Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.5%

Time: 6.1s

Alternatives: 11

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 3.817681005484551e+125
  2. n = -7.286224387105149e+118

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: 99.5% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi + \pi\right) \cdot n\\ \sqrt{\frac{1}{k}} \cdot \frac{\sqrt{t\_0}}{{t\_0}^{\left(0.5 \cdot k\right)}} \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (+ PI PI) n)))
   (* (sqrt (/ 1.0 k)) (/ (sqrt t_0) (pow t_0 (* 0.5 k))))))

C

double code(double k, double n) {
	double t_0 = (((double) M_PI) + ((double) M_PI)) * n;
	return sqrt((1.0 / k)) * (sqrt(t_0) / pow(t_0, (0.5 * k)));
}

Java

public static double code(double k, double n) {
	double t_0 = (Math.PI + Math.PI) * n;
	return Math.sqrt((1.0 / k)) * (Math.sqrt(t_0) / Math.pow(t_0, (0.5 * k)));
}

Python

def code(k, n):
	t_0 = (math.pi + math.pi) * n
	return math.sqrt((1.0 / k)) * (math.sqrt(t_0) / math.pow(t_0, (0.5 * k)))

Julia

function code(k, n)
	t_0 = Float64(Float64(pi + pi) * n)
	return Float64(sqrt(Float64(1.0 / k)) * Float64(sqrt(t_0) / (t_0 ^ Float64(0.5 * k))))
end

MATLAB

function tmp = code(k, n)
	t_0 = (pi + pi) * n;
	tmp = sqrt((1.0 / k)) * (sqrt(t_0) / (t_0 ^ (0.5 * k)));
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision]}, N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Power[t$95$0, N[(0.5 * k), $MachinePrecision]], $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. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. div-sub N/A

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

   8. metadata-eval N/A

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

   9. pow-sub N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-div N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-/.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. count-2-rev N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. pow-sub N/A

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

   12. unpow1 N/A

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

   13. lower-/.f64 N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $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. 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. lower-/.f64 99.5

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

3. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-div N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-/.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-pow2 N/A

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

   5. lower-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift--.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 6: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

Alternative 7: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(\pi + \pi\right) \cdot n}\\ \mathbf{if}\;k \leq 1.05:\\ \;\;\;\;\sqrt{\frac{1}{k}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_0}^{\left(-k\right)}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (sqrt (* (+ PI PI) n))))
   (if (<= k 1.05) (* (sqrt (/ 1.0 k)) t_0) (/ (pow t_0 (- k)) (sqrt k)))))

C

double code(double k, double n) {
	double t_0 = sqrt(((((double) M_PI) + ((double) M_PI)) * n));
	double tmp;
	if (k <= 1.05) {
		tmp = sqrt((1.0 / k)) * t_0;
	} else {
		tmp = pow(t_0, -k) / sqrt(k);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.sqrt(((Math.PI + Math.PI) * n));
	double tmp;
	if (k <= 1.05) {
		tmp = Math.sqrt((1.0 / k)) * t_0;
	} else {
		tmp = Math.pow(t_0, -k) / Math.sqrt(k);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.sqrt(((math.pi + math.pi) * n))
	tmp = 0
	if k <= 1.05:
		tmp = math.sqrt((1.0 / k)) * t_0
	else:
		tmp = math.pow(t_0, -k) / math.sqrt(k)
	return tmp

Julia

function code(k, n)
	t_0 = sqrt(Float64(Float64(pi + pi) * n))
	tmp = 0.0
	if (k <= 1.05)
		tmp = Float64(sqrt(Float64(1.0 / k)) * t_0);
	else
		tmp = Float64((t_0 ^ Float64(-k)) / sqrt(k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = sqrt(((pi + pi) * n));
	tmp = 0.0;
	if (k <= 1.05)
		tmp = sqrt((1.0 / k)) * t_0;
	else
		tmp = (t_0 ^ -k) / sqrt(k);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[k, 1.05], N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Power[t$95$0, (-k)], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.05000000000000004

   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. 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. lower-/.f64 99.5

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in k around 0

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-prod N/A

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

      5. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 49.5

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

   6. Applied rewrites 49.5%

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

   if 1.05000000000000004 < k

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in k around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 53.4

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

   6. Applied rewrites 53.4%

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

Alternative 8: 61.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.9 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 1.9e-14)
   (sqrt (/ (* (+ PI PI) n) k))
   (* (sqrt (/ (+ PI PI) (* n k))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 1.9e-14) {
		tmp = sqrt((((((double) M_PI) + ((double) M_PI)) * n) / k));
	} else {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / (n * k))) * n;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 1.9e-14) {
		tmp = Math.sqrt((((Math.PI + Math.PI) * n) / k));
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (n * k))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 1.9e-14:
		tmp = math.sqrt((((math.pi + math.pi) * n) / k))
	else:
		tmp = math.sqrt(((math.pi + math.pi) / (n * k))) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 1.9e-14)
		tmp = sqrt(Float64(Float64(Float64(pi + pi) * n) / k));
	else
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / Float64(n * k))) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 1.9e-14)
		tmp = sqrt((((pi + pi) * n) / k));
	else
		tmp = sqrt(((pi + pi) / (n * k))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 1.9e-14], N[Sqrt[N[(N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / N[(n * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1.9000000000000001e-14

   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. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 38.4

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

   4. Applied rewrites 38.4%

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

   if 1.9000000000000001e-14 < n

   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. Taylor expanded in k around 0

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

      1. unpow1/2 N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-PI.f64 38.4

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

   4. Applied rewrites 38.4%

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

   5. Taylor expanded in n around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 50.1

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

   7. Applied rewrites 50.1%

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

Alternative 9: 49.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] * n), $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. Taylor expanded in k around 0

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

   1. unpow1/2 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-PI.f64 38.4

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

4. Applied rewrites 38.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. count-2-rev N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. sqrt-undiv N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 49.5%

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

Alternative 10: 38.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision] / 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. Taylor expanded in k around 0

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

   1. unpow1/2 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-PI.f64 38.4

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

4. Applied rewrites 38.4%

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

Alternative 11: 38.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] * N[(n / k), $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. Taylor expanded in k around 0

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

   1. unpow1/2 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-PI.f64 38.4

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

4. Applied rewrites 38.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2-rev N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. count-2-rev N/A

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

   10. lift-+.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-/.f64 38.4

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

6. Applied rewrites 38.4%

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

Reproduce

herbie shell --seed 2025138 
(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))))