Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 6.1s

Alternatives: 11

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 3.51322856506362e+267
  2. n = -9.375545120790664e+232

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, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[t$95$0, (-k)], $MachinePrecision] * t$95$0), $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. Taylor expanded in n around 0

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

   1. sum-log N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 96.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift--.f64 N/A

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

   5. exp-to-pow N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. sqrt-prod N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. count-2-rev N/A

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

   13. sqrt-prod N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. sub-flip N/A

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

   17. +-commutative N/A

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

6. Applied rewrites 99.4%

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

Alternative 2: 99.4% accurate, 0.9× speedup

Math

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

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(Float64(t_0 / (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[(N[(t$95$0 / N[Power[t$95$0, k], $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. Taylor expanded in n around 0

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

   1. sum-log N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 96.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift--.f64 N/A

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

   5. exp-to-pow N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. sqrt-prod N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. count-2-rev N/A

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

   13. sqrt-prod N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

6. Applied rewrites 99.4%

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

Alternative 3: 99.4% 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 99.4%

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

2. Taylor expanded in n around 0

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

   1. sum-log N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 96.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift--.f64 N/A

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

   5. exp-to-pow N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. sqrt-prod N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. count-2-rev N/A

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

   13. sqrt-prod N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

6. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. associate-/l/ N/A

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

8. Applied rewrites 99.4%

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

Alternative 4: 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 99.4%

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

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

   8. pow2 N/A

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

   9. lower-pow.f64 N/A

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

3. Applied rewrites 99.2%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{{\left({\left(\left(\pi + \pi\right) \cdot n\right)}^{\left(\left(1 - k\right) \cdot 0.25\right)}\right)}^{2}} \]

4. Taylor expanded in n around 0

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

6. Applied rewrites 99.4%

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

Alternative 5: 73.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 140000000000:\\ \;\;\;\;\frac{\sqrt{\left(k \cdot \frac{\pi}{n}\right) \cdot 2} \cdot n}{k}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{\left(\left(\pi + \pi\right) \cdot n\right) \cdot \sqrt{\frac{1}{k} \cdot \frac{1}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 140000000000.0)
   (/ (* (sqrt (* (* k (/ PI n)) 2.0)) n) k)
   (if (<= n 1.15e+85)
     (sqrt (* (* (+ PI PI) n) (sqrt (* (/ 1.0 k) (/ 1.0 k)))))
     (* (sqrt (/ (+ PI PI) (* n k))) n))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 140000000000.0) {
		tmp = (sqrt(((k * (((double) M_PI) / n)) * 2.0)) * n) / k;
	} else if (n <= 1.15e+85) {
		tmp = sqrt((((((double) M_PI) + ((double) M_PI)) * n) * sqrt(((1.0 / k) * (1.0 / 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 <= 140000000000.0) {
		tmp = (Math.sqrt(((k * (Math.PI / n)) * 2.0)) * n) / k;
	} else if (n <= 1.15e+85) {
		tmp = Math.sqrt((((Math.PI + Math.PI) * n) * Math.sqrt(((1.0 / k) * (1.0 / k)))));
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (n * k))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 140000000000.0:
		tmp = (math.sqrt(((k * (math.pi / n)) * 2.0)) * n) / k
	elif n <= 1.15e+85:
		tmp = math.sqrt((((math.pi + math.pi) * n) * math.sqrt(((1.0 / k) * (1.0 / k)))))
	else:
		tmp = math.sqrt(((math.pi + math.pi) / (n * k))) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 140000000000.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(k * Float64(pi / n)) * 2.0)) * n) / k);
	elseif (n <= 1.15e+85)
		tmp = sqrt(Float64(Float64(Float64(pi + pi) * n) * sqrt(Float64(Float64(1.0 / k) * Float64(1.0 / 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 <= 140000000000.0)
		tmp = (sqrt(((k * (pi / n)) * 2.0)) * n) / k;
	elseif (n <= 1.15e+85)
		tmp = sqrt((((pi + pi) * n) * sqrt(((1.0 / k) * (1.0 / k)))));
	else
		tmp = sqrt(((pi + pi) / (n * k))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 140000000000.0], N[(N[(N[Sqrt[N[(N[(k * N[(Pi / n), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[n, 1.15e+85], N[Sqrt[N[(N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / N[(n * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1.4e11

   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. 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 37.1

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

   4. Applied rewrites 37.1%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 23.8

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

   6. Applied rewrites 23.8%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 38.0%

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

   10. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-PI.f64 49.7

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

   12. Applied rewrites 49.7%

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

   if 1.4e11 < n < 1.1499999999999999e85

   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. 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 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 37.1

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

   6. Applied rewrites 37.1%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 34.7

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

   8. Applied rewrites 34.7%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. exp-fabs N/A

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

      5. exp-to-pow N/A

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

      6. inv-pow N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 34.0

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

   10. Applied rewrites 34.0%

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

   if 1.1499999999999999e85 < n

   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. 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 37.1

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

   4. Applied rewrites 37.1%

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

   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 49.9

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

   7. Applied rewrites 49.9%

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

Alternative 6: 72.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sqrt{\left(k \cdot \frac{\pi}{n}\right) \cdot 2} \cdot n}{k}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 2e-16)
   (/ (* (sqrt (* (* k (/ PI n)) 2.0)) n) k)
   (* (sqrt (/ (+ PI PI) (* n k))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 2e-16) {
		tmp = (sqrt(((k * (((double) M_PI) / n)) * 2.0)) * 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 <= 2e-16) {
		tmp = (Math.sqrt(((k * (Math.PI / n)) * 2.0)) * n) / k;
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (n * k))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 2e-16:
		tmp = (math.sqrt(((k * (math.pi / n)) * 2.0)) * 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 <= 2e-16)
		tmp = Float64(Float64(sqrt(Float64(Float64(k * Float64(pi / n)) * 2.0)) * 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 <= 2e-16)
		tmp = (sqrt(((k * (pi / n)) * 2.0)) * n) / k;
	else
		tmp = sqrt(((pi + pi) / (n * k))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 2e-16], N[(N[(N[Sqrt[N[(N[(k * N[(Pi / n), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision] / k), $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 < 2e-16

   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. 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 37.1

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

   4. Applied rewrites 37.1%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 23.8

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

   6. Applied rewrites 23.8%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 38.0%

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

   10. Taylor expanded in n around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-PI.f64 49.7

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

   12. Applied rewrites 49.7%

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

   if 2e-16 < n

   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. 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 37.1

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

   4. Applied rewrites 37.1%

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

   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 49.9

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

   7. Applied rewrites 49.9%

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

Alternative 7: 61.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.35 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\left(\pi + \pi\right) \cdot \frac{n}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 1.35e-10)
   (sqrt (* (+ PI PI) (/ n k)))
   (* (sqrt (/ (+ PI PI) (* n k))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 1.35e-10) {
		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 <= 1.35e-10) {
		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 <= 1.35e-10:
		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 <= 1.35e-10)
		tmp = sqrt(Float64(Float64(pi + pi) * Float64(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 <= 1.35e-10)
		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, 1.35e-10], N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] * N[(n / k), $MachinePrecision]), $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 < 1.35e-10

   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. 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. sqr-pow N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

   3. Applied rewrites 99.2%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{{\left({\left(\left(\pi + \pi\right) \cdot n\right)}^{\left(\left(1 - k\right) \cdot 0.25\right)}\right)}^{2}} \]

   4. Taylor expanded in k around 0

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

      1. unpow2 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{4}} \cdot {\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{4}}}{\sqrt{\color{blue}{k}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{4}}}{\sqrt{k}} \]

      3. metadata-eval N/A

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

      4. sqr-pow N/A

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

      5. pow1/2 N/A

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

      6. sqrt-undiv N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. count-2-rev N/A

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

      10. lift-+.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

   6. Applied rewrites 37.1%

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

   if 1.35e-10 < n

   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. 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 37.1

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

   4. Applied rewrites 37.1%

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

   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 49.9

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

   7. Applied rewrites 49.9%

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

Alternative 8: 49.2% 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 99.4%

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

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 37.1

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

4. Applied rewrites 37.1%

   \[\leadsto \color{blue}{\sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}}} \]
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 49.2%

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

Alternative 9: 37.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(Pi + Pi), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n / 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. 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 37.1

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

4. Applied rewrites 37.1%

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

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

   9. sqrt-prod N/A

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

   10. count-2-rev N/A

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

   11. lift-+.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-*.f64 N/A

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

6. Applied rewrites 37.0%

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

Alternative 10: 37.1% accurate, 3.1× speedup

Math

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

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(Float64(pi * 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[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision] * 2.0), $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. 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 37.1

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

4. Applied rewrites 37.1%

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

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

   8. *-commutative N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-PI.f64 37.1

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

6. Applied rewrites 37.1%

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

Alternative 11: 37.0% 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 99.4%

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

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 37.1

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

4. Applied rewrites 37.1%

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

Reproduce

herbie shell --seed 2025142 
(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))))