Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 6.5s

Alternatives: 13

Speedup: 1.1×

• Could not converge on a ground truth

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(n + n), $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-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. div-sub N/A

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

   8. metadata-eval N/A

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

   9. pow-sub N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.5%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{\sqrt{\left(\pi + \pi\right) \cdot n}}{{\left(\left(\pi + \pi\right) \cdot n\right)}^{\left(0.5 \cdot k\right)}}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lift-+.f64 N/A

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

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{1 \cdot {\left(\left(n + n\right) \cdot \pi\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}}{\sqrt{k}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-lft-identity 99.4

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

   3. lift-fma.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. pow-add N/A

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

   6. pow1/2 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. count-2-rev N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lift-PI.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 N/A

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

7. Applied rewrites 99.5%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-div N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 99.4

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

3. Applied rewrites 99.4%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[(N[Sqrt[k], $MachinePrecision] / k), $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. Taylor expanded in k around inf

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. inv-pow N/A

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

   6. sqrt-pow2 N/A

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

   7. metadata-eval N/A

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

   8. pow-neg N/A

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

   9. metadata-eval N/A

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

   10. pow1/2 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-sqrt.f64 99.4

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

4. Applied rewrites 99.4%

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

Alternative 4: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. div-sub N/A

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

   8. metadata-eval N/A

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

   9. pow-sub N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.5%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{\sqrt{\left(\pi + \pi\right) \cdot n}}{{\left(\left(\pi + \pi\right) \cdot n\right)}^{\left(0.5 \cdot k\right)}}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lift-+.f64 N/A

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

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{1 \cdot {\left(\left(n + n\right) \cdot \pi\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}}{\sqrt{k}}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-lft-identity 99.4

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 99.4

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

   8. lift-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 99.4

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

7. Applied rewrites 99.4%

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

Alternative 5: 99.4% 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%

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

3. Applied rewrites 99.4%

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

Alternative 6: 75.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if n < 9.5e9

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. 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 49.9

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

   7. Applied rewrites 49.9%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 50.0

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

   10. Applied rewrites 50.0%

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

   if 9.5e9 < n < 2e180

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. 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 49.9

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

   7. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. div-add N/A

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

      7. frac-add N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 35.2

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

   9. Applied rewrites 35.2%

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

   if 2e180 < n

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. 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 49.9

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

   7. Applied rewrites 49.9%

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

Alternative 7: 73.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\frac{\pi}{k}, n, \frac{\pi \cdot n}{k}\right)}{n \cdot n}} \cdot n\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\sqrt{\frac{1}{k}} \cdot \sqrt{\left(n + n\right) \cdot \pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{\pi \cdot k}{n} \cdot 2}}{k} \cdot n\\ \end{array} \end{array} \]

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 (/ (fma (/ PI k) n (/ (* PI n) k)) (* n n))) n)
     (if (<= t_0 5e+277)
       (* (sqrt (/ 1.0 k)) (sqrt (* (+ n n) PI)))
       (* (/ (sqrt (* (/ (* PI k) n) 2.0)) k) n)))))

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((fma((((double) M_PI) / k), n, ((((double) M_PI) * n) / k)) / (n * n))) * n;
	} else if (t_0 <= 5e+277) {
		tmp = sqrt((1.0 / k)) * sqrt(((n + n) * ((double) M_PI)));
	} else {
		tmp = (sqrt((((((double) M_PI) * k) / n) * 2.0)) / k) * n;
	}
	return tmp;
}

Julia

function code(k, n)
	t_0 = Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(fma(Float64(pi / k), n, Float64(Float64(pi * n) / k)) / Float64(n * n))) * n);
	elseif (t_0 <= 5e+277)
		tmp = Float64(sqrt(Float64(1.0 / k)) * sqrt(Float64(Float64(n + n) * pi)));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(pi * k) / n) * 2.0)) / k) * n);
	end
	return tmp
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(N[(N[(Pi / k), $MachinePrecision] * n + N[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$0, 5e+277], N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(n + n), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[(Pi * k), $MachinePrecision] / n), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision] * n), $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. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. 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 49.9

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

   7. Applied rewrites 49.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. count-2-rev N/A

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

      10. associate-/r* N/A

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

      11. associate-/r* N/A

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

      12. frac-add N/A

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

      13. lower-/.f64 N/A

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

   9. Applied rewrites 38.5%

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

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 4.99999999999999982e277

   1. Initial program 99.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-div N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 99.4

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

   3. Applied rewrites 99.4%

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

   4. 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. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 48.9

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

   6. Applied rewrites 48.9%

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

   if 4.99999999999999982e277 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. 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 49.9

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

   7. Applied rewrites 49.9%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 50.0

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

   10. Applied rewrites 50.0%

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

Alternative 8: 73.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.65 \cdot 10^{-22}:\\ \;\;\;\;\frac{\sqrt{\frac{\pi \cdot k}{n} \cdot 2} \cdot n}{k}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 2.65e-22)
   (/ (* (sqrt (* (/ (* PI k) n) 2.0)) n) k)
   (* (sqrt (/ (+ PI PI) (* n k))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 2.65e-22) {
		tmp = (sqrt((((((double) M_PI) * k) / n) * 2.0)) * 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 <= 2.65e-22) {
		tmp = (Math.sqrt((((Math.PI * k) / n) * 2.0)) * n) / k;
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (n * k))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 2.65e-22:
		tmp = (math.sqrt((((math.pi * k) / n) * 2.0)) * 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 <= 2.65e-22)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(pi * k) / n) * 2.0)) * 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 <= 2.65e-22)
		tmp = (sqrt((((pi * k) / n) * 2.0)) * n) / k;
	else
		tmp = sqrt(((pi + pi) / (n * k))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 2.65e-22], N[(N[(N[Sqrt[N[(N[(N[(Pi * k), $MachinePrecision] / n), $MachinePrecision] * 2.0), $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 < 2.64999999999999986e-22

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. 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 49.9

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

   7. Applied rewrites 49.9%

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

   8. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 50.0

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

   10. Applied rewrites 50.0%

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

   if 2.64999999999999986e-22 < n

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. 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 49.9

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

   7. Applied rewrites 49.9%

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

Alternative 9: 61.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{\frac{\pi + \pi}{n}}}{\sqrt{k}} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 5e-52)
   (* (/ (sqrt (/ (+ PI PI) n)) (sqrt k)) n)
   (* (sqrt (/ (+ PI PI) (* n k))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 5e-52) {
		tmp = (sqrt(((((double) M_PI) + ((double) M_PI)) / n)) / sqrt(k)) * n;
	} else {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / (n * k))) * n;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 5e-52) {
		tmp = (Math.sqrt(((Math.PI + Math.PI) / n)) / Math.sqrt(k)) * n;
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (n * k))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 5e-52:
		tmp = (math.sqrt(((math.pi + math.pi) / n)) / math.sqrt(k)) * n
	else:
		tmp = math.sqrt(((math.pi + math.pi) / (n * k))) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 5e-52)
		tmp = Float64(Float64(sqrt(Float64(Float64(pi + pi) / n)) / sqrt(k)) * n);
	else
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / Float64(n * k))) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 5e-52)
		tmp = (sqrt(((pi + pi) / n)) / sqrt(k)) * n;
	else
		tmp = sqrt(((pi + pi) / (n * k))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 5e-52], N[(N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / N[(n * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 5e-52

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. 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 49.9

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

   7. Applied rewrites 49.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. count-2-rev N/A

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

      9. associate-*r/ N/A

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

      10. sqrt-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. associate-*r/ N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lift-sqrt.f64 49.0

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

   9. Applied rewrites 49.0%

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

   if 5e-52 < n

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. 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 49.9

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

   7. Applied rewrites 49.9%

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

Alternative 10: 60.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2.9 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{\frac{\pi \cdot n}{k} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 2.9e-22)
   (sqrt (* (/ (* PI n) k) 2.0))
   (* (sqrt (/ (+ PI PI) (* n k))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 2.9e-22) {
		tmp = sqrt((((((double) M_PI) * n) / k) * 2.0));
	} else {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / (n * k))) * n;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 2.9e-22) {
		tmp = Math.sqrt((((Math.PI * n) / k) * 2.0));
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (n * k))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 2.9e-22:
		tmp = math.sqrt((((math.pi * n) / k) * 2.0))
	else:
		tmp = math.sqrt(((math.pi + math.pi) / (n * k))) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 2.9e-22)
		tmp = sqrt(Float64(Float64(Float64(pi * n) / k) * 2.0));
	else
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / Float64(n * k))) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 2.9e-22)
		tmp = sqrt((((pi * n) / k) * 2.0));
	else
		tmp = sqrt(((pi + pi) / (n * k))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 2.9e-22], N[Sqrt[N[(N[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / N[(n * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.9000000000000002e-22

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 37.5

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

   6. Applied rewrites 37.5%

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

   if 2.9000000000000002e-22 < n

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 37.5

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. 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 49.9

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

   7. Applied rewrites 49.9%

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

Alternative 11: 48.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. unpow1/2 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-PI.f64 37.5

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

4. Applied rewrites 37.5%

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

   1. lift-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. lower-/.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. associate-*r* N/A

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

   14. count-2-rev N/A

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

   15. lift-+.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lift-PI.f64 N/A

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

   18. lift-sqrt.f64 48.9

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

6. Applied rewrites 48.9%

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

Alternative 12: 37.5% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. unpow1/2 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-PI.f64 37.5

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

4. Applied rewrites 37.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2-rev N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-PI.f64 37.5

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

6. Applied rewrites 37.5%

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

   1. lift-/.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-PI.f64 37.5

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

8. Applied rewrites 37.5%

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

Alternative 13: 37.5% 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. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-PI.f64 37.5

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

4. Applied rewrites 37.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2-rev N/A

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

   7. associate-/l* N/A

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

   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 37.5

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

6. Applied rewrites 37.5%

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

Reproduce

herbie shell --seed 2025139 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))