Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 5.7s

Alternatives: 7

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 5.233817538744218e+186
  2. n = -2.3216342246633717e+22

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{-17}:\\ \;\;\;\;\frac{n \cdot \sqrt{2 \cdot \frac{\pi}{n}}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(\pi + \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.9e-17)
   (/ (* n (sqrt (* 2.0 (/ PI n)))) (sqrt k))
   (sqrt (/ (pow (* (+ PI PI) n) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.9e-17) {
		tmp = (n * sqrt((2.0 * (((double) M_PI) / n)))) / sqrt(k);
	} else {
		tmp = sqrt((pow(((((double) M_PI) + ((double) M_PI)) * n), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.9e-17) {
		tmp = (n * Math.sqrt((2.0 * (Math.PI / n)))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow(((Math.PI + Math.PI) * n), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.9e-17:
		tmp = (n * math.sqrt((2.0 * (math.pi / n)))) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow(((math.pi + math.pi) * n), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.9e-17)
		tmp = Float64(Float64(n * sqrt(Float64(2.0 * Float64(pi / n)))) / sqrt(k));
	else
		tmp = sqrt(Float64((Float64(Float64(pi + pi) * n) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.9e-17)
		tmp = (n * sqrt((2.0 * (pi / n)))) / sqrt(k);
	else
		tmp = sqrt(((((pi + pi) * n) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.9e-17], N[(N[(n * N[Sqrt[N[(2.0 * N[(Pi / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.9000000000000003e-17

   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 50.2

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

   4. Applied rewrites 50.2%

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

   5. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 50.3

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

   7. Applied rewrites 50.3%

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

   if 2.9000000000000003e-17 < k

   1. Initial program 99.4%

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

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

   3. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. count-2-rev N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. metadata-eval N/A

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

      14. mult-flip N/A

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

      15. sub-flip N/A

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

      16. metadata-eval N/A

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

      17. div-sub N/A

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

      18. lift--.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. exp-to-pow N/A

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

      21. lift-log.f64 N/A

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

      22. lift-/.f64 N/A

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

      23. mult-flip N/A

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

   5. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 88.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. count-2 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f64 88.0

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

   7. Applied rewrites 88.0%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.4%

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

Alternative 3: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Applied rewrites 99.5%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. distribute-lft-in N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. count-2-rev N/A

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

   8. lift-*.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. metadata-eval N/A

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

   12. distribute-rgt-neg-out N/A

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

   13. metadata-eval N/A

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

   14. mult-flip N/A

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

   15. sub-flip N/A

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

   16. metadata-eval N/A

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

   17. div-sub N/A

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

   18. lift--.f64 N/A

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

   19. lift-/.f64 N/A

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

   20. exp-to-pow N/A

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

   21. lift-log.f64 N/A

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

   22. lift-/.f64 N/A

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

   23. mult-flip N/A

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

5. Applied rewrites 99.4%

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

Alternative 4: 74.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 20:\\ \;\;\;\;\frac{n \cdot \sqrt{2 \cdot \frac{k \cdot \pi}{n}}}{k}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{\pi}{k \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 20.0)
   (/ (* n (sqrt (* 2.0 (/ (* k PI) n)))) k)
   (* n (sqrt (* 2.0 (/ PI (* k n)))))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 20.0) {
		tmp = (n * sqrt((2.0 * ((k * ((double) M_PI)) / n)))) / 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 <= 20.0) {
		tmp = (n * Math.sqrt((2.0 * ((k * Math.PI) / n)))) / k;
	} else {
		tmp = n * Math.sqrt((2.0 * (Math.PI / (k * n))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 20.0:
		tmp = (n * math.sqrt((2.0 * ((k * math.pi) / n)))) / k
	else:
		tmp = n * math.sqrt((2.0 * (math.pi / (k * n))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 20.0)
		tmp = Float64(Float64(n * sqrt(Float64(2.0 * Float64(Float64(k * pi) / n)))) / 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 <= 20.0)
		tmp = (n * sqrt((2.0 * ((k * pi) / n)))) / k;
	else
		tmp = n * sqrt((2.0 * (pi / (k * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 20.0], N[(N[(n * N[Sqrt[N[(2.0 * N[(N[(k * Pi), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k), $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 < 20

   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 50.2

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

   4. Applied rewrites 50.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 38.8

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

   6. Applied rewrites 38.8%

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

   7. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 38.4

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

   9. Applied rewrites 38.4%

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

   10. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 51.2

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

   12. Applied rewrites 51.2%

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

   if 20 < 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 50.2

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

   4. Applied rewrites 50.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 38.8

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

   6. Applied rewrites 38.8%

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

   7. Taylor expanded in n around inf

      \[\leadsto n \cdot \color{blue}{\sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{k \cdot n}}} \]
   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 49.3

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

   9. Applied rewrites 49.3%

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

Alternative 5: 61.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 10^{+28}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{\pi}{k \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 1e+28)
   (sqrt (* (+ n n) (/ PI k)))
   (* n (sqrt (* 2.0 (/ PI (* k n)))))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 1e+28) {
		tmp = sqrt(((n + n) * (((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 <= 1e+28) {
		tmp = Math.sqrt(((n + n) * (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 <= 1e+28:
		tmp = math.sqrt(((n + n) * (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 <= 1e+28)
		tmp = sqrt(Float64(Float64(n + n) * Float64(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 <= 1e+28)
		tmp = sqrt(((n + n) * (pi / k)));
	else
		tmp = n * sqrt((2.0 * (pi / (k * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 1e+28], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(Pi / k), $MachinePrecision]), $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 < 9.99999999999999958e27

   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 50.2

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

   4. Applied rewrites 50.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 38.8

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

   6. Applied rewrites 38.8%

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

   if 9.99999999999999958e27 < 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 50.2

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

   4. Applied rewrites 50.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 38.8

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

   6. Applied rewrites 38.8%

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

   7. Taylor expanded in n around inf

      \[\leadsto n \cdot \color{blue}{\sqrt{2 \cdot \frac{\mathsf{PI}\left(\right)}{k \cdot n}}} \]
   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 49.3

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

   9. Applied rewrites 49.3%

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

Alternative 6: 50.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. 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 50.2

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

4. Applied rewrites 50.2%

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{0.5}}{\sqrt{k}}} \]
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. lower-sqrt.f64 50.2

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 50.2

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

6. Applied rewrites 50.2%

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

Alternative 7: 38.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(Pi / 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 50.2

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

4. Applied rewrites 50.2%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow1/2 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. sqrt-undiv N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. count-2-rev N/A

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

   13. lower-+.f64 N/A

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

   14. lower-/.f64 38.8

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

6. Applied rewrites 38.8%

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

Reproduce

herbie shell --seed 2025156 
(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))))