Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 9.0s

Alternatives: 7

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 5.794742544065582e+276
  2. n = -2.9077647572972874e+39

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ \frac{\sqrt{t\_0}}{{t\_0}^{\left(0.5 \cdot k\right)} \cdot \sqrt{k}} \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* 2.0 PI) n)))
   (/ (sqrt t_0) (* (pow t_0 (* 0.5 k)) (sqrt k)))))

C

double code(double k, double n) {
	double t_0 = (2.0 * ((double) M_PI)) * n;
	return sqrt(t_0) / (pow(t_0, (0.5 * k)) * sqrt(k));
}

Java

public static double code(double k, double n) {
	double t_0 = (2.0 * Math.PI) * n;
	return Math.sqrt(t_0) / (Math.pow(t_0, (0.5 * k)) * Math.sqrt(k));
}

Python

def code(k, n):
	t_0 = (2.0 * math.pi) * n
	return math.sqrt(t_0) / (math.pow(t_0, (0.5 * k)) * math.sqrt(k))

Julia

function code(k, n)
	t_0 = Float64(Float64(2.0 * pi) * n)
	return Float64(sqrt(t_0) / Float64((t_0 ^ Float64(0.5 * k)) * sqrt(k)))
end

MATLAB

function tmp = code(k, n)
	t_0 = (2.0 * pi) * n;
	tmp = sqrt(t_0) / ((t_0 ^ (0.5 * k)) * sqrt(k));
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision]}, N[(N[Sqrt[t$95$0], $MachinePrecision] / N[(N[Power[t$95$0, N[(0.5 * k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. div-sub N/A

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

   9. metadata-eval N/A

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

   10. pow-sub N/A

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

   11. associate-/l/ N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. div-sub N/A

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

   9. metadata-eval N/A

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

   10. pow-sub N/A

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

   11. associate-/l/ N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

6. Applied rewrites 99.4%

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

Alternative 4: 49.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 39.1

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

5. Applied rewrites 39.1%

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

7. Applied rewrites 39.2%

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

9. Applied rewrites 39.2%

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

11. Applied rewrites 49.8%

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

12. Final simplification 49.8%

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

Alternative 5: 49.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 39.1

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

5. Applied rewrites 39.1%

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

7. Applied rewrites 49.8%

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

8. Final simplification 49.8%

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

Alternative 6: 38.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 39.1

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

5. Applied rewrites 39.1%

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

7. Applied rewrites 39.2%

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

9. Applied rewrites 39.2%

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

10. Final simplification 39.2%

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

Alternative 7: 38.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 39.1

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

5. Applied rewrites 39.1%

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

7. Applied rewrites 39.2%

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

9. Applied rewrites 39.2%

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

10. Final simplification 39.2%

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

Reproduce

herbie shell --seed 2024240 
(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))))