Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 6.5s

Alternatives: 7

Speedup: 1.1×

• could not determine a ground truth

  1. k = 3.8435688582855635e+40
  2. n = -2.2028495095962577e-24

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. 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. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 99.3%

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

5. Final simplification 99.3%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;\frac{\sqrt{\pi} \cdot \sqrt{n}}{\sqrt{k}} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{k}} \cdot {\left(\left(\pi + \pi\right) \cdot n\right)}^{\left(-0.5 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.0)
   (* (/ (* (sqrt PI) (sqrt n)) (sqrt k)) (sqrt 2.0))
   (* (/ 1.0 (sqrt k)) (pow (* (+ PI PI) n) (* -0.5 k)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.0) {
		tmp = ((sqrt(((double) M_PI)) * sqrt(n)) / sqrt(k)) * sqrt(2.0);
	} else {
		tmp = (1.0 / sqrt(k)) * pow(((((double) M_PI) + ((double) M_PI)) * n), (-0.5 * k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.0) {
		tmp = ((Math.sqrt(Math.PI) * Math.sqrt(n)) / Math.sqrt(k)) * Math.sqrt(2.0);
	} else {
		tmp = (1.0 / Math.sqrt(k)) * Math.pow(((Math.PI + Math.PI) * n), (-0.5 * k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.0:
		tmp = ((math.sqrt(math.pi) * math.sqrt(n)) / math.sqrt(k)) * math.sqrt(2.0)
	else:
		tmp = (1.0 / math.sqrt(k)) * math.pow(((math.pi + math.pi) * n), (-0.5 * k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.0)
		tmp = Float64(Float64(Float64(sqrt(pi) * sqrt(n)) / sqrt(k)) * sqrt(2.0));
	else
		tmp = Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(pi + pi) * n) ^ Float64(-0.5 * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.0)
		tmp = ((sqrt(pi) * sqrt(n)) / sqrt(k)) * sqrt(2.0);
	else
		tmp = (1.0 / sqrt(k)) * (((pi + pi) * n) ^ (-0.5 * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.0], N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision], N[(-0.5 * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 98.4%

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

   3. Taylor expanded in k around 0

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 76.2

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

   5. Applied rewrites 76.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-/.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-sqrt.f64 75.8

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

   7. Applied rewrites 75.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-sqrt.f64 98.1

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

   9. Applied rewrites 98.1%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. sqrt-prod N/A

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

       5. lower-*.f64 N/A

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

       6. lower-sqrt.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. lower-sqrt.f64 98.1

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

   11. Applied rewrites 98.1%

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

   if 1 < 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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in k around inf

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

      1. lower-*.f64 99.3

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

   10. Applied rewrites 99.3%

       \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(\pi + \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[k_, n_] := N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision], N[(-0.5 * k + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in k around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.2

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

5. Applied rewrites 99.2%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}} \]
6. Step-by-step derivation

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. count-2-rev N/A

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

   4. lift-+.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-PI.f64 99.2

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

7. Applied rewrites 99.2%

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

Alternative 4: 50.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in k around 0

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

   1. sqrt-unprod N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 40.0

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

5. Applied rewrites 40.0%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqrt-prod N/A

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

   4. lift-/.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-sqrt.f64 39.8

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

7. Applied rewrites 39.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. sqrt-div N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-sqrt.f64 51.1

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

9. Applied rewrites 51.1%

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

    1. lift-sqrt.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. sqrt-prod N/A

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

    5. lower-*.f64 N/A

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

    6. lower-sqrt.f64 N/A

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

    7. lift-PI.f64 N/A

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

    8. lower-sqrt.f64 51.2

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

11. Applied rewrites 51.2%

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

Alternative 5: 50.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in k around 0

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

   1. sqrt-unprod N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 40.0

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

5. Applied rewrites 40.0%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqrt-prod N/A

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

   4. lift-/.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-sqrt.f64 39.8

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

7. Applied rewrites 39.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. sqrt-div N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-sqrt.f64 51.1

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

9. Applied rewrites 51.1%

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

Alternative 6: 50.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. 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. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in k around 0

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

   1. *-lft-identity N/A

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

   2. *-commutative N/A

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

   3. sqrt-unprod N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sqrt.f64 50.9

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 50.9

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

7. Applied rewrites 50.9%

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

Alternative 7: 38.6% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in k around 0

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

   1. sqrt-unprod N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 40.0

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

5. Applied rewrites 40.0%

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

   1. lift-/.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-/.f64 40.0

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

7. Applied rewrites 40.0%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-*.f64 40.0

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

9. Applied rewrites 40.0%

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

Reproduce

herbie shell --seed 2025061 
(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))))