Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 6.2s

Alternatives: 8

Speedup: 1.2×

• Could not converge on a ground truth

  1. k = 2.0643499338907584e+231
  2. n = -1.9678152893513906e+30

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: 99.5% 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 99.5

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

3. Applied rewrites 99.5%

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

4. Evaluated real constant 99.5%

   \[\leadsto \frac{{\left(n \cdot \color{blue}{\frac{884279719003555}{140737488355328}}\right)}^{\left(\mathsf{fma}\left(k, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{k}} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;k \leq 2.3:\\ \;\;\;\;\frac{n \cdot \sqrt{2 \cdot \frac{\pi}{n}}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(n \cdot 6.283185307179586\right)}^{\left(-0.5 \cdot k\right)}}{\sqrt{k}}\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.3)
   (/ (* 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 <= 2.3) {
		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 <= 2.3) {
		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 <= 2.3:
		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 <= 2.3)
		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 <= 2.3)
		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, 2.3], 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 < 2.2999999999999998

   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

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
   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 49.5

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

   4. Applied rewrites 49.5%

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

   5. Taylor expanded in n around inf

      \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{n}}}{\sqrt{\color{blue}{k}}} \]
   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 49.5

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

   7. Applied rewrites 49.5%

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

   if 2.2999999999999998 < 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 99.5

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

   3. Applied rewrites 99.5%

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

   4. Evaluated real constant 99.5%

      \[\leadsto \frac{{\left(n \cdot \color{blue}{\frac{884279719003555}{140737488355328}}\right)}^{\left(\mathsf{fma}\left(k, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{k}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{{\left(n \cdot \frac{884279719003555}{140737488355328}\right)}^{\color{blue}{\left(\frac{-1}{2} \cdot k\right)}}}{\sqrt{k}} \]
   6. Step-by-step derivation

      1. lower-*.f64 53.3

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

   7. Applied rewrites 53.3%

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

Alternative 3: 72.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{\pi}{k \cdot n}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+286}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{n + n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot n} \cdot \frac{\sqrt{\sqrt{\left(k + k\right) \cdot \left(k + k\right)}}}{k}\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0)))))
   (if (<= t_0 0.0)
     (* n (sqrt (* 2.0 (/ PI (* k n)))))
     (if (<= t_0 2e+286)
       (* (sqrt (/ PI k)) (sqrt (+ n n)))
       (* (sqrt (* PI n)) (/ (sqrt (sqrt (* (+ k k) (+ k k)))) k))))))

C

double code(double k, double n) {
	double t_0 = (1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = n * sqrt((2.0 * (((double) M_PI) / (k * n))));
	} else if (t_0 <= 2e+286) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((n + n));
	} else {
		tmp = sqrt((((double) M_PI) * n)) * (sqrt(sqrt(((k + k) * (k + k)))) / k);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = (1.0 / Math.sqrt(k)) * Math.pow(((2.0 * Math.PI) * n), ((1.0 - k) / 2.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = n * Math.sqrt((2.0 * (Math.PI / (k * n))));
	} else if (t_0 <= 2e+286) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((n + n));
	} else {
		tmp = Math.sqrt((Math.PI * n)) * (Math.sqrt(Math.sqrt(((k + k) * (k + k)))) / k);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = (1.0 / math.sqrt(k)) * math.pow(((2.0 * math.pi) * n), ((1.0 - k) / 2.0))
	tmp = 0
	if t_0 <= 0.0:
		tmp = n * math.sqrt((2.0 * (math.pi / (k * n))))
	elif t_0 <= 2e+286:
		tmp = math.sqrt((math.pi / k)) * math.sqrt((n + n))
	else:
		tmp = math.sqrt((math.pi * n)) * (math.sqrt(math.sqrt(((k + k) * (k + k)))) / k)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(n * sqrt(Float64(2.0 * Float64(pi / Float64(k * n)))));
	elseif (t_0 <= 2e+286)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(n + n)));
	else
		tmp = Float64(sqrt(Float64(pi * n)) * Float64(sqrt(sqrt(Float64(Float64(k + k) * Float64(k + k)))) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = (1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = n * sqrt((2.0 * (pi / (k * n))));
	elseif (t_0 <= 2e+286)
		tmp = sqrt((pi / k)) * sqrt((n + n));
	else
		tmp = sqrt((pi * n)) * (sqrt(sqrt(((k + k) * (k + k)))) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 0.0], N[(n * N[Sqrt[N[(2.0 * N[(Pi / N[(k * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+286], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(Pi * n), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[Sqrt[N[(N[(k + k), $MachinePrecision] * N[(k + k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 0.0

   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

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
   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 49.5

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

   4. Applied rewrites 49.5%

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

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

   6. Applied rewrites 37.9%

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

   7. Taylor expanded in n around inf

      \[\leadsto n \cdot \color{blue}{\sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{k \cdot n}}} \]
   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 48.6

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

   9. Applied rewrites 48.6%

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

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 2.00000000000000007e286

   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

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
   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 49.5

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

   4. Applied rewrites 49.5%

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

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

   6. Applied rewrites 37.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sqrt-prod N/A

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

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

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

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

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

      7. lower-unsound-sqrt.f64 49.5

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

   8. Applied rewrites 49.5%

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

   if 2.00000000000000007e286 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
   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 49.5

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

   4. Applied rewrites 49.5%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 49.4

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

   7. Applied rewrites 49.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

   9. Applied rewrites 49.4%

      \[\leadsto \sqrt{\pi \cdot n} \cdot \frac{\sqrt{\frac{2}{\frac{1}{k}}}}{\color{blue}{k}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 38.5

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

       5. lift-/.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. associate-/r/ N/A

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

       8. metadata-eval N/A

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

       9. count-2-rev N/A

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

       10. lower-+.f64 38.5

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

       11. lift-/.f64 N/A

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

       12. lift-/.f64 N/A

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

       13. associate-/r/ N/A

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

       14. metadata-eval N/A

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

       15. count-2-rev N/A

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

       16. lower-+.f64 38.5

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

   11. Applied rewrites 38.5%

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

Alternative 4: 61.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;n \leq 200000000:\\ \;\;\;\;\sqrt{\left|\frac{\pi + \pi}{k} \cdot n\right|}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{\pi}{k \cdot n}}\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 200000000.0)
   (sqrt (fabs (* (/ (+ PI PI) k) n)))
   (* n (sqrt (* 2.0 (/ PI (* k n)))))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 200000000.0) {
		tmp = sqrt(fabs((((((double) M_PI) + ((double) M_PI)) / k) * n)));
	} 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 <= 200000000.0) {
		tmp = Math.sqrt(Math.abs((((Math.PI + Math.PI) / k) * n)));
	} else {
		tmp = n * Math.sqrt((2.0 * (Math.PI / (k * n))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 200000000.0:
		tmp = math.sqrt(math.fabs((((math.pi + math.pi) / k) * n)))
	else:
		tmp = n * math.sqrt((2.0 * (math.pi / (k * n))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 200000000.0)
		tmp = sqrt(abs(Float64(Float64(Float64(pi + pi) / k) * n)));
	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 <= 200000000.0)
		tmp = sqrt(abs((((pi + pi) / k) * n)));
	else
		tmp = n * sqrt((2.0 * (pi / (k * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 200000000.0], N[Sqrt[N[Abs[N[(N[(N[(Pi + Pi), $MachinePrecision] / k), $MachinePrecision] * n), $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 2 regimes

2. if n < 2e8

   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

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
   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 49.5

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

   4. Applied rewrites 49.5%

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

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

   6. Applied rewrites 37.9%

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

   8. Applied rewrites 37.9%

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

   if 2e8 < 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

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
   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 49.5

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

   4. Applied rewrites 49.5%

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

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

   6. Applied rewrites 37.9%

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

   7. Taylor expanded in n around inf

      \[\leadsto n \cdot \color{blue}{\sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{k \cdot n}}} \]
   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 48.6

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

   9. Applied rewrites 48.6%

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

Alternative 5: 49.5% 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

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
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 49.5

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

4. Applied rewrites 49.5%

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

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

6. Applied rewrites 37.9%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. sqrt-prod N/A

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

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

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

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

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

   7. lower-unsound-sqrt.f64 49.5

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

8. Applied rewrites 49.5%

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

Alternative 6: 37.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
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 49.5

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

4. Applied rewrites 49.5%

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

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

6. Applied rewrites 37.9%

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

8. Applied rewrites 37.9%

   \[\leadsto \sqrt{\left|\frac{\pi + \pi}{k} \cdot n\right|} \]
9. Add Preprocessing

Alternative 7: 37.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
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 49.5

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

4. Applied rewrites 49.5%

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

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

6. Applied rewrites 37.9%

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

Alternative 8: 37.9% accurate, 4.0× speedup

Math

\[\sqrt{6.283185307179586 \cdot \frac{n}{k}} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (* 6.283185307179586 (/ n k))))

C

double code(double k, double n) {
	return sqrt((6.283185307179586 * (n / k)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(k, n)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: n
    code = sqrt((6.283185307179586d0 * (n / k)))
end function

Java

public static double code(double k, double n) {
	return Math.sqrt((6.283185307179586 * (n / k)));
}

Python

def code(k, n):
	return math.sqrt((6.283185307179586 * (n / k)))

Julia

function code(k, n)
	return sqrt(Float64(6.283185307179586 * Float64(n / k)))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt((6.283185307179586 * (n / k)));
end

Wolfram

code[k_, n_] := N[Sqrt[N[(6.283185307179586 * N[(n / 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

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
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 49.5

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

4. Applied rewrites 49.5%

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

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

6. Applied rewrites 37.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. distribute-lft-in N/A

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

   5. count-2 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f64 N/A

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

   15. lower-/.f64 37.9

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

8. Applied rewrites 37.9%

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

9. Evaluated real constant 37.9%

   \[\leadsto \sqrt{\frac{884279719003555}{140737488355328} \cdot \frac{n}{k}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025175 
(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))))