53.3%

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

    12. clear-num 53.3%

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

    13. sqrt-unprod 53.4%

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

    14. *-commutative 53.4%

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

    15. associate-*r* 53.4%

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

    16. *-commutative 53.4%

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

    17. associate-*l* 53.4%

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

11. Applied egg-rr 53.4%

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

    1. inv-pow 53.4%

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

    2. sqrt-undiv 39.8%

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

    3. sqrt-pow2 39.9%

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

    4. metadata-eval 39.9%

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

13. Applied egg-rr 39.9%

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

Alternative 10: 36.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.5%

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

   1. associate-/l* 39.6%

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

5. Simplified 39.6%

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

   1. pow1 39.6%

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

   2. sqrt-unprod 39.6%

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

   3. clear-num 39.6%

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

   4. un-div-inv 39.6%

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

7. Applied egg-rr 39.6%

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

   1. unpow1 39.6%

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

   2. *-commutative 39.6%

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

   3. associate-/r/ 39.6%

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

9. Simplified 39.6%

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

10. Final simplification 39.6%

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

Reproduce

herbie shell --seed 2024110 
(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))))