Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 7.8s

Alternatives: 14

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 3.46611332319703e+297
  2. n = -3.809592942999379e+68

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: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. count-2-rev N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lift-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-pow.f64 N/A

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

   11. exp-to-pow N/A

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

   12. distribute-rgt-in N/A

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

   13. log-pow-rev N/A

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

   14. pow1/2 N/A

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

   15. exp-sum N/A

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.4%

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

Alternative 2: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$0, N[(k * -0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. count-2-rev N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lift-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-pow.f64 N/A

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

   11. exp-to-pow N/A

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

   12. distribute-rgt-in N/A

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

   13. log-pow-rev N/A

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

   14. pow1/2 N/A

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

   15. exp-sum N/A

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

5. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-sqrt.f64 N/A

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

7. Applied rewrites 99.4%

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

Alternative 3: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. count-2-rev N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lift-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-pow.f64 N/A

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

   11. exp-to-pow N/A

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

   12. distribute-rgt-in N/A

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

   13. log-pow-rev N/A

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

   14. pow1/2 N/A

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

   15. exp-sum N/A

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

5. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

7. Applied rewrites 99.4%

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

Alternative 4: 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-sqrt.f64 N/A

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

   2. lower-/.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. inv-pow N/A

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

   4. pow1/2 N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. sqrt-pow1 N/A

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

   9. inv-pow N/A

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

   10. lower-sqrt.f64 N/A

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

   11. 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 5: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-sqrt.f64 N/A

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

   2. lower-/.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. inv-pow N/A

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

   4. pow1/2 N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. sqrt-pow1 N/A

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

   9. inv-pow N/A

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

   10. lower-sqrt.f64 N/A

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

   11. 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. Taylor expanded in k around inf

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

6. Applied rewrites 99.3%

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

Alternative 6: 99.3% accurate, 1.1× speedup

Math

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

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((sqrt(Float64(n * Float64(pi + pi))) ^ Float64(1.0 - k)) * Float64(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[Power[N[Sqrt[N[(n * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* 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 38.8

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

4. Applied rewrites 38.8%

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

5. Taylor expanded in k around inf

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

7. Applied rewrites 99.3%

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

9. Applied rewrites 99.3%

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

Alternative 7: 99.3% 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 8: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-sqrt.f64 N/A

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

   2. lower-/.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. inv-pow N/A

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

   4. pow1/2 N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. sqrt-pow1 N/A

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

   9. inv-pow N/A

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

   10. lower-sqrt.f64 N/A

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

   11. 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. Taylor expanded in k around inf

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

6. Applied rewrites 99.3%

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

Alternative 9: 80.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{\frac{1}{k}} \cdot \frac{n}{\sqrt{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\sqrt{\frac{1}{k}} \cdot \sqrt{\left(\pi + \pi\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{\pi + \pi}{n} \cdot k} \cdot n}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0)))))
   (if (<= t_0 0.0)
     (sqrt (* (/ (+ PI PI) (/ 1.0 k)) (/ n (sqrt (* (* k k) (* k k))))))
     (if (<= t_0 2e+303)
       (* (sqrt (/ 1.0 k)) (sqrt (* (+ PI PI) n)))
       (/ (* (sqrt (* (/ (+ PI PI) n) k)) n) k)))))

C

double code(double k, double n) {
	double t_0 = (1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt((((((double) M_PI) + ((double) M_PI)) / (1.0 / k)) * (n / sqrt(((k * k) * (k * k))))));
	} else if (t_0 <= 2e+303) {
		tmp = sqrt((1.0 / k)) * sqrt(((((double) M_PI) + ((double) M_PI)) * n));
	} else {
		tmp = (sqrt((((((double) M_PI) + ((double) M_PI)) / n) * k)) * n) / k;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = (1.0 / Math.sqrt(k)) * Math.pow(((2.0 * Math.PI) * n), ((1.0 - k) / 2.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt((((Math.PI + Math.PI) / (1.0 / k)) * (n / Math.sqrt(((k * k) * (k * k))))));
	} else if (t_0 <= 2e+303) {
		tmp = Math.sqrt((1.0 / k)) * Math.sqrt(((Math.PI + Math.PI) * n));
	} else {
		tmp = (Math.sqrt((((Math.PI + Math.PI) / n) * k)) * n) / k;
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = (1.0 / math.sqrt(k)) * math.pow(((2.0 * math.pi) * n), ((1.0 - k) / 2.0))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt((((math.pi + math.pi) / (1.0 / k)) * (n / math.sqrt(((k * k) * (k * k))))))
	elif t_0 <= 2e+303:
		tmp = math.sqrt((1.0 / k)) * math.sqrt(((math.pi + math.pi) * n))
	else:
		tmp = (math.sqrt((((math.pi + math.pi) / n) * k)) * n) / k
	return tmp

Julia

function code(k, n)
	t_0 = Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = sqrt(Float64(Float64(Float64(pi + pi) / Float64(1.0 / k)) * Float64(n / sqrt(Float64(Float64(k * k) * Float64(k * k))))));
	elseif (t_0 <= 2e+303)
		tmp = Float64(sqrt(Float64(1.0 / k)) * sqrt(Float64(Float64(pi + pi) * n)));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(pi + pi) / n) * k)) * n) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = (1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt((((pi + pi) / (1.0 / k)) * (n / sqrt(((k * k) * (k * k))))));
	elseif (t_0 <= 2e+303)
		tmp = sqrt((1.0 / k)) * sqrt(((pi + pi) * n));
	else
		tmp = (sqrt((((pi + pi) / n) * k)) * n) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[Sqrt[N[(N[(N[(Pi + Pi), $MachinePrecision] / N[(1.0 / k), $MachinePrecision]), $MachinePrecision] * N[(n / N[Sqrt[N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 2e+303], N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[(Pi + Pi), $MachinePrecision] / n), $MachinePrecision] * k), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision] / k), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. sqrt-undiv N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* 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 38.8

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

   4. Applied rewrites 38.8%

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

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-+.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-/.f64 38.8

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

   6. Applied rewrites 38.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.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. rem-square-sqrt N/A

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

      5. pow2 N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. pow-div N/A

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

      9. unpow1 N/A

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

      10. pow1/2 N/A

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

      11. mult-flip N/A

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

      12. inv-pow N/A

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

      13. pow1/2 N/A

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

      14. pow-pow N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-pow1 N/A

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

      18. inv-pow N/A

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

      19. *-commutative N/A

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

      20. pow-prod-down N/A

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

      21. unpow2 N/A

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

      22. rem-square-sqrt N/A

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

   8. Applied rewrites 32.6%

      \[\leadsto \sqrt{\frac{\pi + \pi}{\frac{1}{k}} \cdot \frac{n}{k \cdot k}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. fabs-sqr N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 31.0

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

   10. Applied rewrites 31.0%

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

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 2e303

   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-sqrt.f64 N/A

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

      2. lower-/.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. inv-pow N/A

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

      4. pow1/2 N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. inv-pow N/A

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

      10. lower-sqrt.f64 N/A

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

      11. 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. Taylor expanded in k around 0

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-prod N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 50.4

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

   6. Applied rewrites 50.4%

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

   if 2e303 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   1. Initial program 99.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lower-/.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. inv-pow N/A

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

      4. pow1/2 N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. inv-pow N/A

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

      10. lower-sqrt.f64 N/A

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

      11. 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. Taylor expanded in k around 0

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

      1. inv-pow N/A

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

      2. sqrt-pow1 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. pow1/2 N/A

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

      7. inv-pow N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 39.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. count-2-rev N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 50.4%

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

   9. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. sqrt-unprod N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. associate-*r/ N/A

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

       8. count-2-rev N/A

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

       9. lift-+.f64 N/A

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

       10. lift-PI.f64 N/A

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

       11. lift-PI.f64 N/A

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

       12. lower-/.f64 50.9

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

   11. Applied rewrites 50.9%

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

Alternative 10: 74.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sqrt{\frac{\pi + \pi}{n} \cdot k} \cdot n}{k}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 2.4e-15)
   (/ (* (sqrt (* (/ (+ PI PI) n) k)) n) k)
   (* (sqrt (/ (+ PI PI) (* n k))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 2.4e-15) {
		tmp = (sqrt((((((double) M_PI) + ((double) M_PI)) / n) * k)) * n) / k;
	} else {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / (n * k))) * n;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 2.4e-15) {
		tmp = (Math.sqrt((((Math.PI + Math.PI) / n) * k)) * n) / k;
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (n * k))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 2.4e-15:
		tmp = (math.sqrt((((math.pi + math.pi) / n) * k)) * n) / k
	else:
		tmp = math.sqrt(((math.pi + math.pi) / (n * k))) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 2.4e-15)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(pi + pi) / n) * k)) * n) / k);
	else
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / Float64(n * k))) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 2.4e-15)
		tmp = (sqrt((((pi + pi) / n) * k)) * n) / k;
	else
		tmp = sqrt(((pi + pi) / (n * k))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 2.4e-15], N[(N[(N[Sqrt[N[(N[(N[(Pi + Pi), $MachinePrecision] / n), $MachinePrecision] * k), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision] / k), $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 < 2.39999999999999995e-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. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lower-/.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. inv-pow N/A

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

      4. pow1/2 N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. inv-pow N/A

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

      10. lower-sqrt.f64 N/A

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

      11. 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. Taylor expanded in k around 0

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

      1. inv-pow N/A

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

      2. sqrt-pow1 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. pow1/2 N/A

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

      7. inv-pow N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 39.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. count-2-rev N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   8. Applied rewrites 50.4%

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

   9. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. sqrt-unprod N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. associate-*r/ N/A

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

       8. count-2-rev N/A

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

       9. lift-+.f64 N/A

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

       10. lift-PI.f64 N/A

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

       11. lift-PI.f64 N/A

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

       12. lower-/.f64 50.9

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

   11. Applied rewrites 50.9%

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

   if 2.39999999999999995e-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. sqrt-undiv N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* 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 38.8

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

   4. Applied rewrites 38.8%

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

   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. lower-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-PI.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 50.5

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

   7. Applied rewrites 50.5%

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

Alternative 11: 65.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{1}{k}} \cdot \sqrt{\left(\pi + \pi\right) \cdot n}\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\pi \cdot k\right) \cdot 2\right) \cdot \frac{n}{k \cdot k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.55e-11)
   (* (sqrt (/ 1.0 k)) (sqrt (* (+ PI PI) n)))
   (if (<= k 1.35e+154)
     (* (sqrt (/ (+ PI PI) (* n k))) n)
     (sqrt (* (* (* PI k) 2.0) (/ n (* k k)))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.55e-11) {
		tmp = sqrt((1.0 / k)) * sqrt(((((double) M_PI) + ((double) M_PI)) * n));
	} else if (k <= 1.35e+154) {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / (n * k))) * n;
	} else {
		tmp = sqrt((((((double) M_PI) * k) * 2.0) * (n / (k * k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.55e-11) {
		tmp = Math.sqrt((1.0 / k)) * Math.sqrt(((Math.PI + Math.PI) * n));
	} else if (k <= 1.35e+154) {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (n * k))) * n;
	} else {
		tmp = Math.sqrt((((Math.PI * k) * 2.0) * (n / (k * k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.55e-11:
		tmp = math.sqrt((1.0 / k)) * math.sqrt(((math.pi + math.pi) * n))
	elif k <= 1.35e+154:
		tmp = math.sqrt(((math.pi + math.pi) / (n * k))) * n
	else:
		tmp = math.sqrt((((math.pi * k) * 2.0) * (n / (k * k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.55e-11)
		tmp = Float64(sqrt(Float64(1.0 / k)) * sqrt(Float64(Float64(pi + pi) * n)));
	elseif (k <= 1.35e+154)
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / Float64(n * k))) * n);
	else
		tmp = sqrt(Float64(Float64(Float64(pi * k) * 2.0) * Float64(n / Float64(k * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.55e-11)
		tmp = sqrt((1.0 / k)) * sqrt(((pi + pi) * n));
	elseif (k <= 1.35e+154)
		tmp = sqrt(((pi + pi) / (n * k))) * n;
	else
		tmp = sqrt((((pi * k) * 2.0) * (n / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.55e-11], N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.35e+154], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / N[(n * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision], N[Sqrt[N[(N[(N[(Pi * k), $MachinePrecision] * 2.0), $MachinePrecision] * N[(n / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.55000000000000014e-11

   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-sqrt.f64 N/A

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

      2. lower-/.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. inv-pow N/A

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

      4. pow1/2 N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. inv-pow N/A

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

      10. lower-sqrt.f64 N/A

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

      11. 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. Taylor expanded in k around 0

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-prod N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 50.4

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

   6. Applied rewrites 50.4%

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

   if 1.55000000000000014e-11 < k < 1.35000000000000003e154

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. sqrt-undiv N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* 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 38.8

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

   4. Applied rewrites 38.8%

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

   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. lower-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-PI.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 50.5

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

   7. Applied rewrites 50.5%

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

   if 1.35000000000000003e154 < k

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. sqrt-undiv N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* 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 38.8

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

   4. Applied rewrites 38.8%

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

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-+.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-/.f64 38.8

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

   6. Applied rewrites 38.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.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. rem-square-sqrt N/A

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

      5. pow2 N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. pow-div N/A

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

      9. unpow1 N/A

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

      10. pow1/2 N/A

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

      11. mult-flip N/A

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

      12. inv-pow N/A

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

      13. pow1/2 N/A

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

      14. pow-pow N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. sqrt-pow1 N/A

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

      18. inv-pow N/A

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

      19. *-commutative N/A

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

      20. pow-prod-down N/A

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

      21. unpow2 N/A

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

      22. rem-square-sqrt N/A

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

   8. Applied rewrites 32.6%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lift-PI.f64 32.6

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

   11. Applied rewrites 32.6%

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

Alternative 12: 62.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 0.05:\\ \;\;\;\;\sqrt{\left(\pi + \pi\right) \cdot \frac{n}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 0.05)
   (sqrt (* (+ PI PI) (/ n k)))
   (* (sqrt (/ (+ PI PI) (* n k))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 0.05) {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) * (n / k)));
	} else {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / (n * k))) * n;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 0.05) {
		tmp = Math.sqrt(((Math.PI + Math.PI) * (n / k)));
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (n * k))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 0.05:
		tmp = math.sqrt(((math.pi + math.pi) * (n / k)))
	else:
		tmp = math.sqrt(((math.pi + math.pi) / (n * k))) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 0.05)
		tmp = sqrt(Float64(Float64(pi + pi) * Float64(n / k)));
	else
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / Float64(n * k))) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 0.05)
		tmp = sqrt(((pi + pi) * (n / k)));
	else
		tmp = sqrt(((pi + pi) / (n * k))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 0.05], N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] * N[(n / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / N[(n * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 0.050000000000000003

   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. sqrt-undiv N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* 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 38.8

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

   4. Applied rewrites 38.8%

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

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-+.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-/.f64 38.8

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

   6. Applied rewrites 38.8%

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

   if 0.050000000000000003 < 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. sqrt-undiv N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* 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 38.8

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

   4. Applied rewrites 38.8%

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

   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. lower-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-PI.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 50.5

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

   7. Applied rewrites 50.5%

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

Alternative 13: 50.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* 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 38.8

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

4. Applied rewrites 38.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. count-2-rev N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. sqrt-undiv N/A

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

   11. pow1/2 N/A

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

   12. lower-/.f64 N/A

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

6. Applied rewrites 50.4%

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

Alternative 14: 38.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. unpow1/2 N/A

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

   2. sqrt-undiv N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* 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 38.8

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

4. Applied rewrites 38.8%

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

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

   8. lower-*.f64 N/A

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

   9. count-2-rev N/A

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

   10. lift-+.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-/.f64 38.8

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

6. Applied rewrites 38.8%

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

Reproduce

herbie shell --seed 2025134 
(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))))