Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 6.0s

Alternatives: 10

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 1.745017235057577e+68
  2. n = -1.6566390987967273e+63

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-pow.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. +-commutative N/A

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

   6. div-add N/A

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

   7. metadata-eval N/A

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

   8. pow-add N/A

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

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

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[k_, n_] := N[(N[Power[N[(n * N[(Pi + Pi), $MachinePrecision]), $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. Add Preprocessing

Alternative 3: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

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

   6. Applied rewrites 38.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

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

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

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

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

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

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 49.4

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

   8. Applied rewrites 49.4%

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

   if 1 < k

   1. Initial program 99.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. +-commutative N/A

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

      6. div-add N/A

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

      7. metadata-eval N/A

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

      8. pow-add N/A

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

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

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

   3. Applied rewrites 99.5%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in k around inf

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

      1. lower-*.f64 53.6

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

   8. Applied rewrites 53.6%

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

Alternative 4: 73.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(\pi + \pi\right)\\ \mathbf{if}\;k \leq 0.026:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\frac{\pi + \pi}{k}}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{n \cdot \sqrt{\frac{t\_0}{n \cdot n}}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{k} \cdot \sqrt{\frac{t\_0}{k \cdot k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* n (+ PI PI))))
   (if (<= k 0.026)
     (* (sqrt n) (sqrt (/ (+ PI PI) k)))
     (if (<= k 1.5e+154)
       (/ (* n (sqrt (/ t_0 (* n n)))) (sqrt k))
       (* (sqrt k) (sqrt (/ t_0 (* k k))))))))

C

double code(double k, double n) {
	double t_0 = n * (((double) M_PI) + ((double) M_PI));
	double tmp;
	if (k <= 0.026) {
		tmp = sqrt(n) * sqrt(((((double) M_PI) + ((double) M_PI)) / k));
	} else if (k <= 1.5e+154) {
		tmp = (n * sqrt((t_0 / (n * n)))) / sqrt(k);
	} else {
		tmp = sqrt(k) * sqrt((t_0 / (k * k)));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = n * (Math.PI + Math.PI);
	double tmp;
	if (k <= 0.026) {
		tmp = Math.sqrt(n) * Math.sqrt(((Math.PI + Math.PI) / k));
	} else if (k <= 1.5e+154) {
		tmp = (n * Math.sqrt((t_0 / (n * n)))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt(k) * Math.sqrt((t_0 / (k * k)));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = n * (math.pi + math.pi)
	tmp = 0
	if k <= 0.026:
		tmp = math.sqrt(n) * math.sqrt(((math.pi + math.pi) / k))
	elif k <= 1.5e+154:
		tmp = (n * math.sqrt((t_0 / (n * n)))) / math.sqrt(k)
	else:
		tmp = math.sqrt(k) * math.sqrt((t_0 / (k * k)))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(n * Float64(pi + pi))
	tmp = 0.0
	if (k <= 0.026)
		tmp = Float64(sqrt(n) * sqrt(Float64(Float64(pi + pi) / k)));
	elseif (k <= 1.5e+154)
		tmp = Float64(Float64(n * sqrt(Float64(t_0 / Float64(n * n)))) / sqrt(k));
	else
		tmp = Float64(sqrt(k) * sqrt(Float64(t_0 / Float64(k * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = n * (pi + pi);
	tmp = 0.0;
	if (k <= 0.026)
		tmp = sqrt(n) * sqrt(((pi + pi) / k));
	elseif (k <= 1.5e+154)
		tmp = (n * sqrt((t_0 / (n * n)))) / sqrt(k);
	else
		tmp = sqrt(k) * sqrt((t_0 / (k * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 0.026], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.5e+154], N[(N[(n * N[Sqrt[N[(t$95$0 / N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[k], $MachinePrecision] * N[Sqrt[N[(t$95$0 / N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 0.0259999999999999988

   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

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

   6. Applied rewrites 38.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

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

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

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

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

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

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 49.4

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

   8. Applied rewrites 49.4%

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

   if 0.0259999999999999988 < k < 1.50000000000000013e154

   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

      \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{n}}}{\color{blue}{\sqrt{k}}} \]
   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-*.f64 N/A

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

      3. lower-sqrt.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-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-sqrt.f64 49.5

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

   7. Applied rewrites 49.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-add N/A

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

      6. *-commutative N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. distribute-lft-out N/A

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

      12. lift-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. sqr-neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. sqr-neg-rev N/A

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

      21. lower-*.f64 50.3

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

   9. Applied rewrites 50.3%

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

   if 1.50000000000000013e154 < k

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

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

   6. Applied rewrites 38.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 37.9

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

   8. Applied rewrites 37.9%

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

   10. Applied rewrites 33.2%

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

Alternative 5: 65.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\frac{\pi + \pi}{k}}\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\sqrt{k} \cdot \sqrt{\frac{n \cdot \left(\pi + \pi\right)}{k \cdot k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.8e-13)
   (* (sqrt n) (sqrt (/ (+ PI PI) k)))
   (if (<= k 1.35e+154)
     (* (sqrt (/ (+ PI PI) (* n k))) n)
     (* (sqrt k) (sqrt (/ (* n (+ PI PI)) (* k k)))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.8e-13) {
		tmp = sqrt(n) * sqrt(((((double) M_PI) + ((double) M_PI)) / k));
	} else if (k <= 1.35e+154) {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / (n * k))) * n;
	} else {
		tmp = sqrt(k) * sqrt(((n * (((double) M_PI) + ((double) M_PI))) / (k * k)));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.8e-13) {
		tmp = Math.sqrt(n) * Math.sqrt(((Math.PI + Math.PI) / k));
	} else if (k <= 1.35e+154) {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (n * k))) * n;
	} else {
		tmp = Math.sqrt(k) * Math.sqrt(((n * (Math.PI + Math.PI)) / (k * k)));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 4.8e-13:
		tmp = math.sqrt(n) * math.sqrt(((math.pi + math.pi) / k))
	elif k <= 1.35e+154:
		tmp = math.sqrt(((math.pi + math.pi) / (n * k))) * n
	else:
		tmp = math.sqrt(k) * math.sqrt(((n * (math.pi + math.pi)) / (k * k)))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.8e-13)
		tmp = Float64(sqrt(n) * sqrt(Float64(Float64(pi + pi) / k)));
	elseif (k <= 1.35e+154)
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / Float64(n * k))) * n);
	else
		tmp = Float64(sqrt(k) * sqrt(Float64(Float64(n * Float64(pi + pi)) / Float64(k * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 4.8e-13)
		tmp = sqrt(n) * sqrt(((pi + pi) / k));
	elseif (k <= 1.35e+154)
		tmp = sqrt(((pi + pi) / (n * k))) * n;
	else
		tmp = sqrt(k) * sqrt(((n * (pi + pi)) / (k * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.8e-13], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.35e+154], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / N[(n * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision], N[(N[Sqrt[k], $MachinePrecision] * N[Sqrt[N[(N[(n * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.7999999999999997e-13

   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

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

   6. Applied rewrites 38.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

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

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

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

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

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

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 49.4

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

   8. Applied rewrites 49.4%

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

   if 4.7999999999999997e-13 < k < 1.35000000000000003e154

   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

      \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{n}}}{\color{blue}{\sqrt{k}}} \]
   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-*.f64 N/A

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

      3. lower-sqrt.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-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-sqrt.f64 49.5

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

   7. Applied rewrites 49.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-undiv N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. count-2-rev N/A

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

      14. lift-+.f64 N/A

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

      15. associate-/l/ N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 50.0

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

   9. Applied rewrites 50.0%

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

   if 1.35000000000000003e154 < k

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

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

   6. Applied rewrites 38.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 37.9

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

   8. Applied rewrites 37.9%

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

   10. Applied rewrites 33.2%

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

Alternative 6: 62.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\frac{\pi + \pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.8e-13)
   (* (sqrt n) (sqrt (/ (+ PI PI) k)))
   (* (sqrt (/ (+ PI PI) (* n k))) n)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.8e-13) {
		tmp = sqrt(n) * sqrt(((((double) M_PI) + ((double) M_PI)) / k));
	} else {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / (n * k))) * n;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.8e-13) {
		tmp = Math.sqrt(n) * Math.sqrt(((Math.PI + Math.PI) / k));
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (n * k))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 4.8e-13:
		tmp = math.sqrt(n) * math.sqrt(((math.pi + math.pi) / k))
	else:
		tmp = math.sqrt(((math.pi + math.pi) / (n * k))) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.8e-13)
		tmp = Float64(sqrt(n) * sqrt(Float64(Float64(pi + pi) / k)));
	else
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / Float64(n * k))) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 4.8e-13)
		tmp = sqrt(n) * sqrt(((pi + pi) / k));
	else
		tmp = sqrt(((pi + pi) / (n * k))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.8e-13], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / N[(n * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.7999999999999997e-13

   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

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

   6. Applied rewrites 38.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

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

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

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

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

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

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 49.4

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

   8. Applied rewrites 49.4%

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

   if 4.7999999999999997e-13 < k

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      \[\leadsto \frac{n \cdot \sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{n}}}{\color{blue}{\sqrt{k}}} \]
   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-*.f64 N/A

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

      3. lower-sqrt.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-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-sqrt.f64 49.5

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

   7. Applied rewrites 49.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. sqrt-undiv N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. count-2-rev N/A

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

      14. lift-+.f64 N/A

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

      15. associate-/l/ N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 50.0

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

   9. Applied rewrites 50.0%

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

Alternative 7: 49.4% accurate, 2.7× speedup

Math

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

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

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

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

6. Applied rewrites 38.0%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. sqrt-prod N/A

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

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

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

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

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

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

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

   11. mult-flip-rev N/A

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

   12. lower-/.f64 49.4

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

8. Applied rewrites 49.4%

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

Alternative 8: 38.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

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

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

6. Applied rewrites 38.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. count-2-rev N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lift-PI.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 38.0

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

8. Applied rewrites 38.0%

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

Alternative 9: 38.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

6. Applied rewrites 38.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 38.0

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

8. Applied rewrites 38.0%

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

Alternative 10: 37.9% accurate, 3.1× speedup

Math

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

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

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

6. Applied rewrites 38.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-+.f64 N/A

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

   6. count-2-rev N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 37.9

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

8. Applied rewrites 37.9%

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

Reproduce

herbie shell --seed 2025164 
(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))))