Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 6.1s

Alternatives: 11

Speedup: 1.2×

• could not determine a ground truth

  1. k = 1.1284539860997781e+85
  2. n = -5.0686141046512555e-27

Specification

Math

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

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

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

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, 1.2× speedup

Math

\[\frac{{\left(n \cdot 6.283185307179586\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}{\sqrt{k}} \]

FPCore

(FPCore (k n)
 :precision binary64
 (/ (pow (* n 6.283185307179586) (fma k -0.5 0.5)) (sqrt k)))

C

double code(double k, double n) {
	return pow((n * 6.283185307179586), fma(k, -0.5, 0.5)) / sqrt(k);
}

Julia

function code(k, n)
	return Float64((Float64(n * 6.283185307179586) ^ fma(k, -0.5, 0.5)) / sqrt(k))
end

Wolfram

code[k_, n_] := N[(N[Power[N[(n * 6.283185307179586), $MachinePrecision], N[(k * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lower-/.f64 99.4%

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

3. Applied rewrites 99.4%

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

4. Evaluated real constant 99.4%

   \[\leadsto \frac{{\left(n \cdot \color{blue}{6.283185307179586}\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}{\sqrt{k}} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;k \leq 1.9:\\ \;\;\;\;\frac{n \cdot \sqrt{2 \cdot \frac{\pi}{n}}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(n \cdot 6.283185307179586\right)}^{\left(-0.5 \cdot k\right)}}{\sqrt{k}}\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.9)
   (/ (* n (sqrt (* 2.0 (/ PI n)))) (sqrt k))
   (/ (pow (* n 6.283185307179586) (* -0.5 k)) (sqrt k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.9) {
		tmp = (n * sqrt((2.0 * (((double) M_PI) / n)))) / sqrt(k);
	} else {
		tmp = pow((n * 6.283185307179586), (-0.5 * k)) / sqrt(k);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.9) {
		tmp = (n * Math.sqrt((2.0 * (Math.PI / n)))) / Math.sqrt(k);
	} else {
		tmp = Math.pow((n * 6.283185307179586), (-0.5 * k)) / Math.sqrt(k);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.9:
		tmp = (n * math.sqrt((2.0 * (math.pi / n)))) / math.sqrt(k)
	else:
		tmp = math.pow((n * 6.283185307179586), (-0.5 * k)) / math.sqrt(k)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.9)
		tmp = Float64(Float64(n * sqrt(Float64(2.0 * Float64(pi / n)))) / sqrt(k));
	else
		tmp = Float64((Float64(n * 6.283185307179586) ^ Float64(-0.5 * k)) / sqrt(k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.9)
		tmp = (n * sqrt((2.0 * (pi / n)))) / sqrt(k);
	else
		tmp = ((n * 6.283185307179586) ^ (-0.5 * k)) / sqrt(k);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.9], N[(N[(n * N[Sqrt[N[(2.0 * N[(Pi / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(n * 6.283185307179586), $MachinePrecision], N[(-0.5 * k), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.8999999999999999

   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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 50.6%

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 50.6%

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

   7. Applied rewrites 50.6%

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

   if 1.8999999999999999 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 99.4%

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

   3. Applied rewrites 99.4%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{{\left(n \cdot \color{blue}{6.283185307179586}\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}{\sqrt{k}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{{\left(n \cdot 6.283185307179586\right)}^{\color{blue}{\left(\frac{-1}{2} \cdot k\right)}}}{\sqrt{k}} \]
   6. Step-by-step derivation

      1. lower-*.f64 52.2%

         \[\leadsto \frac{{\left(n \cdot 6.283185307179586\right)}^{\left(-0.5 \cdot \color{blue}{k}\right)}}{\sqrt{k}} \]

   7. Applied rewrites 52.2%

      \[\leadsto \frac{{\left(n \cdot 6.283185307179586\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}}}{\sqrt{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;n \leq 13200000:\\ \;\;\;\;\frac{n \cdot \sqrt{2 \cdot \frac{k \cdot \pi}{n}}}{k}\\ \mathbf{elif}\;n \leq 2.05 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{\log \left(e^{\frac{n + n}{k} \cdot \pi}\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{\pi}{k \cdot n}}\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 13200000.0)
   (/ (* n (sqrt (* 2.0 (/ (* k PI) n)))) k)
   (if (<= n 2.05e+26)
     (sqrt (log (exp (* (/ (+ n n) k) PI))))
     (* n (sqrt (* 2.0 (/ PI (* k n))))))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 13200000.0) {
		tmp = (n * sqrt((2.0 * ((k * ((double) M_PI)) / n)))) / k;
	} else if (n <= 2.05e+26) {
		tmp = sqrt(log(exp((((n + n) / k) * ((double) M_PI)))));
	} else {
		tmp = n * sqrt((2.0 * (((double) M_PI) / (k * n))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 13200000.0) {
		tmp = (n * Math.sqrt((2.0 * ((k * Math.PI) / n)))) / k;
	} else if (n <= 2.05e+26) {
		tmp = Math.sqrt(Math.log(Math.exp((((n + n) / k) * Math.PI))));
	} else {
		tmp = n * Math.sqrt((2.0 * (Math.PI / (k * n))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 13200000.0:
		tmp = (n * math.sqrt((2.0 * ((k * math.pi) / n)))) / k
	elif n <= 2.05e+26:
		tmp = math.sqrt(math.log(math.exp((((n + n) / k) * math.pi))))
	else:
		tmp = n * math.sqrt((2.0 * (math.pi / (k * n))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 13200000.0)
		tmp = Float64(Float64(n * sqrt(Float64(2.0 * Float64(Float64(k * pi) / n)))) / k);
	elseif (n <= 2.05e+26)
		tmp = sqrt(log(exp(Float64(Float64(Float64(n + n) / k) * pi))));
	else
		tmp = Float64(n * sqrt(Float64(2.0 * Float64(pi / Float64(k * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 13200000.0)
		tmp = (n * sqrt((2.0 * ((k * pi) / n)))) / k;
	elseif (n <= 2.05e+26)
		tmp = sqrt(log(exp((((n + n) / k) * pi))));
	else
		tmp = n * sqrt((2.0 * (pi / (k * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 13200000.0], N[(N[(n * N[Sqrt[N[(2.0 * N[(N[(k * Pi), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[n, 2.05e+26], N[Sqrt[N[Log[N[Exp[N[(N[(N[(n + n), $MachinePrecision] / k), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(n * N[Sqrt[N[(2.0 * N[(Pi / N[(k * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.32e7

   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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 50.6%

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

   4. Applied rewrites 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 38.9%

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

   6. Applied rewrites 38.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 38.8%

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

   9. Applied rewrites 38.8%

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

   10. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 51.5%

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

   12. Applied rewrites 51.5%

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

   if 1.32e7 < n < 2.04999999999999992e26

   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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 50.6%

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

   4. Applied rewrites 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 38.9%

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

   6. Applied rewrites 38.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 38.9%

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

   8. Applied rewrites 38.9%

      \[\leadsto \sqrt{\pi \cdot \frac{n + n}{k}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-log-exp N/A

         \[\leadsto \sqrt{\frac{n + n}{k} \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)} \]

      5. log-pow-rev N/A

         \[\leadsto \sqrt{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{n + n}{k}\right)}\right)} \]

      6. lower-log.f64 N/A

         \[\leadsto \sqrt{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{n + n}{k}\right)}\right)} \]

      7. lift-PI.f64 N/A

         \[\leadsto \sqrt{\log \left({\left(e^{\pi}\right)}^{\left(\frac{n + n}{k}\right)}\right)} \]

      8. pow-exp N/A

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

      9. lift-*.f64 N/A

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

      10. lower-exp.f64 14.9%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 14.9%

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

   10. Applied rewrites 14.9%

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

   if 2.04999999999999992e26 < 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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 50.6%

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

   4. Applied rewrites 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 38.9%

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

   6. Applied rewrites 38.9%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 50.3%

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

   9. Applied rewrites 50.3%

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

Alternative 4: 74.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;n \leq 14000000:\\ \;\;\;\;\frac{n \cdot \sqrt{2 \cdot \frac{k \cdot \pi}{n}}}{k}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{\frac{k \cdot \left(\pi \cdot \left(n + n\right)\right)}{k \cdot k}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{\pi}{k \cdot n}}\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 14000000.0)
   (/ (* n (sqrt (* 2.0 (/ (* k PI) n)))) k)
   (if (<= n 3e+29)
     (sqrt (/ (* k (* PI (+ n n))) (* k k)))
     (* n (sqrt (* 2.0 (/ PI (* k n))))))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 14000000.0) {
		tmp = (n * sqrt((2.0 * ((k * ((double) M_PI)) / n)))) / k;
	} else if (n <= 3e+29) {
		tmp = sqrt(((k * (((double) M_PI) * (n + n))) / (k * k)));
	} else {
		tmp = n * sqrt((2.0 * (((double) M_PI) / (k * n))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 14000000.0) {
		tmp = (n * Math.sqrt((2.0 * ((k * Math.PI) / n)))) / k;
	} else if (n <= 3e+29) {
		tmp = Math.sqrt(((k * (Math.PI * (n + n))) / (k * k)));
	} else {
		tmp = n * Math.sqrt((2.0 * (Math.PI / (k * n))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 14000000.0:
		tmp = (n * math.sqrt((2.0 * ((k * math.pi) / n)))) / k
	elif n <= 3e+29:
		tmp = math.sqrt(((k * (math.pi * (n + n))) / (k * k)))
	else:
		tmp = n * math.sqrt((2.0 * (math.pi / (k * n))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 14000000.0)
		tmp = Float64(Float64(n * sqrt(Float64(2.0 * Float64(Float64(k * pi) / n)))) / k);
	elseif (n <= 3e+29)
		tmp = sqrt(Float64(Float64(k * Float64(pi * Float64(n + n))) / Float64(k * k)));
	else
		tmp = Float64(n * sqrt(Float64(2.0 * Float64(pi / Float64(k * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 14000000.0)
		tmp = (n * sqrt((2.0 * ((k * pi) / n)))) / k;
	elseif (n <= 3e+29)
		tmp = sqrt(((k * (pi * (n + n))) / (k * k)));
	else
		tmp = n * sqrt((2.0 * (pi / (k * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 14000000.0], N[(N[(n * N[Sqrt[N[(2.0 * N[(N[(k * Pi), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[n, 3e+29], N[Sqrt[N[(N[(k * N[(Pi * N[(n + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(n * N[Sqrt[N[(2.0 * N[(Pi / N[(k * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.4e7

   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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 50.6%

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

   4. Applied rewrites 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 38.9%

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

   6. Applied rewrites 38.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 38.8%

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

   9. Applied rewrites 38.8%

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

   10. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 51.5%

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

   12. Applied rewrites 51.5%

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

   if 1.4e7 < n < 2.9999999999999999e29

   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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 50.6%

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

   4. Applied rewrites 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 38.9%

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

   6. Applied rewrites 38.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 38.9%

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

   8. Applied rewrites 38.9%

      \[\leadsto \sqrt{\pi \cdot \frac{n + n}{k}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. div-add N/A

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

      9. frac-add N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. count-2-rev N/A

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

      20. lift-*.f64 N/A

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

   10. Applied rewrites 23.7%

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

   if 2.9999999999999999e29 < 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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 50.6%

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

   4. Applied rewrites 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 38.9%

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

   6. Applied rewrites 38.9%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 50.3%

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

   9. Applied rewrites 50.3%

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

Alternative 5: 74.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;n \leq 14000000:\\ \;\;\;\;\frac{n \cdot \sqrt{2 \cdot \frac{k \cdot \pi}{n}}}{k}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\pi \cdot n, k, \left(\pi \cdot n\right) \cdot k\right)}{k \cdot k}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{\pi}{k \cdot n}}\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 14000000.0)
   (/ (* n (sqrt (* 2.0 (/ (* k PI) n)))) k)
   (if (<= n 3e+29)
     (sqrt (/ (fma (* PI n) k (* (* PI n) k)) (* k k)))
     (* n (sqrt (* 2.0 (/ PI (* k n))))))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 14000000.0) {
		tmp = (n * sqrt((2.0 * ((k * ((double) M_PI)) / n)))) / k;
	} else if (n <= 3e+29) {
		tmp = sqrt((fma((((double) M_PI) * n), k, ((((double) M_PI) * n) * k)) / (k * k)));
	} else {
		tmp = n * sqrt((2.0 * (((double) M_PI) / (k * n))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 14000000.0)
		tmp = Float64(Float64(n * sqrt(Float64(2.0 * Float64(Float64(k * pi) / n)))) / k);
	elseif (n <= 3e+29)
		tmp = sqrt(Float64(fma(Float64(pi * n), k, Float64(Float64(pi * n) * k)) / Float64(k * k)));
	else
		tmp = Float64(n * sqrt(Float64(2.0 * Float64(pi / Float64(k * n)))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[n, 14000000.0], N[(N[(n * N[Sqrt[N[(2.0 * N[(N[(k * Pi), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[n, 3e+29], N[Sqrt[N[(N[(N[(Pi * n), $MachinePrecision] * k + N[(N[(Pi * n), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(n * N[Sqrt[N[(2.0 * N[(Pi / N[(k * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.4e7

   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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 50.6%

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

   4. Applied rewrites 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 38.9%

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

   6. Applied rewrites 38.9%

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

   7. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 38.8%

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

   9. Applied rewrites 38.8%

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

   10. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 51.5%

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

   12. Applied rewrites 51.5%

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

   if 1.4e7 < n < 2.9999999999999999e29

   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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 50.6%

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

   4. Applied rewrites 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 38.9%

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

   6. Applied rewrites 38.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. count-2-rev N/A

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

      8. div-add N/A

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

      9. common-denominator N/A

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

      10. lower-/.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 23.7%

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

   8. Applied rewrites 23.7%

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

   if 2.9999999999999999e29 < 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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 50.6%

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

   4. Applied rewrites 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 38.9%

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

   6. Applied rewrites 38.9%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 50.3%

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

   9. Applied rewrites 50.3%

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

Alternative 6: 62.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(n + n\right)\\ \mathbf{if}\;n \leq 14000000:\\ \;\;\;\;\sqrt{\frac{\left|t\_0\right|}{k}}\\ \mathbf{elif}\;n \leq 3 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{\frac{k \cdot t\_0}{k \cdot k}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{\pi}{k \cdot n}}\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (+ n n))))
   (if (<= n 14000000.0)
     (sqrt (/ (fabs t_0) k))
     (if (<= n 3e+29)
       (sqrt (/ (* k t_0) (* k k)))
       (* n (sqrt (* 2.0 (/ PI (* k n)))))))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (n + n);
	double tmp;
	if (n <= 14000000.0) {
		tmp = sqrt((fabs(t_0) / k));
	} else if (n <= 3e+29) {
		tmp = sqrt(((k * t_0) / (k * k)));
	} else {
		tmp = n * sqrt((2.0 * (((double) M_PI) / (k * n))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (n + n);
	double tmp;
	if (n <= 14000000.0) {
		tmp = Math.sqrt((Math.abs(t_0) / k));
	} else if (n <= 3e+29) {
		tmp = Math.sqrt(((k * t_0) / (k * k)));
	} else {
		tmp = n * Math.sqrt((2.0 * (Math.PI / (k * n))));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.pi * (n + n)
	tmp = 0
	if n <= 14000000.0:
		tmp = math.sqrt((math.fabs(t_0) / k))
	elif n <= 3e+29:
		tmp = math.sqrt(((k * t_0) / (k * k)))
	else:
		tmp = n * math.sqrt((2.0 * (math.pi / (k * n))))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(n + n))
	tmp = 0.0
	if (n <= 14000000.0)
		tmp = sqrt(Float64(abs(t_0) / k));
	elseif (n <= 3e+29)
		tmp = sqrt(Float64(Float64(k * t_0) / Float64(k * k)));
	else
		tmp = Float64(n * sqrt(Float64(2.0 * Float64(pi / Float64(k * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = pi * (n + n);
	tmp = 0.0;
	if (n <= 14000000.0)
		tmp = sqrt((abs(t_0) / k));
	elseif (n <= 3e+29)
		tmp = sqrt(((k * t_0) / (k * k)));
	else
		tmp = n * sqrt((2.0 * (pi / (k * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(n + n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, 14000000.0], N[Sqrt[N[(N[Abs[t$95$0], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 3e+29], N[Sqrt[N[(N[(k * t$95$0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(n * N[Sqrt[N[(2.0 * N[(Pi / N[(k * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.4e7

   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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 50.6%

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

   4. Applied rewrites 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 38.9%

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

   6. Applied rewrites 38.9%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

      5. mul-fabs N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-fabs.f64 38.9%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 38.9%

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

   8. Applied rewrites 38.9%

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

   if 1.4e7 < n < 2.9999999999999999e29

   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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 50.6%

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

   4. Applied rewrites 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 38.9%

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

   6. Applied rewrites 38.9%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 38.9%

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

   8. Applied rewrites 38.9%

      \[\leadsto \sqrt{\pi \cdot \frac{n + n}{k}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. div-add N/A

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

      9. frac-add N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. count-2-rev N/A

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

      20. lift-*.f64 N/A

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

   10. Applied rewrites 23.7%

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

   if 2.9999999999999999e29 < 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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 50.6%

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

   4. Applied rewrites 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 38.9%

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

   6. Applied rewrites 38.9%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 50.3%

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

   9. Applied rewrites 50.3%

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

Alternative 7: 62.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{\frac{\left|\pi \cdot \left(n + n\right)\right|}{k}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{\pi}{k \cdot n}}\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 2e-20)
   (sqrt (/ (fabs (* PI (+ n n))) k))
   (* n (sqrt (* 2.0 (/ PI (* k n)))))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 2e-20) {
		tmp = sqrt((fabs((((double) M_PI) * (n + n))) / k));
	} else {
		tmp = n * sqrt((2.0 * (((double) M_PI) / (k * n))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 2e-20) {
		tmp = Math.sqrt((Math.abs((Math.PI * (n + n))) / k));
	} else {
		tmp = n * Math.sqrt((2.0 * (Math.PI / (k * n))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 2e-20:
		tmp = math.sqrt((math.fabs((math.pi * (n + n))) / k))
	else:
		tmp = n * math.sqrt((2.0 * (math.pi / (k * n))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 2e-20)
		tmp = sqrt(Float64(abs(Float64(pi * Float64(n + n))) / k));
	else
		tmp = Float64(n * sqrt(Float64(2.0 * Float64(pi / Float64(k * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 2e-20)
		tmp = sqrt((abs((pi * (n + n))) / k));
	else
		tmp = n * sqrt((2.0 * (pi / (k * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 2e-20], N[Sqrt[N[(N[Abs[N[(Pi * N[(n + n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision], N[(n * N[Sqrt[N[(2.0 * N[(Pi / N[(k * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1.99999999999999989e-20

   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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 50.6%

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

   4. Applied rewrites 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 38.9%

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

   6. Applied rewrites 38.9%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

      5. mul-fabs N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-fabs.f64 38.9%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 38.9%

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

   8. Applied rewrites 38.9%

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

   if 1.99999999999999989e-20 < 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

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 50.6%

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

   4. Applied rewrites 50.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 38.9%

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

   6. Applied rewrites 38.9%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 50.3%

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

   9. Applied rewrites 50.3%

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

Alternative 8: 50.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 50.6%

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

4. Applied rewrites 50.6%

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

   1. lift-pow.f64 N/A

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

   2. unpow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. count-2-rev N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-lft-in N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-sqrt.f64 50.6%

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

   11. lift-*.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. distribute-lft-in N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. count-2-rev N/A

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

   17. lift-*.f64 N/A

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

   18. associate-*r* N/A

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

   19. lower-*.f64 N/A

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

   20. count-2-rev N/A

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

   21. lower-+.f64 50.6%

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

6. Applied rewrites 50.6%

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

Alternative 9: 50.6% accurate, 2.7× speedup

Math

\[\sqrt{\frac{\pi}{k}} \cdot \sqrt{n + n} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 50.6%

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

4. Applied rewrites 50.6%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow1/2 N/A

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

   4. lift-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. sqrt-undiv N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-/.f64 38.9%

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

6. Applied rewrites 38.9%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. sqrt-prod N/A

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

   7. lower-unsound-*.f64 N/A

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

   8. lower-unsound-sqrt.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-unsound-sqrt.f64 50.6%

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

8. Applied rewrites 50.6%

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

Alternative 10: 50.6% accurate, 2.7× speedup

Math

\[\sqrt{n} \cdot \sqrt{\frac{\pi + \pi}{k}} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 50.6%

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

4. Applied rewrites 50.6%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow1/2 N/A

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

   4. lift-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. sqrt-undiv N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-/.f64 38.9%

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

6. Applied rewrites 38.9%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*l* N/A

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

   12. sqrt-prod N/A

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

   13. lower-unsound-*.f64 N/A

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

   14. lower-unsound-sqrt.f64 N/A

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

   15. lower-unsound-sqrt.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. mult-flip-rev N/A

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

   18. lower-/.f64 N/A

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

   19. lift-PI.f64 N/A

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

   20. count-2-rev N/A

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

   21. lower-+.f64 50.6%

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

8. Applied rewrites 50.6%

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

Alternative 11: 38.9% accurate, 3.1× speedup

Math

\[\sqrt{\pi \cdot \frac{n + n}{k}} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 50.6%

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

4. Applied rewrites 50.6%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow1/2 N/A

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

   4. lift-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. sqrt-undiv N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-/.f64 38.9%

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

6. Applied rewrites 38.9%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. mult-flip-rev N/A

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

   10. lower-/.f64 38.9%

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

8. Applied rewrites 38.9%

   \[\leadsto \sqrt{\pi \cdot \frac{n + n}{k}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(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))))