Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 7.4s

Alternatives: 14

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 4.67444209155444e+118
  2. n = -6.3246239397115604e+75

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in n around 0

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

   1. sum-log N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 96.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift--.f64 N/A

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

   5. exp-to-pow N/A

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

   6. lift-sqrt.f64 N/A

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

   7. sqrt-pow2 N/A

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

   8. div-sub N/A

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

   9. metadata-eval N/A

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

   10. pow-sub N/A

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

6. Applied rewrites 99.5%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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. metadata-eval N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-div N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 99.4

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

3. Applied rewrites 99.4%

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

Alternative 3: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. associate-/r* N/A

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

   4. mult-flip N/A

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

   5. div-flip N/A

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

   6. /-rgt-identity N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 90.9%

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

6. Applied rewrites 99.4%

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

Alternative 4: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[k_, n_] := N[(N[Power[N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision], N[(-0.5 * k + 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. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. mult-flip-rev N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.4%

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

Alternative 5: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in n around 0

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

   1. sum-log N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 96.2%

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

6. Applied rewrites 99.4%

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

Alternative 6: 75.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 9500000000:\\ \;\;\;\;\frac{\sqrt{\frac{\pi \cdot k}{n} \cdot 2} \cdot n}{k}\\ \mathbf{elif}\;n \leq 2 \cdot 10^{+180}:\\ \;\;\;\;\sqrt{\frac{\left(2 \cdot k\right) \cdot \left(n \cdot \pi\right)}{\left(n \cdot k\right) \cdot \left(n \cdot k\right)}} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 9500000000.0)
   (/ (* (sqrt (* (/ (* PI k) n) 2.0)) n) k)
   (if (<= n 2e+180)
     (* (sqrt (/ (* (* 2.0 k) (* n PI)) (* (* n k) (* n k)))) n)
     (* (/ 1.0 (sqrt (/ (* n k) (+ PI PI)))) n))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 9500000000.0) {
		tmp = (sqrt((((((double) M_PI) * k) / n) * 2.0)) * n) / k;
	} else if (n <= 2e+180) {
		tmp = sqrt((((2.0 * k) * (n * ((double) M_PI))) / ((n * k) * (n * k)))) * n;
	} else {
		tmp = (1.0 / sqrt(((n * k) / (((double) M_PI) + ((double) M_PI))))) * n;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 9500000000.0) {
		tmp = (Math.sqrt((((Math.PI * k) / n) * 2.0)) * n) / k;
	} else if (n <= 2e+180) {
		tmp = Math.sqrt((((2.0 * k) * (n * Math.PI)) / ((n * k) * (n * k)))) * n;
	} else {
		tmp = (1.0 / Math.sqrt(((n * k) / (Math.PI + Math.PI)))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 9500000000.0:
		tmp = (math.sqrt((((math.pi * k) / n) * 2.0)) * n) / k
	elif n <= 2e+180:
		tmp = math.sqrt((((2.0 * k) * (n * math.pi)) / ((n * k) * (n * k)))) * n
	else:
		tmp = (1.0 / math.sqrt(((n * k) / (math.pi + math.pi)))) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 9500000000.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(pi * k) / n) * 2.0)) * n) / k);
	elseif (n <= 2e+180)
		tmp = Float64(sqrt(Float64(Float64(Float64(2.0 * k) * Float64(n * pi)) / Float64(Float64(n * k) * Float64(n * k)))) * n);
	else
		tmp = Float64(Float64(1.0 / sqrt(Float64(Float64(n * k) / Float64(pi + pi)))) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 9500000000.0)
		tmp = (sqrt((((pi * k) / n) * 2.0)) * n) / k;
	elseif (n <= 2e+180)
		tmp = sqrt((((2.0 * k) * (n * pi)) / ((n * k) * (n * k)))) * n;
	else
		tmp = (1.0 / sqrt(((n * k) / (pi + pi)))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 9500000000.0], N[(N[(N[Sqrt[N[(N[(N[(Pi * k), $MachinePrecision] / n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[n, 2e+180], N[(N[Sqrt[N[(N[(N[(2.0 * k), $MachinePrecision] * N[(n * Pi), $MachinePrecision]), $MachinePrecision] / N[(N[(n * k), $MachinePrecision] * N[(n * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision], N[(N[(1.0 / N[Sqrt[N[(N[(n * k), $MachinePrecision] / N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 9.5e9

   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. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.9

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

   7. Applied rewrites 49.9%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 50.0

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

   10. Applied rewrites 50.0%

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

   if 9.5e9 < n < 2e180

   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. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.9

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

   7. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2-rev N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. count-2-rev N/A

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

      12. lift-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-/.f64 48.8

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

   9. Applied rewrites 48.8%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l/ N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-PI.f64 N/A

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

       6. lift-+.f64 N/A

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

       7. *-commutative N/A

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

       8. div-add-rev N/A

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

       9. frac-add N/A

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

   11. Applied rewrites 35.1%

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

   if 2e180 < 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. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.9

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

   7. Applied rewrites 49.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 50.4

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

   9. Applied rewrites 50.4%

      \[\leadsto \frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 73.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 9500000000:\\ \;\;\;\;\frac{\sqrt{\frac{\pi \cdot k}{n} \cdot 2} \cdot n}{k}\\ \mathbf{elif}\;n \leq 2 \cdot 10^{+180}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\pi, n \cdot k, \left(n \cdot \pi\right) \cdot k\right)}{\left(n \cdot k\right) \cdot \left(n \cdot k\right)}} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 9500000000.0)
   (/ (* (sqrt (* (/ (* PI k) n) 2.0)) n) k)
   (if (<= n 2e+180)
     (* (sqrt (/ (fma PI (* n k) (* (* n PI) k)) (* (* n k) (* n k)))) n)
     (* (/ 1.0 (sqrt (/ (* n k) (+ PI PI)))) n))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 9500000000.0) {
		tmp = (sqrt((((((double) M_PI) * k) / n) * 2.0)) * n) / k;
	} else if (n <= 2e+180) {
		tmp = sqrt((fma(((double) M_PI), (n * k), ((n * ((double) M_PI)) * k)) / ((n * k) * (n * k)))) * n;
	} else {
		tmp = (1.0 / sqrt(((n * k) / (((double) M_PI) + ((double) M_PI))))) * n;
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 9500000000.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(pi * k) / n) * 2.0)) * n) / k);
	elseif (n <= 2e+180)
		tmp = Float64(sqrt(Float64(fma(pi, Float64(n * k), Float64(Float64(n * pi) * k)) / Float64(Float64(n * k) * Float64(n * k)))) * n);
	else
		tmp = Float64(Float64(1.0 / sqrt(Float64(Float64(n * k) / Float64(pi + pi)))) * n);
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[n, 9500000000.0], N[(N[(N[Sqrt[N[(N[(N[(Pi * k), $MachinePrecision] / n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[n, 2e+180], N[(N[Sqrt[N[(N[(Pi * N[(n * k), $MachinePrecision] + N[(N[(n * Pi), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / N[(N[(n * k), $MachinePrecision] * N[(n * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision], N[(N[(1.0 / N[Sqrt[N[(N[(n * k), $MachinePrecision] / N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 9.5e9

   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. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.9

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

   7. Applied rewrites 49.9%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 50.0

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

   10. Applied rewrites 50.0%

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

   if 9.5e9 < n < 2e180

   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. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.9

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

   7. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-add-rev N/A

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

      8. frac-add N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 35.2

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

   9. Applied rewrites 35.2%

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

   if 2e180 < 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. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.9

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

   7. Applied rewrites 49.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 50.4

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

   9. Applied rewrites 50.4%

      \[\leadsto \frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 73.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.000235:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\frac{\pi + \pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{\left(\pi + \pi\right) \cdot n}{n \cdot n}}{k}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 0.000235)
   (* (sqrt n) (sqrt (/ (+ PI PI) k)))
   (* (sqrt (/ (/ (* (+ PI PI) n) (* n n)) k)) n)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 0.000235) {
		tmp = sqrt(n) * sqrt(((((double) M_PI) + ((double) M_PI)) / k));
	} else {
		tmp = sqrt(((((((double) M_PI) + ((double) M_PI)) * n) / (n * n)) / k)) * n;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 0.000235) {
		tmp = Math.sqrt(n) * Math.sqrt(((Math.PI + Math.PI) / k));
	} else {
		tmp = Math.sqrt(((((Math.PI + Math.PI) * n) / (n * n)) / k)) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 0.000235:
		tmp = math.sqrt(n) * math.sqrt(((math.pi + math.pi) / k))
	else:
		tmp = math.sqrt(((((math.pi + math.pi) * n) / (n * n)) / k)) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 0.000235)
		tmp = Float64(sqrt(n) * sqrt(Float64(Float64(pi + pi) / k)));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(pi + pi) * n) / Float64(n * n)) / k)) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 0.000235)
		tmp = sqrt(n) * sqrt(((pi + pi) / k));
	else
		tmp = sqrt(((((pi + pi) * n) / (n * n)) / k)) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 0.000235], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.34999999999999993e-4

   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. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}}} \]
   5. 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. lower-*.f64 N/A

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

      5. lower-/.f64 37.5

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

   6. Applied rewrites 37.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. sqrt-div N/A

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

   8. Applied rewrites 49.0%

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

   if 2.34999999999999993e-4 < 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. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.9

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

   7. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2-rev N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. count-2-rev N/A

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

      12. lift-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-/.f64 48.8

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

   9. Applied rewrites 48.8%

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

       1. lift-/.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. div-add N/A

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

       6. frac-add N/A

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

       7. *-commutative N/A

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

       8. count-2-rev N/A

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

       9. lower-/.f64 N/A

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

       10. count-2-rev N/A

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

       11. distribute-lft-in N/A

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

       12. lift-+.f64 N/A

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

       13. lift-PI.f64 N/A

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

       14. lift-PI.f64 N/A

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

       15. *-commutative N/A

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

       16. lift-*.f64 N/A

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

       17. lower-*.f64 49.9

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

   11. Applied rewrites 49.9%

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

Alternative 9: 73.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{\sqrt{\frac{\pi \cdot k}{n} \cdot 2} \cdot n}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 2.8e-22)
   (/ (* (sqrt (* (/ (* PI k) n) 2.0)) n) k)
   (* (/ 1.0 (sqrt (/ (* n k) (+ PI PI)))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 2.8e-22) {
		tmp = (sqrt((((((double) M_PI) * k) / n) * 2.0)) * n) / k;
	} else {
		tmp = (1.0 / sqrt(((n * k) / (((double) M_PI) + ((double) M_PI))))) * n;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 2.8e-22) {
		tmp = (Math.sqrt((((Math.PI * k) / n) * 2.0)) * n) / k;
	} else {
		tmp = (1.0 / Math.sqrt(((n * k) / (Math.PI + Math.PI)))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 2.8e-22:
		tmp = (math.sqrt((((math.pi * k) / n) * 2.0)) * n) / k
	else:
		tmp = (1.0 / math.sqrt(((n * k) / (math.pi + math.pi)))) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 2.8e-22)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(pi * k) / n) * 2.0)) * n) / k);
	else
		tmp = Float64(Float64(1.0 / sqrt(Float64(Float64(n * k) / Float64(pi + pi)))) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 2.8e-22)
		tmp = (sqrt((((pi * k) / n) * 2.0)) * n) / k;
	else
		tmp = (1.0 / sqrt(((n * k) / (pi + pi)))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 2.8e-22], N[(N[(N[Sqrt[N[(N[(N[(Pi * k), $MachinePrecision] / n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision] / k), $MachinePrecision], N[(N[(1.0 / N[Sqrt[N[(N[(n * k), $MachinePrecision] / N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.79999999999999995e-22

   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. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.9

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

   7. Applied rewrites 49.9%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 50.0

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

   10. Applied rewrites 50.0%

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

   if 2.79999999999999995e-22 < 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. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.9

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

   7. Applied rewrites 49.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 50.4

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

   9. Applied rewrites 50.4%

      \[\leadsto \frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 61.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.75 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{\frac{\pi + \pi}{n}}}{\sqrt{k}} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 1.75e-51)
   (* (/ (sqrt (/ (+ PI PI) n)) (sqrt k)) n)
   (* (sqrt (/ (+ PI PI) (* n k))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 1.75e-51) {
		tmp = (sqrt(((((double) M_PI) + ((double) M_PI)) / n)) / sqrt(k)) * n;
	} 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 (n <= 1.75e-51) {
		tmp = (Math.sqrt(((Math.PI + Math.PI) / n)) / Math.sqrt(k)) * n;
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (n * k))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 1.75e-51:
		tmp = (math.sqrt(((math.pi + math.pi) / n)) / math.sqrt(k)) * n
	else:
		tmp = math.sqrt(((math.pi + math.pi) / (n * k))) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 1.75e-51)
		tmp = Float64(Float64(sqrt(Float64(Float64(pi + pi) / n)) / sqrt(k)) * n);
	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 (n <= 1.75e-51)
		tmp = (sqrt(((pi + pi) / n)) / sqrt(k)) * n;
	else
		tmp = sqrt(((pi + pi) / (n * k))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 1.75e-51], N[(N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * n), $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 n < 1.7499999999999999e-51

   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. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.9

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

   7. Applied rewrites 49.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. count-2-rev N/A

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

      9. associate-*r/ N/A

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

      10. sqrt-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. associate-*r/ N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lift-sqrt.f64 49.0

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

   9. Applied rewrites 49.0%

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

   if 1.7499999999999999e-51 < 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. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.9

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

   7. Applied rewrites 49.9%

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

Alternative 11: 61.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 10^{-14}:\\ \;\;\;\;\frac{\sqrt{\frac{\pi + \pi}{n}}}{\sqrt{k}} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 1e-14)
   (* (/ (sqrt (/ (+ PI PI) n)) (sqrt k)) n)
   (* (/ 1.0 (sqrt (/ (* n k) (+ PI PI)))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 1e-14) {
		tmp = (sqrt(((((double) M_PI) + ((double) M_PI)) / n)) / sqrt(k)) * n;
	} else {
		tmp = (1.0 / sqrt(((n * k) / (((double) M_PI) + ((double) M_PI))))) * n;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 1e-14) {
		tmp = (Math.sqrt(((Math.PI + Math.PI) / n)) / Math.sqrt(k)) * n;
	} else {
		tmp = (1.0 / Math.sqrt(((n * k) / (Math.PI + Math.PI)))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 1e-14:
		tmp = (math.sqrt(((math.pi + math.pi) / n)) / math.sqrt(k)) * n
	else:
		tmp = (1.0 / math.sqrt(((n * k) / (math.pi + math.pi)))) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 1e-14)
		tmp = Float64(Float64(sqrt(Float64(Float64(pi + pi) / n)) / sqrt(k)) * n);
	else
		tmp = Float64(Float64(1.0 / sqrt(Float64(Float64(n * k) / Float64(pi + pi)))) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 1e-14)
		tmp = (sqrt(((pi + pi) / n)) / sqrt(k)) * n;
	else
		tmp = (1.0 / sqrt(((n * k) / (pi + pi)))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 1e-14], N[(N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(N[(1.0 / N[Sqrt[N[(N[(n * k), $MachinePrecision] / N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 9.99999999999999999e-15

   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. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.9

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

   7. Applied rewrites 49.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. count-2-rev N/A

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

      9. associate-*r/ N/A

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

      10. sqrt-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. associate-*r/ N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lift-sqrt.f64 49.0

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

   9. Applied rewrites 49.0%

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

   if 9.99999999999999999e-15 < 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. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.9

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

   7. Applied rewrites 49.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 50.4

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

   9. Applied rewrites 50.4%

      \[\leadsto \frac{1}{\sqrt{\frac{n \cdot k}{\pi + \pi}}} \cdot n \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 60.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.5 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{\frac{\pi \cdot n}{k} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 2.5e-22)
   (sqrt (* (/ (* PI n) k) 2.0))
   (* (sqrt (/ (+ PI PI) (* n k))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 2.5e-22) {
		tmp = sqrt((((((double) M_PI) * n) / k) * 2.0));
	} 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 (n <= 2.5e-22) {
		tmp = Math.sqrt((((Math.PI * n) / k) * 2.0));
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (n * k))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 2.5e-22:
		tmp = math.sqrt((((math.pi * n) / k) * 2.0))
	else:
		tmp = math.sqrt(((math.pi + math.pi) / (n * k))) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 2.5e-22)
		tmp = sqrt(Float64(Float64(Float64(pi * n) / k) * 2.0));
	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 (n <= 2.5e-22)
		tmp = sqrt((((pi * n) / k) * 2.0));
	else
		tmp = sqrt(((pi + pi) / (n * k))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 2.5e-22], N[Sqrt[N[(N[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision] * 2.0), $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 n < 2.49999999999999977e-22

   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. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

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

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 37.5

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

   6. Applied rewrites 37.5%

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

   if 2.49999999999999977e-22 < 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. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 49.9

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

   7. Applied rewrites 49.9%

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

Alternative 13: 49.0% 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. unpow1/2 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-PI.f64 37.5

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

4. Applied rewrites 37.5%

   \[\leadsto \color{blue}{\sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}}} \]
5. 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. lower-*.f64 N/A

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

   5. lower-/.f64 37.5

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

6. Applied rewrites 37.5%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. sqrt-div N/A

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

8. Applied rewrites 49.0%

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

Alternative 14: 37.5% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(n * 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. unpow1/2 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-PI.f64 37.5

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

4. Applied rewrites 37.5%

   \[\leadsto \color{blue}{\sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}}} \]
5. 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. lower-*.f64 N/A

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

   5. lower-/.f64 37.5

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

6. Applied rewrites 37.5%

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

   1. lift-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. count-2-rev N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. metadata-eval N/A

       \[\leadsto \sqrt{\frac{\left(\sqrt{4} \cdot \mathsf{PI}\left(\right)\right) \cdot n}{k}} \]

   13. add-sqr-sqrt N/A

       \[\leadsto \sqrt{\frac{\left(\sqrt{4} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot n}{k}} \]

   14. sqrt-prod N/A

       \[\leadsto \sqrt{\frac{\left(\sqrt{4} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}\right) \cdot n}{k}} \]

   15. unpow2 N/A

       \[\leadsto \sqrt{\frac{\left(\sqrt{4} \cdot \sqrt{{\mathsf{PI}\left(\right)}^{2}}\right) \cdot n}{k}} \]

   16. sqrt-prod N/A

       \[\leadsto \sqrt{\frac{\sqrt{4 \cdot {\mathsf{PI}\left(\right)}^{2}} \cdot n}{k}} \]

   17. *-commutative N/A

       \[\leadsto \sqrt{\frac{n \cdot \sqrt{4 \cdot {\mathsf{PI}\left(\right)}^{2}}}{k}} \]

   18. associate-/l* N/A

       \[\leadsto \sqrt{n \cdot \frac{\sqrt{4 \cdot {\mathsf{PI}\left(\right)}^{2}}}{k}} \]

   19. lower-*.f64 N/A

       \[\leadsto \sqrt{n \cdot \frac{\sqrt{4 \cdot {\mathsf{PI}\left(\right)}^{2}}}{k}} \]

8. Applied rewrites 37.5%

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

Reproduce

herbie shell --seed 2025139 
(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))))