Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 17.6s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (/ (pow (* 2.0 (* PI n)) (- 0.5 (/ k 2.0))) (sqrt k)))

C

double code(double k, double n) {
	return pow((2.0 * (((double) M_PI) * n)), (0.5 - (k / 2.0))) / sqrt(k);
}

Java

public static double code(double k, double n) {
	return Math.pow((2.0 * (Math.PI * n)), (0.5 - (k / 2.0))) / Math.sqrt(k);
}

Python

def code(k, n):
	return math.pow((2.0 * (math.pi * n)), (0.5 - (k / 2.0))) / math.sqrt(k)

Julia

function code(k, n)
	return Float64((Float64(2.0 * Float64(pi * n)) ^ Float64(0.5 - Float64(k / 2.0))) / sqrt(k))
end

MATLAB

function tmp = code(k, n)
	tmp = ((2.0 * (pi * n)) ^ (0.5 - (k / 2.0))) / sqrt(k);
end

Wolfram

code[k_, n_] := N[(N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(0.5 - N[(k / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. associate-*l* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.2e-77)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.2e-77) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = sqrt((pow((((double) M_PI) * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.2e-77) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((2.0 * n));
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.2e-77:
		tmp = math.sqrt((math.pi / k)) * math.sqrt((2.0 * n))
	else:
		tmp = math.sqrt((math.pow((math.pi * (2.0 * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.2e-77)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(2.0 * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.2e-77)
		tmp = sqrt((pi / k)) * sqrt((2.0 * n));
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.2e-77], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.2e-77

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 72.1%

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

      1. associate-/l* 72.2%

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

   5. Simplified 72.2%

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

      1. pow1 72.2%

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

      2. *-commutative 72.2%

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

      3. sqrt-unprod 72.3%

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

      4. associate-*r/ 72.2%

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

      5. *-commutative 72.2%

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

   7. Applied egg-rr 72.2%

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

      1. unpow1 72.2%

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

      2. *-commutative 72.2%

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

      3. associate-/l* 72.4%

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

      4. associate-*l* 72.4%

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

   9. Simplified 72.4%

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

       1. associate-*r* 72.4%

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

       2. sqrt-prod 72.2%

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

       3. div-inv 72.2%

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

       4. associate-*r* 72.2%

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

       5. *-commutative 72.2%

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

       6. associate-*r* 72.2%

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

       7. div-inv 72.2%

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

       8. *-commutative 72.2%

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

       9. sqrt-unprod 99.1%

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

       10. associate-*l* 99.1%

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

       11. sqrt-unprod 99.4%

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

   11. Applied egg-rr 99.4%

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

   if 3.2e-77 < k

   1. Initial program 99.7%

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

   3. Taylor expanded in k around 0 99.7%

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

   5. Applied egg-rr 99.2%

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

      1. distribute-lft-in 99.2%

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

      2. metadata-eval 99.2%

         \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{1} + 2 \cdot \left(k \cdot -0.5\right)\right)}}{k}} \]

      3. *-commutative 99.2%

         \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 + 2 \cdot \color{blue}{\left(-0.5 \cdot k\right)}\right)}}{k}} \]

      4. associate-*r* 99.2%

         \[\leadsto \sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 + \color{blue}{\left(2 \cdot -0.5\right) \cdot k}\right)}}{k}} \]

      5. metadata-eval 99.2%

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

      6. neg-mul-1 99.2%

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

      7. sub-neg 99.2%

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

      8. *-commutative 99.2%

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

      9. associate-*l* 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \mathsf{fma}\left(\pi, \frac{2 \cdot n}{k}, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.85e+87)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (sqrt (+ 1.0 (fma PI (/ (* 2.0 n) k) -1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.85e+87) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = sqrt((1.0 + fma(((double) M_PI), ((2.0 * n) / k), -1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.85e+87)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = sqrt(Float64(1.0 + fma(pi, Float64(Float64(2.0 * n) / k), -1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.85e+87], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(1.0 + N[(Pi * N[(N[(2.0 * n), $MachinePrecision] / k), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.85000000000000001e87

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 56.5%

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

      1. associate-/l* 56.5%

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

   5. Simplified 56.5%

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

      1. pow1 56.5%

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

      2. *-commutative 56.5%

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

      3. sqrt-unprod 56.7%

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

      4. associate-*r/ 56.6%

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

      5. *-commutative 56.6%

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

   7. Applied egg-rr 56.6%

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

      1. unpow1 56.6%

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

      2. *-commutative 56.6%

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

      3. associate-/l* 56.7%

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

      4. associate-*l* 56.7%

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

   9. Simplified 56.7%

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

       1. associate-*r* 56.7%

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

       2. sqrt-prod 56.5%

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

       3. div-inv 56.6%

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

       4. associate-*r* 56.6%

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

       5. *-commutative 56.6%

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

       6. associate-*r* 56.5%

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

       7. div-inv 56.5%

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

       8. *-commutative 56.5%

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

       9. sqrt-unprod 72.3%

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

       10. associate-*l* 72.3%

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

       11. sqrt-unprod 72.5%

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

   11. Applied egg-rr 72.5%

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

   if 1.85000000000000001e87 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.8%

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

      1. associate-/l* 2.8%

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

   5. Simplified 2.8%

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

      1. pow1 2.8%

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

      2. *-commutative 2.8%

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

      3. sqrt-unprod 2.8%

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

      4. associate-*r/ 2.8%

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

      5. *-commutative 2.8%

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

   7. Applied egg-rr 2.8%

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

      1. unpow1 2.8%

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

      2. *-commutative 2.8%

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

      3. associate-/l* 2.8%

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

      4. associate-*l* 2.8%

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

   9. Simplified 2.8%

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

       1. *-commutative 2.8%

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

       2. clear-num 2.8%

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

       3. un-div-inv 2.8%

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

   11. Applied egg-rr 2.8%

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

       1. expm1-log1p-u 2.8%

          \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \frac{2}{\frac{k}{n}}\right)\right)}} \]

       2. expm1-undefine 42.4%

          \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\pi \cdot \frac{2}{\frac{k}{n}}\right)} - 1}} \]

       3. *-commutative 42.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\color{blue}{\frac{2}{\frac{k}{n}} \cdot \pi}\right)} - 1} \]

       4. div-inv 42.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\color{blue}{\left(2 \cdot \frac{1}{\frac{k}{n}}\right)} \cdot \pi\right)} - 1} \]

       5. clear-num 42.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\left(2 \cdot \color{blue}{\frac{n}{k}}\right) \cdot \pi\right)} - 1} \]

       6. associate-*r* 42.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \left(\frac{n}{k} \cdot \pi\right)}\right)} - 1} \]

       7. associate-/r/ 42.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(2 \cdot \color{blue}{\frac{n}{\frac{k}{\pi}}}\right)} - 1} \]

       8. *-commutative 42.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\color{blue}{\frac{n}{\frac{k}{\pi}} \cdot 2}\right)} - 1} \]

       9. div-inv 42.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\color{blue}{\left(n \cdot \frac{1}{\frac{k}{\pi}}\right)} \cdot 2\right)} - 1} \]

       10. clear-num 42.4%

           \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\left(n \cdot \color{blue}{\frac{\pi}{k}}\right) \cdot 2\right)} - 1} \]

       11. associate-*l* 42.4%

           \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\color{blue}{n \cdot \left(\frac{\pi}{k} \cdot 2\right)}\right)} - 1} \]

   13. Applied egg-rr 42.4%

       \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(n \cdot \left(\frac{\pi}{k} \cdot 2\right)\right)} - 1}} \]
   14. Step-by-step derivation

       1. log1p-undefine 42.4%

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

       2. rem-exp-log 42.4%

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

       3. associate-+r- 42.4%

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

       4. associate-*r* 42.4%

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

       5. *-commutative 42.4%

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

       6. associate-*r/ 42.4%

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

       7. associate-*r/ 42.4%

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

       8. associate-*r* 42.4%

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

       9. *-commutative 42.4%

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

       10. associate-/l* 42.4%

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

       11. associate-*l/ 42.4%

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

       12. *-commutative 42.4%

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

       13. fma-neg 42.4%

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

       14. metadata-eval 42.4%

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

       15. associate-*r/ 42.4%

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

   15. Simplified 42.4%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \mathsf{fma}\left(\pi, \frac{2 \cdot n}{k}, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, 2 \cdot \frac{\pi}{k}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.85e+87)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (sqrt (+ -1.0 (fma n (* 2.0 (/ PI k)) 1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.85e+87) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = sqrt((-1.0 + fma(n, (2.0 * (((double) M_PI) / k)), 1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.85e+87)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = sqrt(Float64(-1.0 + fma(n, Float64(2.0 * Float64(pi / k)), 1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.85e+87], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-1.0 + N[(n * N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.85000000000000001e87

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 56.5%

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

      1. associate-/l* 56.5%

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

   5. Simplified 56.5%

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

      1. pow1 56.5%

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

      2. *-commutative 56.5%

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

      3. sqrt-unprod 56.7%

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

      4. associate-*r/ 56.6%

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

      5. *-commutative 56.6%

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

   7. Applied egg-rr 56.6%

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

      1. unpow1 56.6%

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

      2. *-commutative 56.6%

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

      3. associate-/l* 56.7%

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

      4. associate-*l* 56.7%

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

   9. Simplified 56.7%

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

       1. associate-*r* 56.7%

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

       2. sqrt-prod 56.5%

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

       3. div-inv 56.6%

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

       4. associate-*r* 56.6%

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

       5. *-commutative 56.6%

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

       6. associate-*r* 56.5%

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

       7. div-inv 56.5%

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

       8. *-commutative 56.5%

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

       9. sqrt-unprod 72.3%

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

       10. associate-*l* 72.3%

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

       11. sqrt-unprod 72.5%

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

   11. Applied egg-rr 72.5%

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

   if 1.85000000000000001e87 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.8%

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

      1. associate-/l* 2.8%

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

   5. Simplified 2.8%

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

      1. pow1 2.8%

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

      2. *-commutative 2.8%

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

      3. sqrt-unprod 2.8%

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

      4. associate-*r/ 2.8%

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

      5. *-commutative 2.8%

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

   7. Applied egg-rr 2.8%

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

      1. unpow1 2.8%

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

      2. *-commutative 2.8%

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

      3. associate-/l* 2.8%

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

      4. associate-*l* 2.8%

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

   9. Simplified 2.8%

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

       1. expm1-log1p-u 2.8%

          \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(\frac{n}{k} \cdot 2\right)\right)\right)}} \]

       2. expm1-undefine 42.4%

          \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\pi \cdot \left(\frac{n}{k} \cdot 2\right)\right)} - 1}} \]

       3. *-commutative 42.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\color{blue}{\left(\frac{n}{k} \cdot 2\right) \cdot \pi}\right)} - 1} \]

       4. *-commutative 42.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\color{blue}{\left(2 \cdot \frac{n}{k}\right)} \cdot \pi\right)} - 1} \]

       5. associate-*l* 42.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \left(\frac{n}{k} \cdot \pi\right)}\right)} - 1} \]

       6. div-inv 42.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(2 \cdot \left(\color{blue}{\left(n \cdot \frac{1}{k}\right)} \cdot \pi\right)\right)} - 1} \]

       7. associate-*r* 42.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(2 \cdot \color{blue}{\left(n \cdot \left(\frac{1}{k} \cdot \pi\right)\right)}\right)} - 1} \]

       8. *-commutative 42.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(2 \cdot \left(n \cdot \color{blue}{\left(\pi \cdot \frac{1}{k}\right)}\right)\right)} - 1} \]

       9. div-inv 42.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(2 \cdot \left(n \cdot \color{blue}{\frac{\pi}{k}}\right)\right)} - 1} \]

   11. Applied egg-rr 42.4%

       \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)} - 1}} \]
   12. Step-by-step derivation

       1. sub-neg 42.4%

          \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)} + \left(-1\right)}} \]

       2. metadata-eval 42.4%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)} + \color{blue}{-1}} \]

       3. +-commutative 42.4%

          \[\leadsto \sqrt{\color{blue}{-1 + e^{\mathsf{log1p}\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}}} \]

       4. log1p-undefine 42.4%

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

       5. rem-exp-log 42.4%

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

       6. +-commutative 42.4%

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

       7. *-commutative 42.4%

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

       8. associate-*l* 42.4%

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

       9. fma-define 42.4%

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

   13. Simplified 42.4%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, 2 \cdot \frac{\pi}{k}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot \left(-1 + \mathsf{fma}\left(2, \frac{n}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.85e+87)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (sqrt (* PI (+ -1.0 (fma 2.0 (/ n k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.85e+87) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = sqrt((((double) M_PI) * (-1.0 + fma(2.0, (n / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.85e+87)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = sqrt(Float64(pi * Float64(-1.0 + fma(2.0, Float64(n / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.85e+87], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(Pi * N[(-1.0 + N[(2.0 * N[(n / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.85000000000000001e87

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 56.5%

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

      1. associate-/l* 56.5%

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

   5. Simplified 56.5%

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

      1. pow1 56.5%

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

      2. *-commutative 56.5%

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

      3. sqrt-unprod 56.7%

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

      4. associate-*r/ 56.6%

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

      5. *-commutative 56.6%

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

   7. Applied egg-rr 56.6%

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

      1. unpow1 56.6%

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

      2. *-commutative 56.6%

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

      3. associate-/l* 56.7%

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

      4. associate-*l* 56.7%

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

   9. Simplified 56.7%

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

       1. associate-*r* 56.7%

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

       2. sqrt-prod 56.5%

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

       3. div-inv 56.6%

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

       4. associate-*r* 56.6%

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

       5. *-commutative 56.6%

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

       6. associate-*r* 56.5%

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

       7. div-inv 56.5%

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

       8. *-commutative 56.5%

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

       9. sqrt-unprod 72.3%

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

       10. associate-*l* 72.3%

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

       11. sqrt-unprod 72.5%

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

   11. Applied egg-rr 72.5%

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

   if 1.85000000000000001e87 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.8%

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

      1. associate-/l* 2.8%

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

   5. Simplified 2.8%

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

      1. pow1 2.8%

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

      2. *-commutative 2.8%

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

      3. sqrt-unprod 2.8%

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

      4. associate-*r/ 2.8%

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

      5. *-commutative 2.8%

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

   7. Applied egg-rr 2.8%

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

      1. unpow1 2.8%

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

      2. *-commutative 2.8%

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

      3. associate-/l* 2.8%

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

      4. associate-*l* 2.8%

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

   9. Simplified 2.8%

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

       1. expm1-log1p-u 2.8%

          \[\leadsto \sqrt{\pi \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n}{k} \cdot 2\right)\right)}} \]

       2. expm1-undefine 42.4%

          \[\leadsto \sqrt{\pi \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{n}{k} \cdot 2\right)} - 1\right)}} \]

       3. associate-*l/ 42.4%

          \[\leadsto \sqrt{\pi \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{n \cdot 2}{k}}\right)} - 1\right)} \]

   11. Applied egg-rr 42.4%

       \[\leadsto \sqrt{\pi \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{n \cdot 2}{k}\right)} - 1\right)}} \]
   12. Step-by-step derivation

       1. sub-neg 42.4%

          \[\leadsto \sqrt{\pi \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{n \cdot 2}{k}\right)} + \left(-1\right)\right)}} \]

       2. metadata-eval 42.4%

          \[\leadsto \sqrt{\pi \cdot \left(e^{\mathsf{log1p}\left(\frac{n \cdot 2}{k}\right)} + \color{blue}{-1}\right)} \]

       3. +-commutative 42.4%

          \[\leadsto \sqrt{\pi \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(\frac{n \cdot 2}{k}\right)}\right)}} \]

       4. log1p-undefine 42.4%

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

       5. rem-exp-log 42.4%

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

       6. +-commutative 42.4%

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

       7. *-commutative 42.4%

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

       8. associate-*r/ 42.4%

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

       9. fma-define 42.4%

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

   13. Simplified 42.4%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\pi \cdot \left(-1 + \mathsf{fma}\left(2, \frac{n}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{+256}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\pi}{k} \cdot \left(2 \cdot n\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6.5e+256)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (cbrt (pow (* (/ PI k) (* 2.0 n)) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6.5e+256) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = cbrt(pow(((((double) M_PI) / k) * (2.0 * n)), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 6.5e+256) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((2.0 * n));
	} else {
		tmp = Math.cbrt(Math.pow(((Math.PI / k) * (2.0 * n)), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 6.5e+256)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = cbrt((Float64(Float64(pi / k) * Float64(2.0 * n)) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6.5e+256], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(Pi / k), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.50000000000000053e256

   1. Initial program 99.5%

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

   3. Taylor expanded in k around 0 42.5%

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

      1. associate-/l* 42.5%

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

   5. Simplified 42.5%

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

      1. pow1 42.5%

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

      2. *-commutative 42.5%

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

      3. sqrt-unprod 42.6%

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

      4. associate-*r/ 42.5%

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

      5. *-commutative 42.5%

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

   7. Applied egg-rr 42.5%

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

      1. unpow1 42.5%

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

      2. *-commutative 42.5%

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

      3. associate-/l* 42.6%

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

      4. associate-*l* 42.6%

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

   9. Simplified 42.6%

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

       1. associate-*r* 42.6%

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

       2. sqrt-prod 42.5%

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

       3. div-inv 42.5%

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

       4. associate-*r* 42.5%

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

       5. *-commutative 42.5%

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

       6. associate-*r* 42.5%

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

       7. div-inv 42.5%

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

       8. *-commutative 42.5%

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

       9. sqrt-unprod 54.1%

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

       10. associate-*l* 54.1%

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

       11. sqrt-unprod 54.3%

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

   11. Applied egg-rr 54.3%

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

   if 6.50000000000000053e256 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.9%

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

      1. associate-/l* 2.9%

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

   5. Simplified 2.9%

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

      1. pow1 2.9%

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

      2. *-commutative 2.9%

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

      3. sqrt-unprod 2.9%

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

      4. associate-*r/ 2.9%

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

      5. *-commutative 2.9%

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

   7. Applied egg-rr 2.9%

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

      1. unpow1 2.9%

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

      2. *-commutative 2.9%

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

      3. associate-/l* 2.9%

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

      4. associate-*l* 2.9%

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

   9. Simplified 2.9%

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

       1. add-cbrt-cube 31.1%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\pi \cdot \left(\frac{n}{k} \cdot 2\right)} \cdot \sqrt{\pi \cdot \left(\frac{n}{k} \cdot 2\right)}\right) \cdot \sqrt{\pi \cdot \left(\frac{n}{k} \cdot 2\right)}}} \]

       2. pow1/3 31.1%

          \[\leadsto \color{blue}{{\left(\left(\sqrt{\pi \cdot \left(\frac{n}{k} \cdot 2\right)} \cdot \sqrt{\pi \cdot \left(\frac{n}{k} \cdot 2\right)}\right) \cdot \sqrt{\pi \cdot \left(\frac{n}{k} \cdot 2\right)}\right)}^{0.3333333333333333}} \]

   11. Applied egg-rr 31.1%

       \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   12. Step-by-step derivation

       1. unpow1/3 31.1%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1.5}}} \]

       2. associate-*r* 31.1%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(2 \cdot n\right) \cdot \frac{\pi}{k}\right)}}^{1.5}} \]

       3. *-commutative 31.1%

          \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(n \cdot 2\right)} \cdot \frac{\pi}{k}\right)}^{1.5}} \]

   13. Simplified 31.1%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(n \cdot 2\right) \cdot \frac{\pi}{k}\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{+256}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\pi}{k} \cdot \left(2 \cdot n\right)\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 37.2%

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

   1. associate-/l* 37.2%

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

5. Simplified 37.2%

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

   1. pow1 37.2%

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

   2. *-commutative 37.2%

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

   3. sqrt-unprod 37.3%

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

   4. associate-*r/ 37.3%

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

   5. *-commutative 37.3%

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

7. Applied egg-rr 37.3%

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

   1. unpow1 37.3%

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

   2. *-commutative 37.3%

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

   3. associate-/l* 37.3%

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

   4. associate-*l* 37.3%

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

9. Simplified 37.3%

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

    1. associate-*r* 37.3%

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

    2. sqrt-prod 37.2%

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

    3. div-inv 37.2%

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

    4. associate-*r* 37.2%

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

    5. *-commutative 37.2%

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

    6. associate-*r* 37.2%

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

    7. div-inv 37.2%

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

    8. *-commutative 37.2%

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

    9. sqrt-unprod 47.4%

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

    10. associate-*l* 47.3%

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

    11. sqrt-unprod 47.5%

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

11. Applied egg-rr 47.5%

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

12. Final simplification 47.5%

    \[\leadsto \sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n} \]
13. Add Preprocessing

Alternative 8: 38.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 37.2%

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

   1. associate-/l* 37.2%

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

5. Simplified 37.2%

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

   1. pow1 37.2%

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

   2. *-commutative 37.2%

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

   3. sqrt-unprod 37.3%

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

   4. associate-*r/ 37.3%

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

   5. *-commutative 37.3%

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

7. Applied egg-rr 37.3%

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

   1. unpow1 37.3%

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

   2. *-commutative 37.3%

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

   3. associate-/l* 37.3%

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

   4. associate-*l* 37.3%

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

9. Simplified 37.3%

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

10. Final simplification 37.3%

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

Alternative 9: 38.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 37.2%

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

   1. associate-/l* 37.2%

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

5. Simplified 37.2%

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

   1. pow1 37.2%

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

   2. *-commutative 37.2%

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

   3. sqrt-unprod 37.3%

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

   4. associate-*r/ 37.3%

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

   5. *-commutative 37.3%

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

7. Applied egg-rr 37.3%

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

   1. unpow1 37.3%

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

   2. *-commutative 37.3%

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

   3. associate-/l* 37.3%

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

   4. associate-*l* 37.3%

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

9. Simplified 37.3%

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

    1. *-commutative 37.3%

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

    2. clear-num 37.3%

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

    3. un-div-inv 37.3%

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

11. Applied egg-rr 37.3%

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

    1. associate-/r/ 37.3%

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

13. Applied egg-rr 37.3%

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

14. Final simplification 37.3%

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

Reproduce

herbie shell --seed 2024113 
(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))))