Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 4.5s

Alternatives: 7

Speedup: 0.9×

• Could not converge on a ground truth

  1. k = 1.6819611763365171e+97
  2. n = -3496258765311778000.0

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi + \pi\right) \cdot n\\ \frac{1 \cdot \sqrt{t\_0}}{\sqrt{k} \cdot {t\_0}^{\left(0.5 \cdot k\right)}} \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (+ PI PI) n)))
   (/ (* 1.0 (sqrt t_0)) (* (sqrt k) (pow t_0 (* 0.5 k))))))

C

double code(double k, double n) {
	double t_0 = (((double) M_PI) + ((double) M_PI)) * n;
	return (1.0 * sqrt(t_0)) / (sqrt(k) * pow(t_0, (0.5 * k)));
}

Java

public static double code(double k, double n) {
	double t_0 = (Math.PI + Math.PI) * n;
	return (1.0 * Math.sqrt(t_0)) / (Math.sqrt(k) * Math.pow(t_0, (0.5 * k)));
}

Python

def code(k, n):
	t_0 = (math.pi + math.pi) * n
	return (1.0 * math.sqrt(t_0)) / (math.sqrt(k) * math.pow(t_0, (0.5 * k)))

Julia

function code(k, n)
	t_0 = Float64(Float64(pi + pi) * n)
	return Float64(Float64(1.0 * sqrt(t_0)) / Float64(sqrt(k) * (t_0 ^ Float64(0.5 * k))))
end

MATLAB

function tmp = code(k, n)
	t_0 = (pi + pi) * n;
	tmp = (1.0 * sqrt(t_0)) / (sqrt(k) * (t_0 ^ (0.5 * k)));
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision]}, N[(N[(1.0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[k], $MachinePrecision] * N[Power[t$95$0, N[(0.5 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.4%

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

   1. lift-pow.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. 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-/.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-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-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. lower-*.f64 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow1/2 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-pow.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. 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-/.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-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-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. lower-*.f64 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow1/2 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-lft-identity 99.5

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. sqrt-prod N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-sqrt.f64 99.3

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

7. Applied rewrites 99.3%

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

Alternative 3: 99.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (k n)
 :precision binary64
 (/ (pow (* n (+ PI PI)) (* 0.5 (- 1.0 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[(0.5 * N[(1.0 - k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lower-/.f64 99.5

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

   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. div-flip N/A

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

   14. associate-/r/ N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f64 99.5

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

3. Applied rewrites 99.5%

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

Alternative 4: 61.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.55 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{\frac{n \cdot \left(\pi + \pi\right)}{k}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{\pi}{k \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 1.55e-47)
   (sqrt (/ (* n (+ PI PI)) k))
   (* n (sqrt (* 2.0 (/ PI (* k n)))))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 1.55e-47) {
		tmp = sqrt(((n * (((double) M_PI) + ((double) M_PI))) / k));
	} 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 <= 1.55e-47) {
		tmp = Math.sqrt(((n * (Math.PI + Math.PI)) / k));
	} else {
		tmp = n * Math.sqrt((2.0 * (Math.PI / (k * n))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 1.55e-47:
		tmp = math.sqrt(((n * (math.pi + math.pi)) / k))
	else:
		tmp = n * math.sqrt((2.0 * (math.pi / (k * n))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 1.55e-47)
		tmp = sqrt(Float64(Float64(n * Float64(pi + pi)) / k));
	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 <= 1.55e-47)
		tmp = sqrt(((n * (pi + pi)) / k));
	else
		tmp = n * sqrt((2.0 * (pi / (k * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 1.55e-47], N[Sqrt[N[(N[(n * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision], N[(n * N[Sqrt[N[(2.0 * N[(Pi / N[(k * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1.5499999999999999e-47

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.6

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

   4. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-sqrt.f64 49.6

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 49.6

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

   6. Applied rewrites 49.6%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 38.1

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

   8. Applied rewrites 38.1%

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

   if 1.5499999999999999e-47 < 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. 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 49.6

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

   4. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-sqrt.f64 49.6

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 49.6

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

   6. Applied rewrites 49.6%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 38.1

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

   8. Applied rewrites 38.1%

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

   9. Taylor expanded in n around inf

      \[\leadsto n \cdot \color{blue}{\sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{k \cdot n}}} \]
   10. 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 49.7

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

   11. Applied rewrites 49.7%

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

Alternative 5: 49.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. 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 49.6

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

4. Applied rewrites 49.6%

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. count-2-rev N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-lft-in N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-sqrt.f64 49.6

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 49.6

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

6. Applied rewrites 49.6%

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

Alternative 6: 38.1% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. 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 49.6

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

4. Applied rewrites 49.6%

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. count-2-rev N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-lft-in N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-sqrt.f64 49.6

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 49.6

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

6. Applied rewrites 49.6%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-/.f64 38.1

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

8. Applied rewrites 38.1%

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

Alternative 7: 38.1% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. 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 49.6

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}} \]

4. Applied rewrites 49.6%

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. count-2-rev N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-lft-in N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-sqrt.f64 49.6

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 49.6

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

6. Applied rewrites 49.6%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-/.f64 38.1

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

8. Applied rewrites 38.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 38.1

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

10. Applied rewrites 38.1%

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

Reproduce

herbie shell --seed 2025151 
(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))))