Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 6.1s

Alternatives: 10

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 2.5801701475585045e+248
  2. n = -2.26573000750019e+87

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lower-/.f64 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-pow.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-neg-out N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. sub-flip N/A

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

   9. pow-sub N/A

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

   10. lower-unsound-pow.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. distribute-lft-in N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. count-2-rev N/A

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

   17. lift-*.f64 N/A

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

   18. lower-unsound-/.f64 N/A

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

5. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lower-/.f64 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-pow.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. pow-add N/A

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

   4. lower-unsound-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. distribute-lft-in N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. count-2-rev N/A

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

   11. lift-*.f64 N/A

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

   12. lower-unsound-*.f64 N/A

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

5. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lower-/.f64 99.5

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

3. Applied rewrites 99.5%

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

Alternative 4: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   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. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 37.6

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 37.6

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

   6. Applied rewrites 37.6%

      \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]
   7. 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. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. sqrt-prod N/A

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

      9. lower-unsound-*.f64 N/A

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

      10. lower-unsound-sqrt.f64 N/A

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

      11. lower-unsound-sqrt.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 49.2

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

   8. Applied rewrites 49.2%

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

   if 1 < k

   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-*.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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 99.5

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in k around inf

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

      1. lower-*.f64 53.7

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

   6. Applied rewrites 53.7%

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

Alternative 5: 70.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{\pi}{k \cdot n}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+240}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\frac{\pi + \pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(n \cdot \pi\right)}}{k \cdot \sqrt{\sqrt{\frac{1}{k} \cdot \frac{1}{k}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0)))))
   (if (<= t_0 0.0)
     (* n (sqrt (* 2.0 (/ PI (* k n)))))
     (if (<= t_0 4e+240)
       (* (sqrt n) (sqrt (/ (+ PI PI) k)))
       (/
        (sqrt (* 2.0 (* n PI)))
        (* k (sqrt (sqrt (* (/ 1.0 k) (/ 1.0 k))))))))))

C

double code(double k, double n) {
	double t_0 = (1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = n * sqrt((2.0 * (((double) M_PI) / (k * n))));
	} else if (t_0 <= 4e+240) {
		tmp = sqrt(n) * sqrt(((((double) M_PI) + ((double) M_PI)) / k));
	} else {
		tmp = sqrt((2.0 * (n * ((double) M_PI)))) / (k * sqrt(sqrt(((1.0 / k) * (1.0 / k)))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = (1.0 / Math.sqrt(k)) * Math.pow(((2.0 * Math.PI) * n), ((1.0 - k) / 2.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = n * Math.sqrt((2.0 * (Math.PI / (k * n))));
	} else if (t_0 <= 4e+240) {
		tmp = Math.sqrt(n) * Math.sqrt(((Math.PI + Math.PI) / k));
	} else {
		tmp = Math.sqrt((2.0 * (n * Math.PI))) / (k * Math.sqrt(Math.sqrt(((1.0 / k) * (1.0 / k)))));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = (1.0 / math.sqrt(k)) * math.pow(((2.0 * math.pi) * n), ((1.0 - k) / 2.0))
	tmp = 0
	if t_0 <= 0.0:
		tmp = n * math.sqrt((2.0 * (math.pi / (k * n))))
	elif t_0 <= 4e+240:
		tmp = math.sqrt(n) * math.sqrt(((math.pi + math.pi) / k))
	else:
		tmp = math.sqrt((2.0 * (n * math.pi))) / (k * math.sqrt(math.sqrt(((1.0 / k) * (1.0 / k)))))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(n * sqrt(Float64(2.0 * Float64(pi / Float64(k * n)))));
	elseif (t_0 <= 4e+240)
		tmp = Float64(sqrt(n) * sqrt(Float64(Float64(pi + pi) / k)));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(n * pi))) / Float64(k * sqrt(sqrt(Float64(Float64(1.0 / k) * Float64(1.0 / k))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = (1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = n * sqrt((2.0 * (pi / (k * n))));
	elseif (t_0 <= 4e+240)
		tmp = sqrt(n) * sqrt(((pi + pi) / k));
	else
		tmp = sqrt((2.0 * (n * pi))) / (k * sqrt(sqrt(((1.0 / k) * (1.0 / k)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 0.0], N[(n * N[Sqrt[N[(2.0 * N[(Pi / N[(k * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+240], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(n * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(k * N[Sqrt[N[Sqrt[N[(N[(1.0 / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 0.0

   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. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 37.6

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 37.6

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

   6. Applied rewrites 37.6%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 49.8

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

   9. Applied rewrites 49.8%

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

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 4.00000000000000006e240

   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. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 37.6

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 37.6

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

   6. Applied rewrites 37.6%

      \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]
   7. 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. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. sqrt-prod N/A

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

      9. lower-unsound-*.f64 N/A

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

      10. lower-unsound-sqrt.f64 N/A

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

      11. lower-unsound-sqrt.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 49.2

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

   8. Applied rewrites 49.2%

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

   if 4.00000000000000006e240 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   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. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 49.1

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

   7. Applied rewrites 49.1%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 37.6

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

   9. Applied rewrites 37.6%

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

Alternative 6: 60.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 1.8 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\frac{\pi + \pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{\pi}{k \cdot n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 1.8e+50)
   (* (sqrt n) (sqrt (/ (+ PI PI) k)))
   (* n (sqrt (* 2.0 (/ PI (* k n)))))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 1.8e+50) {
		tmp = sqrt(n) * sqrt(((((double) M_PI) + ((double) M_PI)) / k));
	} else {
		tmp = n * sqrt((2.0 * (((double) M_PI) / (k * n))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 1.8e+50) {
		tmp = Math.sqrt(n) * Math.sqrt(((Math.PI + Math.PI) / k));
	} else {
		tmp = n * Math.sqrt((2.0 * (Math.PI / (k * n))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 1.8e+50:
		tmp = math.sqrt(n) * math.sqrt(((math.pi + math.pi) / k))
	else:
		tmp = n * math.sqrt((2.0 * (math.pi / (k * n))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 1.8e+50)
		tmp = Float64(sqrt(n) * sqrt(Float64(Float64(pi + pi) / k)));
	else
		tmp = Float64(n * sqrt(Float64(2.0 * Float64(pi / Float64(k * n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 1.8e+50)
		tmp = sqrt(n) * sqrt(((pi + pi) / k));
	else
		tmp = n * sqrt((2.0 * (pi / (k * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 1.8e+50], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(n * N[Sqrt[N[(2.0 * N[(Pi / N[(k * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 1.79999999999999993e50

   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. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 37.6

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 37.6

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

   6. Applied rewrites 37.6%

      \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]
   7. 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. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. sqrt-prod N/A

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

      9. lower-unsound-*.f64 N/A

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

      10. lower-unsound-sqrt.f64 N/A

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

      11. lower-unsound-sqrt.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. lower-/.f64 49.2

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

   8. Applied rewrites 49.2%

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

   if 1.79999999999999993e50 < 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. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.2

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

   4. Applied rewrites 49.2%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 37.6

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 37.6

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

   6. Applied rewrites 37.6%

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

   7. Taylor expanded in n around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-*.f64 49.8

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

   9. Applied rewrites 49.8%

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

Alternative 7: 49.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / 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. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 49.2

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

4. Applied rewrites 49.2%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow1/2 N/A

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

   4. lift-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. sqrt-undiv N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-/.f64 37.6

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 37.6

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

6. Applied rewrites 37.6%

   \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]
7. 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. mult-flip N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l* N/A

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

   8. sqrt-prod N/A

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

   9. lower-unsound-*.f64 N/A

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

   10. lower-unsound-sqrt.f64 N/A

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

   11. lower-unsound-sqrt.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. mult-flip-rev N/A

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

   14. lower-/.f64 49.2

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

8. Applied rewrites 49.2%

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

Alternative 8: 37.6% 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. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 49.2

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

4. Applied rewrites 49.2%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow1/2 N/A

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

   4. lift-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. sqrt-undiv N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-/.f64 37.6

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 37.6

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

6. Applied rewrites 37.6%

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

Alternative 9: 37.6% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. 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. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 49.2

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

4. Applied rewrites 49.2%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow1/2 N/A

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

   4. lift-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. sqrt-undiv N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-/.f64 37.6

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 37.6

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

6. Applied rewrites 37.6%

   \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]
7. 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. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 37.6

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

8. Applied rewrites 37.6%

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

Alternative 10: 37.6% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(Pi / 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. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 49.2

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

4. Applied rewrites 49.2%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow1/2 N/A

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

   4. lift-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. sqrt-undiv N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-/.f64 37.6

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 37.6

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

6. Applied rewrites 37.6%

   \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]
7. 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-+.f64 N/A

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

   4. count-2-rev N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. associate-*r* N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. count-2-rev N/A

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

   13. lower-+.f64 N/A

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

   14. lower-/.f64 37.6

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

8. Applied rewrites 37.6%

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

Reproduce

herbie shell --seed 2025154 
(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))))