Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.5%

Time: 53.5s

Alternatives: 7

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 1.212819373580092e+288
  2. n = -3.276350087596002e+52

Specification

Math

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

FPCore

(FPCore (k n)
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))

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 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(N[(1 - k), $MachinePrecision] / 2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (k n)
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))

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 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(N[(1 - k), $MachinePrecision] / 2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(\pi + \pi\right) \cdot n\\ \frac{\sqrt{t\_0}}{{t\_0}^{\left(\frac{1}{2} \cdot k\right)} \cdot \sqrt{k}} \end{array} \]

FPCore

(FPCore (k n)
  :precision binary64
  (let* ((t_0 (* (+ PI PI) n)))
  (/ (sqrt t_0) (* (pow t_0 (* 1/2 k)) (sqrt k)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision]}, N[(N[Sqrt[t$95$0], $MachinePrecision] / N[(N[Power[t$95$0, N[(1/2 * k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.5%

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

   1. lift-pow.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. metadata-eval N/A

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

   6. pow-sub N/A

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

   7. lower-unsound-/.f64 N/A

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

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

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f64 N/A

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

   15. lower-unsound-pow.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. count-2-rev N/A

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

   21. lower-+.f64 N/A

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

   22. mult-flip N/A

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

   23. metadata-eval N/A

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

   24. *-commutative N/A

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

   25. lower-*.f64 99.5%

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

3. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. *-lft-identity N/A

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

   6. lower-/.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow1/2 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

5. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (k n)
  :precision binary64
  (* (pow (* (+ PI PI) n) (* (- 1 k) 1/2)) (sqrt (/ 1 k))))

C

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

Java

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

Python

def code(k, n):
	return math.pow(((math.pi + math.pi) * n), ((1.0 - k) * 0.5)) * math.sqrt((1.0 / k))

Julia

function code(k, n)
	return Float64((Float64(Float64(pi + pi) * n) ^ Float64(Float64(1.0 - k) * 0.5)) * sqrt(Float64(1.0 / k)))
end

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision], N[(N[(1 - k), $MachinePrecision] * 1/2), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1 / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. lift-pow.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. metadata-eval N/A

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

   6. pow-sub N/A

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

   7. lower-unsound-/.f64 N/A

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

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

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f64 N/A

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

   15. lower-unsound-pow.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. count-2-rev N/A

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

   21. lower-+.f64 N/A

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

   22. mult-flip N/A

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

   23. metadata-eval N/A

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

   24. *-commutative N/A

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

   25. lower-*.f64 99.5%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. metadata-eval N/A

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

   4. sqrt-fabs-rev N/A

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

   5. lift-sqrt.f64 N/A

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

   6. fabs-div N/A

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

   7. lift-/.f64 N/A

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

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

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

   9. lift-*.f64 N/A

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

   10. lower-sqrt.f64 99.4%

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

   11. lift-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. frac-2neg N/A

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

   14. metadata-eval N/A

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

   15. lift-/.f64 N/A

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

   16. frac-2neg N/A

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

   17. metadata-eval N/A

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

   18. frac-times N/A

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

   19. metadata-eval N/A

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

   20. sqr-neg-rev N/A

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

   21. lift-sqrt.f64 N/A

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

   22. lift-sqrt.f64 N/A

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

   23. rem-square-sqrt N/A

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

   24. lower-/.f64 99.5%

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

7. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (k n)
  :precision binary64
  (/ (pow (* n (+ PI PI)) (* 1/2 (- 1 k))) (sqrt k)))

C

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

Java

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

Python

def code(k, n):
	return math.pow((n * (math.pi + math.pi)), (0.5 * (1.0 - k))) / math.sqrt(k)

Julia

function code(k, n)
	return Float64((Float64(n * Float64(pi + pi)) ^ Float64(0.5 * Float64(1.0 - k))) / sqrt(k))
end

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(n * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision], N[(1/2 * N[(1 - k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lower-/.f64 99.5%

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.5%

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

   9. lift-*.f64 N/A

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

   10. count-2-rev N/A

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

   11. lower-+.f64 99.5%

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

   12. lift-/.f64 N/A

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

   13. mult-flip N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.5%

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

3. Applied rewrites 99.5%

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

Alternative 4: 63.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;k \leq \frac{7482888383134223}{748288838313422294120286634350736906063837462003712}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\frac{\pi + \pi}{k}}\\ \mathbf{elif}\;k \leq 1949999999999999958289089547254969003180158784162376815230088976785689798552796854866284000535520384908499348776794345879253213054967868666923284326497919619731811089038308278272:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{\pi}{k \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\frac{\frac{1}{k}}{k}} \cdot \left(n \cdot \left(\pi + \pi\right)\right)}\\ \end{array} \]

FPCore

(FPCore (k n)
  :precision binary64
  (if (<=
     k
     7482888383134223/748288838313422294120286634350736906063837462003712)
  (* (sqrt n) (sqrt (/ (+ PI PI) k)))
  (if (<=
       k
       1949999999999999958289089547254969003180158784162376815230088976785689798552796854866284000535520384908499348776794345879253213054967868666923284326497919619731811089038308278272)
    (* n (sqrt (* 2 (/ PI (* k n)))))
    (sqrt (* (sqrt (/ (/ 1 k) k)) (* n (+ PI PI)))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1e-35) {
		tmp = sqrt(n) * sqrt(((((double) M_PI) + ((double) M_PI)) / k));
	} else if (k <= 1.95e+177) {
		tmp = n * sqrt((2.0 * (((double) M_PI) / (k * n))));
	} else {
		tmp = sqrt((sqrt(((1.0 / k) / k)) * (n * (((double) M_PI) + ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1e-35) {
		tmp = Math.sqrt(n) * Math.sqrt(((Math.PI + Math.PI) / k));
	} else if (k <= 1.95e+177) {
		tmp = n * Math.sqrt((2.0 * (Math.PI / (k * n))));
	} else {
		tmp = Math.sqrt((Math.sqrt(((1.0 / k) / k)) * (n * (Math.PI + Math.PI))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1e-35:
		tmp = math.sqrt(n) * math.sqrt(((math.pi + math.pi) / k))
	elif k <= 1.95e+177:
		tmp = n * math.sqrt((2.0 * (math.pi / (k * n))))
	else:
		tmp = math.sqrt((math.sqrt(((1.0 / k) / k)) * (n * (math.pi + math.pi))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1e-35)
		tmp = Float64(sqrt(n) * sqrt(Float64(Float64(pi + pi) / k)));
	elseif (k <= 1.95e+177)
		tmp = Float64(n * sqrt(Float64(2.0 * Float64(pi / Float64(k * n)))));
	else
		tmp = sqrt(Float64(sqrt(Float64(Float64(1.0 / k) / k)) * Float64(n * Float64(pi + pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1e-35)
		tmp = sqrt(n) * sqrt(((pi + pi) / k));
	elseif (k <= 1.95e+177)
		tmp = n * sqrt((2.0 * (pi / (k * n))));
	else
		tmp = sqrt((sqrt(((1.0 / k) / k)) * (n * (pi + pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 7482888383134223/748288838313422294120286634350736906063837462003712], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1949999999999999958289089547254969003180158784162376815230088976785689798552796854866284000535520384908499348776794345879253213054967868666923284326497919619731811089038308278272], N[(n * N[Sqrt[N[(2 * N[(Pi / N[(k * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Sqrt[N[(N[(1 / k), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision] * N[(n * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1e-35

   1. Initial program 99.5%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.5%

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

   4. Applied rewrites 49.5%

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

      1. lift-/.f64 N/A

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

   6. Applied rewrites 38.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

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

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

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

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

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

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 49.4%

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

   8. Applied rewrites 49.4%

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

   if 1e-35 < k < 1.95e177

   1. Initial program 99.5%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.5%

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

   4. Applied rewrites 49.5%

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

      1. lift-/.f64 N/A

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

   6. Applied rewrites 38.4%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 49.1%

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

   9. Applied rewrites 49.1%

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

   if 1.95e177 < k

   1. Initial program 99.5%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.5%

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

   4. Applied rewrites 49.5%

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

      1. lift-/.f64 N/A

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

   6. Applied rewrites 38.4%

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

      1. rem-square-sqrt N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-undiv N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. div-flip-rev N/A

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

      11. associate-/r/ N/A

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

      12. lift-sqrt.f64 N/A

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

      13. div-flip-rev N/A

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

   8. Applied rewrites 38.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lift-/.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-fabs-rev N/A

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

      9. lift-sqrt.f64 N/A

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

      10. fabs-div N/A

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

      11. lift-/.f64 N/A

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

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

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

      13. lift-*.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. sqrt-undiv N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 34.5%

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

   10. Applied rewrites 34.5%

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

Alternative 5: 60.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;n \leq 9999999999999999455752309870428160:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{k} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{\pi}{k \cdot n}}\\ \end{array} \]

FPCore

(FPCore (k n)
  :precision binary64
  (if (<= n 9999999999999999455752309870428160)
  (sqrt (* (/ (+ PI PI) k) n))
  (* n (sqrt (* 2 (/ PI (* k n)))))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 1e+34) {
		tmp = sqrt((((((double) M_PI) + ((double) M_PI)) / k) * n));
	} else {
		tmp = n * sqrt((2.0 * (((double) M_PI) / (k * n))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 1e+34) {
		tmp = Math.sqrt((((Math.PI + Math.PI) / k) * n));
	} else {
		tmp = n * Math.sqrt((2.0 * (Math.PI / (k * n))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 1e+34:
		tmp = math.sqrt((((math.pi + math.pi) / k) * n))
	else:
		tmp = n * math.sqrt((2.0 * (math.pi / (k * n))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 1e+34)
		tmp = sqrt(Float64(Float64(Float64(pi + pi) / k) * n));
	else
		tmp = Float64(n * sqrt(Float64(2.0 * Float64(pi / Float64(k * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 1e+34)
		tmp = sqrt((((pi + pi) / k) * n));
	else
		tmp = n * sqrt((2.0 * (pi / (k * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 9999999999999999455752309870428160], N[Sqrt[N[(N[(N[(Pi + Pi), $MachinePrecision] / k), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], N[(n * N[Sqrt[N[(2 * N[(Pi / N[(k * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 9.9999999999999995e33

   1. Initial program 99.5%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.5%

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

   4. Applied rewrites 49.5%

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

      1. lift-/.f64 N/A

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

   6. Applied rewrites 38.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 38.4%

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

   8. Applied rewrites 38.4%

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

   if 9.9999999999999995e33 < n

   1. Initial program 99.5%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.5%

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

   4. Applied rewrites 49.5%

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

      1. lift-/.f64 N/A

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

   6. Applied rewrites 38.4%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 49.1%

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

   9. Applied rewrites 49.1%

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

Alternative 6: 38.4% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 49.5%

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

4. Applied rewrites 49.5%

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

   1. lift-/.f64 N/A

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

6. Applied rewrites 38.4%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. mult-flip-rev N/A

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

   9. lower-/.f64 38.4%

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

8. Applied rewrites 38.4%

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

Alternative 7: 38.4% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 49.5%

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

4. Applied rewrites 49.5%

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

   1. lift-/.f64 N/A

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

6. Applied rewrites 38.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 38.4%

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

8. Applied rewrites 38.4%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))