Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.4%

Time: 16.2s

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*r* 99.3%

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

   2. div-sub 99.3%

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

   3. metadata-eval 99.3%

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

   4. pow-sub 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. associate-*r* 99.7%

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

   11. *-commutative 99.7%

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

   12. associate-*l* 99.7%

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

   13. div-inv 99.7%

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

   14. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (2.0 * 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 = Math.PI * (2.0 * 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 = math.pi * (2.0 * 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(pi * Float64(2.0 * n))
	return Float64((k ^ -0.5) * Float64(sqrt(t_0) * (t_0 ^ Float64(k * -0.5))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*r* 99.3%

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

   2. div-sub 99.3%

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

   3. metadata-eval 99.3%

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

   4. sub-neg 99.3%

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

   5. unpow-prod-up 99.6%

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

   6. pow1/2 99.6%

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

   7. associate-*r* 99.6%

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

   8. *-commutative 99.6%

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

   9. associate-*l* 99.6%

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

   10. associate-*r* 99.6%

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

   11. *-commutative 99.6%

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

   12. associate-*l* 99.6%

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

   13. div-inv 99.6%

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

   14. metadata-eval 99.6%

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

   15. distribute-rgt-neg-in 99.6%

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

   16. metadata-eval 99.6%

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

3. Applied egg-rr 99.6%

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

   1. expm1-log1p-u 96.3%

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

   2. expm1-udef 77.2%

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

   3. pow1/2 77.2%

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

   4. pow-flip 77.2%

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

   5. metadata-eval 77.2%

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

5. Applied egg-rr 77.2%

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

   1. expm1-def 96.3%

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

   2. expm1-log1p 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot n\right)\\ \mathbf{if}\;k \leq 10^{-39}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{k}}{\sqrt{t_0}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{t_0}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* 2.0 n))))
   (if (<= k 1e-39)
     (/ 1.0 (/ (sqrt k) (sqrt t_0)))
     (pow (/ k (pow t_0 (- 1.0 k))) -0.5))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (2.0 * n);
	double tmp;
	if (k <= 1e-39) {
		tmp = 1.0 / (sqrt(k) / sqrt(t_0));
	} else {
		tmp = pow((k / pow(t_0, (1.0 - k))), -0.5);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (2.0 * n);
	double tmp;
	if (k <= 1e-39) {
		tmp = 1.0 / (Math.sqrt(k) / Math.sqrt(t_0));
	} else {
		tmp = Math.pow((k / Math.pow(t_0, (1.0 - k))), -0.5);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.pi * (2.0 * n)
	tmp = 0
	if k <= 1e-39:
		tmp = 1.0 / (math.sqrt(k) / math.sqrt(t_0))
	else:
		tmp = math.pow((k / math.pow(t_0, (1.0 - k))), -0.5)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(2.0 * n))
	tmp = 0.0
	if (k <= 1e-39)
		tmp = Float64(1.0 / Float64(sqrt(k) / sqrt(t_0)));
	else
		tmp = Float64(k / (t_0 ^ Float64(1.0 - k))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = pi * (2.0 * n);
	tmp = 0.0;
	if (k <= 1e-39)
		tmp = 1.0 / (sqrt(k) / sqrt(t_0));
	else
		tmp = (k / (t_0 ^ (1.0 - k))) ^ -0.5;
	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-39], N[(1.0 / N[(N[Sqrt[k], $MachinePrecision] / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(k / N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 9.99999999999999929e-40

   1. Initial program 99.3%

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

      1. add-sqr-sqrt 98.9%

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

      2. sqrt-unprod 75.5%

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

      3. *-commutative 75.5%

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

      4. div-inv 75.5%

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

      5. *-commutative 75.5%

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

      6. div-inv 75.6%

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

      7. frac-times 75.5%

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

   3. Applied egg-rr 75.6%

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

   5. Simplified 75.7%

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

      1. clear-num 75.7%

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

      2. sqrt-div 76.3%

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

      3. metadata-eval 76.3%

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

      4. associate-*r* 76.3%

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

      5. *-commutative 76.3%

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

      6. *-commutative 76.3%

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

   7. Applied egg-rr 76.3%

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

   8. Taylor expanded in k around 0 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-*l* 76.3%

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

      3. *-commutative 76.3%

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

   10. Simplified 76.3%

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

       1. sqrt-div 99.3%

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

       2. *-commutative 99.3%

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

   12. Applied egg-rr 99.3%

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

       1. *-commutative 99.3%

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

       2. associate-*r* 99.3%

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

   14. Simplified 99.3%

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

   if 9.99999999999999929e-40 < k

   1. Initial program 99.3%

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

      1. associate-*r* 99.3%

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

      2. div-sub 99.3%

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

      3. metadata-eval 99.3%

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

      4. sub-neg 99.3%

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

      5. unpow-prod-up 99.9%

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

      6. pow1/2 99.9%

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

      7. associate-*r* 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-*l* 99.9%

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

      10. associate-*r* 99.9%

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

      11. *-commutative 99.9%

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

      12. associate-*l* 99.9%

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

      13. div-inv 99.9%

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

      14. metadata-eval 99.9%

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

      15. distribute-rgt-neg-in 99.9%

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

      16. metadata-eval 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. div-inv 99.8%

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

   5. Applied egg-rr 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-39}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{k}}{\sqrt{\pi \cdot \left(2 \cdot n\right)}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \]

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. sqr-pow 99.2%

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

   4. pow-sqr 99.4%

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

   5. associate-*l* 99.4%

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

   6. *-commutative 99.4%

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

   7. associate-*l/ 99.4%

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

   8. associate-/l* 99.4%

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

   9. metadata-eval 99.4%

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

   10. /-rgt-identity 99.4%

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

   11. div-sub 99.4%

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

   12. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. div-inv 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*l* 99.3%

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

   5. div-inv 99.3%

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

   6. metadata-eval 99.3%

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

   7. pow1/2 99.3%

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

   8. pow-flip 99.4%

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

   9. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-39}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{k}}{\sqrt{\pi \cdot \left(2 \cdot n\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5e-39)
   (/ 1.0 (/ (sqrt k) (sqrt (* PI (* 2.0 n)))))
   (sqrt (/ (pow (* n (* PI 2.0)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5e-39) {
		tmp = 1.0 / (sqrt(k) / sqrt((((double) M_PI) * (2.0 * n))));
	} else {
		tmp = sqrt((pow((n * (((double) M_PI) * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 5e-39) {
		tmp = 1.0 / (Math.sqrt(k) / Math.sqrt((Math.PI * (2.0 * n))));
	} else {
		tmp = Math.sqrt((Math.pow((n * (Math.PI * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 5e-39:
		tmp = 1.0 / (math.sqrt(k) / math.sqrt((math.pi * (2.0 * n))))
	else:
		tmp = math.sqrt((math.pow((n * (math.pi * 2.0)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5e-39)
		tmp = Float64(1.0 / Float64(sqrt(k) / sqrt(Float64(pi * Float64(2.0 * n)))));
	else
		tmp = sqrt(Float64((Float64(n * Float64(pi * 2.0)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 5e-39)
		tmp = 1.0 / (sqrt(k) / sqrt((pi * (2.0 * n))));
	else
		tmp = sqrt((((n * (pi * 2.0)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5e-39], N[(1.0 / N[(N[Sqrt[k], $MachinePrecision] / N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.9999999999999998e-39

   1. Initial program 99.3%

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

      1. add-sqr-sqrt 98.9%

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

      2. sqrt-unprod 75.5%

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

      3. *-commutative 75.5%

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

      4. div-inv 75.5%

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

      5. *-commutative 75.5%

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

      6. div-inv 75.6%

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

      7. frac-times 75.5%

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

   3. Applied egg-rr 75.6%

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

   5. Simplified 75.7%

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

      1. clear-num 75.7%

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

      2. sqrt-div 76.3%

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

      3. metadata-eval 76.3%

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

      4. associate-*r* 76.3%

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

      5. *-commutative 76.3%

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

      6. *-commutative 76.3%

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

   7. Applied egg-rr 76.3%

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

   8. Taylor expanded in k around 0 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-*l* 76.3%

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

      3. *-commutative 76.3%

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

   10. Simplified 76.3%

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

       1. sqrt-div 99.3%

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

       2. *-commutative 99.3%

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

   12. Applied egg-rr 99.3%

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

       1. *-commutative 99.3%

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

       2. associate-*r* 99.3%

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

   14. Simplified 99.3%

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

   if 4.9999999999999998e-39 < k

   1. Initial program 99.3%

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

      1. add-sqr-sqrt 99.3%

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

      2. sqrt-unprod 99.3%

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

      3. *-commutative 99.3%

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

      4. div-inv 99.3%

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

      5. *-commutative 99.3%

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

      6. div-inv 99.3%

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

      7. frac-times 99.3%

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

   3. Applied egg-rr 99.3%

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

   5. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 6: 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 99.3%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. sqr-pow 99.2%

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

   4. pow-sqr 99.4%

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

   5. associate-*l* 99.4%

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

   6. *-commutative 99.4%

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

   7. associate-*l/ 99.4%

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

   8. associate-/l* 99.4%

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

   9. metadata-eval 99.4%

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

   10. /-rgt-identity 99.4%

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

   11. div-sub 99.4%

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

   12. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 7: 48.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 89.5%

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

   3. *-commutative 89.5%

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

   4. div-inv 89.5%

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

   5. *-commutative 89.5%

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

   6. div-inv 89.5%

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

   7. frac-times 89.5%

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

3. Applied egg-rr 89.5%

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

5. Simplified 89.6%

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

   1. clear-num 89.6%

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

   2. sqrt-div 89.8%

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

   3. metadata-eval 89.8%

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

   4. associate-*r* 89.8%

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

   5. *-commutative 89.8%

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

   6. *-commutative 89.8%

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

7. Applied egg-rr 89.8%

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

8. Taylor expanded in k around 0 40.8%

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

   1. *-commutative 40.8%

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

   2. associate-*l* 40.8%

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

   3. *-commutative 40.8%

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

10. Simplified 40.8%

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

    1. pow1/2 40.8%

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

    2. div-inv 40.8%

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

    3. unpow-prod-down 50.4%

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

    4. pow1/2 50.4%

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

    5. *-commutative 50.4%

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

12. Applied egg-rr 50.4%

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

    1. unpow1/2 50.4%

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

    2. *-commutative 50.4%

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

    3. associate-/r* 50.4%

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

    4. *-commutative 50.4%

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

    5. associate-/r* 50.4%

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

    6. metadata-eval 50.4%

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

14. Simplified 50.4%

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

15. Final simplification 50.4%

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

Alternative 8: 48.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 89.5%

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

   3. *-commutative 89.5%

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

   4. div-inv 89.5%

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

   5. *-commutative 89.5%

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

   6. div-inv 89.5%

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

   7. frac-times 89.5%

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

3. Applied egg-rr 89.5%

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

5. Simplified 89.6%

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

   1. clear-num 89.6%

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

   2. sqrt-div 89.8%

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

   3. metadata-eval 89.8%

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

   4. associate-*r* 89.8%

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

   5. *-commutative 89.8%

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

   6. *-commutative 89.8%

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

7. Applied egg-rr 89.8%

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

8. Taylor expanded in k around 0 40.8%

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

   1. *-commutative 40.8%

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

   2. associate-*l* 40.8%

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

   3. *-commutative 40.8%

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

10. Simplified 40.8%

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

    1. sqrt-div 50.4%

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

    2. *-commutative 50.4%

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

12. Applied egg-rr 50.4%

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

    1. *-commutative 50.4%

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

    2. associate-*r* 50.4%

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

14. Simplified 50.4%

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

15. Final simplification 50.4%

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

Alternative 9: 37.5% accurate, 1.5× 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 99.3%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 89.5%

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

   3. *-commutative 89.5%

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

   4. div-inv 89.5%

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

   5. *-commutative 89.5%

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

   6. div-inv 89.5%

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

   7. frac-times 89.5%

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

3. Applied egg-rr 89.5%

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

5. Simplified 89.6%

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

   1. clear-num 89.6%

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

   2. sqrt-div 89.8%

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

   3. metadata-eval 89.8%

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

   4. associate-*r* 89.8%

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

   5. *-commutative 89.8%

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

   6. *-commutative 89.8%

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

7. Applied egg-rr 89.8%

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

8. Taylor expanded in k around 0 40.8%

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

   1. *-commutative 40.8%

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

   2. associate-*l* 40.8%

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

   3. *-commutative 40.8%

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

10. Simplified 40.8%

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

    1. expm1-log1p-u 39.3%

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

    2. expm1-udef 39.1%

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

    3. pow1/2 39.1%

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

    4. pow-flip 39.1%

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

    5. *-commutative 39.1%

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

    6. metadata-eval 39.1%

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

12. Applied egg-rr 39.1%

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

    1. expm1-def 39.3%

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

    2. expm1-log1p 40.9%

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

    3. *-lft-identity 40.9%

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

    4. associate-*r* 40.9%

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

    5. *-commutative 40.9%

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

    6. *-commutative 40.9%

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

    7. times-frac 40.9%

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

    8. metadata-eval 40.9%

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

    9. associate-/r* 40.9%

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

14. Simplified 40.9%

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

15. Final simplification 40.9%

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

Alternative 10: 37.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 89.5%

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

   3. *-commutative 89.5%

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

   4. div-inv 89.5%

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

   5. *-commutative 89.5%

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

   6. div-inv 89.5%

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

   7. frac-times 89.5%

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

3. Applied egg-rr 89.5%

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

5. Simplified 89.6%

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

   1. clear-num 89.6%

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

   2. sqrt-div 89.8%

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

   3. metadata-eval 89.8%

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

   4. associate-*r* 89.8%

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

   5. *-commutative 89.8%

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

   6. *-commutative 89.8%

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

7. Applied egg-rr 89.8%

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

8. Taylor expanded in k around 0 40.8%

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

   1. *-commutative 40.8%

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

   2. associate-*l* 40.8%

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

   3. *-commutative 40.8%

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

10. Simplified 40.8%

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

    1. expm1-log1p-u 39.3%

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

    2. expm1-udef 39.1%

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

    3. pow1/2 39.1%

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

    4. pow-flip 39.1%

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

    5. *-commutative 39.1%

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

    6. metadata-eval 39.1%

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

12. Applied egg-rr 39.1%

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

    1. expm1-def 39.3%

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

    2. expm1-log1p 40.9%

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

    3. associate-*r* 40.9%

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

    4. *-commutative 40.9%

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

    5. *-commutative 40.9%

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

    6. *-commutative 40.9%

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

    7. associate-*r* 40.9%

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

    8. *-commutative 40.9%

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

14. Simplified 40.9%

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

15. Final simplification 40.9%

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

Reproduce

herbie shell --seed 2023314 
(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))))