Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 5.6s

Alternatives: 7

Speedup: 1.0×

• Could not converge on a ground truth

  1. k = 1.5881786826437104e+243
  2. n = -2.2913663160639355e+211

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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in k around 0

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

   1. lower-+.f64 N/A

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

   2. lift-*.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ \mathbf{if}\;k \leq 1:\\ \;\;\;\;\sqrt{\frac{1}{k}} \cdot {t\_0}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{k}} \cdot {t\_0}^{\left(-0.5 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(k, n)
	t_0 = (2.0 * pi) * n;
	tmp = 0.0;
	if (k <= 1.0)
		tmp = sqrt((1.0 / k)) * (t_0 ^ 0.5);
	else
		tmp = (1.0 / sqrt(k)) * (t_0 ^ (-0.5 * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[k, 1.0], N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$0, 0.5], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, N[(-0.5 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 48.6%

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

   if 1 < k

   1. Initial program 99.4%

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

   2. Taylor expanded in k around inf

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

      1. lower-*.f64 54.2

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

   4. Applied rewrites 54.2%

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

Alternative 3: 61.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\frac{1}{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \left(\sqrt{\frac{k \cdot \pi}{n}} \cdot \sqrt{2}\right)}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.7e-26)
   (* (sqrt (/ 1.0 k)) (pow (* (* 2.0 PI) n) 0.5))
   (/ (* n (* (sqrt (/ (* k PI) n)) (sqrt 2.0))) k)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.7e-26) {
		tmp = sqrt((1.0 / k)) * pow(((2.0 * ((double) M_PI)) * n), 0.5);
	} else {
		tmp = (n * (sqrt(((k * ((double) M_PI)) / n)) * sqrt(2.0))) / k;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.7e-26) {
		tmp = Math.sqrt((1.0 / k)) * Math.pow(((2.0 * Math.PI) * n), 0.5);
	} else {
		tmp = (n * (Math.sqrt(((k * Math.PI) / n)) * Math.sqrt(2.0))) / k;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.7e-26:
		tmp = math.sqrt((1.0 / k)) * math.pow(((2.0 * math.pi) * n), 0.5)
	else:
		tmp = (n * (math.sqrt(((k * math.pi) / n)) * math.sqrt(2.0))) / k
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.7e-26)
		tmp = Float64(sqrt(Float64(1.0 / k)) * (Float64(Float64(2.0 * pi) * n) ^ 0.5));
	else
		tmp = Float64(Float64(n * Float64(sqrt(Float64(Float64(k * pi) / n)) * sqrt(2.0))) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.7e-26)
		tmp = sqrt((1.0 / k)) * (((2.0 * pi) * n) ^ 0.5);
	else
		tmp = (n * (sqrt(((k * pi) / n)) * sqrt(2.0))) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.7e-26], N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[(N[(n * N[(N[Sqrt[N[(N[(k * Pi), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.6999999999999999e-26

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 48.6%

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

   if 3.6999999999999999e-26 < k

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{k \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)\right) + \frac{1}{8} \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left({\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \sqrt{2}\right)\right)\right) + \sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{k}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{k \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)\right) + \frac{1}{8} \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left({\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \sqrt{2}\right)\right)\right) + \sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{\color{blue}{k}} \]

   4. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(k, \mathsf{fma}\left(-0.5, \sqrt{k \cdot \left(n \cdot \pi\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \pi\right)\right) \cdot \sqrt{2}\right), 0.125 \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \pi\right)} \cdot \left({\log \left(2 \cdot \left(n \cdot \pi\right)\right)}^{2} \cdot \sqrt{2}\right)\right)\right), \sqrt{k \cdot \left(n \cdot \pi\right)} \cdot \sqrt{2}\right)}{k}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \frac{n \cdot \left(k \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{k \cdot \mathsf{PI}\left(\right)}{n}} \cdot \left(\sqrt{2} \cdot \left(\log \left(2 \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \log \left(\frac{1}{n}\right)\right)\right)\right) + \frac{1}{8} \cdot \left(\sqrt{\frac{{k}^{3} \cdot \mathsf{PI}\left(\right)}{n}} \cdot \left(\sqrt{2} \cdot {\left(\log \left(2 \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \log \left(\frac{1}{n}\right)\right)}^{2}\right)\right)\right) + \sqrt{\frac{k \cdot \mathsf{PI}\left(\right)}{n}} \cdot \sqrt{2}\right)}{k} \]

   7. Applied rewrites 57.8%

      \[\leadsto \frac{n \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(-0.5, \sqrt{\frac{k \cdot \pi}{n}} \cdot \left(\sqrt{2} \cdot \left(\log \left(2 \cdot \pi\right) + -1 \cdot \log \left(\frac{1}{n}\right)\right)\right), 0.125 \cdot \left(\sqrt{\frac{{k}^{3} \cdot \pi}{n}} \cdot \left(\sqrt{2} \cdot {\left(\log \left(2 \cdot \pi\right) + -1 \cdot \log \left(\frac{1}{n}\right)\right)}^{2}\right)\right)\right), \sqrt{\frac{k \cdot \pi}{n}} \cdot \sqrt{2}\right)}{k} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{n \cdot \left(\sqrt{\frac{k \cdot \mathsf{PI}\left(\right)}{n}} \cdot \sqrt{2}\right)}{k} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 50.0

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

   10. Applied rewrites 50.0%

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

Alternative 4: 61.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 56000000:\\ \;\;\;\;\frac{n \cdot \left(\sqrt{\frac{k \cdot \pi}{n}} \cdot \sqrt{2}\right)}{k}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{k}} \cdot \left(\sqrt{n \cdot \pi} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 56000000.0)
   (/ (* n (* (sqrt (/ (* k PI) n)) (sqrt 2.0))) k)
   (* (sqrt (/ 1.0 k)) (* (sqrt (* n PI)) (sqrt 2.0)))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 56000000.0) {
		tmp = (n * (sqrt(((k * ((double) M_PI)) / n)) * sqrt(2.0))) / k;
	} else {
		tmp = sqrt((1.0 / k)) * (sqrt((n * ((double) M_PI))) * sqrt(2.0));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 56000000.0) {
		tmp = (n * (Math.sqrt(((k * Math.PI) / n)) * Math.sqrt(2.0))) / k;
	} else {
		tmp = Math.sqrt((1.0 / k)) * (Math.sqrt((n * Math.PI)) * Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 56000000.0:
		tmp = (n * (math.sqrt(((k * math.pi) / n)) * math.sqrt(2.0))) / k
	else:
		tmp = math.sqrt((1.0 / k)) * (math.sqrt((n * math.pi)) * math.sqrt(2.0))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 56000000.0)
		tmp = Float64(Float64(n * Float64(sqrt(Float64(Float64(k * pi) / n)) * sqrt(2.0))) / k);
	else
		tmp = Float64(sqrt(Float64(1.0 / k)) * Float64(sqrt(Float64(n * pi)) * sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 56000000.0)
		tmp = (n * (sqrt(((k * pi) / n)) * sqrt(2.0))) / k;
	else
		tmp = sqrt((1.0 / k)) * (sqrt((n * pi)) * sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 56000000.0], N[(N[(n * N[(N[Sqrt[N[(N[(k * Pi), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(n * Pi), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 5.6e7

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{k \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)\right) + \frac{1}{8} \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left({\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \sqrt{2}\right)\right)\right) + \sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{k}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{k \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)\right) + \frac{1}{8} \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left({\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \sqrt{2}\right)\right)\right) + \sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{\color{blue}{k}} \]

   4. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(k, \mathsf{fma}\left(-0.5, \sqrt{k \cdot \left(n \cdot \pi\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \pi\right)\right) \cdot \sqrt{2}\right), 0.125 \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \pi\right)} \cdot \left({\log \left(2 \cdot \left(n \cdot \pi\right)\right)}^{2} \cdot \sqrt{2}\right)\right)\right), \sqrt{k \cdot \left(n \cdot \pi\right)} \cdot \sqrt{2}\right)}{k}} \]

   5. Taylor expanded in n around inf

      \[\leadsto \frac{n \cdot \left(k \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{k \cdot \mathsf{PI}\left(\right)}{n}} \cdot \left(\sqrt{2} \cdot \left(\log \left(2 \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \log \left(\frac{1}{n}\right)\right)\right)\right) + \frac{1}{8} \cdot \left(\sqrt{\frac{{k}^{3} \cdot \mathsf{PI}\left(\right)}{n}} \cdot \left(\sqrt{2} \cdot {\left(\log \left(2 \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \log \left(\frac{1}{n}\right)\right)}^{2}\right)\right)\right) + \sqrt{\frac{k \cdot \mathsf{PI}\left(\right)}{n}} \cdot \sqrt{2}\right)}{k} \]

   7. Applied rewrites 57.8%

      \[\leadsto \frac{n \cdot \mathsf{fma}\left(k, \mathsf{fma}\left(-0.5, \sqrt{\frac{k \cdot \pi}{n}} \cdot \left(\sqrt{2} \cdot \left(\log \left(2 \cdot \pi\right) + -1 \cdot \log \left(\frac{1}{n}\right)\right)\right), 0.125 \cdot \left(\sqrt{\frac{{k}^{3} \cdot \pi}{n}} \cdot \left(\sqrt{2} \cdot {\left(\log \left(2 \cdot \pi\right) + -1 \cdot \log \left(\frac{1}{n}\right)\right)}^{2}\right)\right)\right), \sqrt{\frac{k \cdot \pi}{n}} \cdot \sqrt{2}\right)}{k} \]

   8. Taylor expanded in k around 0

      \[\leadsto \frac{n \cdot \left(\sqrt{\frac{k \cdot \mathsf{PI}\left(\right)}{n}} \cdot \sqrt{2}\right)}{k} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 50.0

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

   10. Applied rewrites 50.0%

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

   if 5.6e7 < n

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in k around 0

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 48.6

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

   7. Applied rewrites 48.6%

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

Alternative 5: 48.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 5 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{k \cdot \left(n \cdot \pi\right)} \cdot \sqrt{2}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))) 5e+95)
   (* (sqrt (/ (* n PI) k)) (sqrt 2.0))
   (/ (* (sqrt (* k (* n PI))) (sqrt 2.0)) k)))

C

double code(double k, double n) {
	double tmp;
	if (((1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0))) <= 5e+95) {
		tmp = sqrt(((n * ((double) M_PI)) / k)) * sqrt(2.0);
	} else {
		tmp = (sqrt((k * (n * ((double) M_PI)))) * sqrt(2.0)) / k;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (((1.0 / Math.sqrt(k)) * Math.pow(((2.0 * Math.PI) * n), ((1.0 - k) / 2.0))) <= 5e+95) {
		tmp = Math.sqrt(((n * Math.PI) / k)) * Math.sqrt(2.0);
	} else {
		tmp = (Math.sqrt((k * (n * Math.PI))) * Math.sqrt(2.0)) / k;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if ((1.0 / math.sqrt(k)) * math.pow(((2.0 * math.pi) * n), ((1.0 - k) / 2.0))) <= 5e+95:
		tmp = math.sqrt(((n * math.pi) / k)) * math.sqrt(2.0)
	else:
		tmp = (math.sqrt((k * (n * math.pi))) * math.sqrt(2.0)) / k
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0))) <= 5e+95)
		tmp = Float64(sqrt(Float64(Float64(n * pi) / k)) * sqrt(2.0));
	else
		tmp = Float64(Float64(sqrt(Float64(k * Float64(n * pi))) * sqrt(2.0)) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (((1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0))) <= 5e+95)
		tmp = sqrt(((n * pi) / k)) * sqrt(2.0);
	else
		tmp = (sqrt((k * (n * pi))) * sqrt(2.0)) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[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], 5e+95], N[(N[Sqrt[N[(N[(n * Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(k * N[(n * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 5.00000000000000025e95

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lower-sqrt.f64 37.4

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

   4. Applied rewrites 37.4%

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

   if 5.00000000000000025e95 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{k \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)\right) + \frac{1}{8} \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left({\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \sqrt{2}\right)\right)\right) + \sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{k}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{k \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)\right) + \frac{1}{8} \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left({\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \sqrt{2}\right)\right)\right) + \sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{\color{blue}{k}} \]

   4. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(k, \mathsf{fma}\left(-0.5, \sqrt{k \cdot \left(n \cdot \pi\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \pi\right)\right) \cdot \sqrt{2}\right), 0.125 \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \pi\right)} \cdot \left({\log \left(2 \cdot \left(n \cdot \pi\right)\right)}^{2} \cdot \sqrt{2}\right)\right)\right), \sqrt{k \cdot \left(n \cdot \pi\right)} \cdot \sqrt{2}\right)}{k}} \]

   5. Taylor expanded in k around 0

      \[\leadsto \frac{\sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{k} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 37.3

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

   7. Applied rewrites 37.3%

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

Alternative 6: 47.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in k around 0

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 48.6

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

7. Applied rewrites 48.6%

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

Alternative 7: 37.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lower-sqrt.f64 37.4

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

4. Applied rewrites 37.4%

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

Reproduce

herbie shell --seed 2025132 
(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))))