Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 5.2s

Alternatives: 8

Speedup: 1.1×

• could not determine a ground truth

  1. k = 1.0035355486222125e+105
  2. n = -2.612958418836041e+68

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, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(\left(n + n\right) \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {k}^{-0.5} \end{array} \]

FPCore

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

C

double code(double k, double n) {
	return pow(((n + n) * ((double) M_PI)), ((1.0 - k) / 2.0)) * pow(k, -0.5);
}

Java

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

Python

def code(k, n):
	return math.pow(((n + n) * math.pi), ((1.0 - k) / 2.0)) * math.pow(k, -0.5)

Julia

function code(k, n)
	return Float64((Float64(Float64(n + n) * pi) ^ Float64(Float64(1.0 - k) / 2.0)) * (k ^ -0.5))
end

MATLAB

function tmp = code(k, n)
	tmp = (((n + n) * pi) ^ ((1.0 - k) / 2.0)) * (k ^ -0.5);
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-div N/A

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

   5. lower-sqrt.f64 N/A

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

   6. inv-pow N/A

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

   7. lower-pow.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. inv-pow N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

6. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 99.5

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

8. Applied rewrites 99.5%

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

Alternative 2: 98.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.0)
   (* (sqrt (* (* n 2.0) PI)) (pow k -0.5))
   (* (/ 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(((n * 2.0) * ((double) M_PI))) * pow(k, -0.5);
	} 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(((n * 2.0) * Math.PI)) * Math.pow(k, -0.5);
	} 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(((n * 2.0) * math.pi)) * math.pow(k, -0.5)
	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(sqrt(Float64(Float64(n * 2.0) * pi)) * (k ^ -0.5));
	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(((n * 2.0) * pi)) * (k ^ -0.5);
	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[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[Power[k, -0.5], $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 98.9%

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

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-div N/A

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

      5. lower-sqrt.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 99.0

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

   4. Applied rewrites 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. inv-pow N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-lft-identity N/A

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

      4. sqrt-unprod N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 97.3

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

   9. Applied rewrites 97.3%

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

   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 inf

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

      1. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 100.0

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

   7. Applied rewrites 100.0%

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

Alternative 3: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (fma -0.5 k 0.5))))

C

double code(double k, double n) {
	return (1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), fma(-0.5, k, 0.5));
}

Julia

function code(k, n)
	return Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * 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[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(-0.5 * k + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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 99.5

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

5. Applied rewrites 99.5%

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

Alternative 4: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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 99.5%

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

5. Taylor expanded in k around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

   \[\leadsto \frac{{\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}}{\sqrt{k}} \]
9. Add Preprocessing

Alternative 5: 50.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-div N/A

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

   5. lower-sqrt.f64 N/A

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

   6. inv-pow N/A

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

   7. lower-pow.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. inv-pow N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-lft-identity N/A

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

   4. sqrt-unprod N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 48.9

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

9. Applied rewrites 48.9%

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

Alternative 6: 50.1% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-div N/A

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

   5. lower-sqrt.f64 N/A

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

   6. inv-pow N/A

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

   7. lower-pow.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. *-lft-identity N/A

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

   3. sqrt-unprod N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-sqrt.f64 48.9

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

   11. lift-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. associate-*r* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lift-PI.f64 48.9

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

7. Applied rewrites 48.9%

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

   1. lift-pow.f64 N/A

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

   2. inv-pow N/A

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

   3. lower-/.f64 48.9

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

9. Applied rewrites 48.9%

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

Alternative 7: 38.4% accurate, 4.8× speedup

Math

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

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(Float64(pi * 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[(N[(Pi * N[(n / k), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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 37.2

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

5. Applied rewrites 37.2%

   \[\leadsto \color{blue}{\sqrt{\frac{\pi \cdot n}{k} \cdot 2}} \]
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 37.2

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

7. Applied rewrites 37.2%

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

Alternative 8: 38.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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 37.2

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

5. Applied rewrites 37.2%

   \[\leadsto \color{blue}{\sqrt{\frac{\pi \cdot n}{k} \cdot 2}} \]
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. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-PI.f64 36.9

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

7. Applied rewrites 36.9%

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

Reproduce

herbie shell --seed 2025085 
(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))))