Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.6%

Time: 6.4s

Alternatives: 7

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 9.634355365278647e+27
  2. n = -6.146704296039295e+44

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: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Applied rewrites 99.5%

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

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. pow-sub N/A

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

5. Applied rewrites 99.6%

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

Alternative 2: 99.5% 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%

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

3. Applied rewrites 99.5%

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

Alternative 3: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := If[LessEqual[k, 1.0], N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision], (-k)], $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. 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 99.3%

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

   4. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-sqrt.f64 50.1

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 50.1

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

   6. Applied rewrites 50.1%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-div N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 50.2

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

   8. Applied rewrites 50.2%

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

   if 1 < k

   1. Initial program 99.4%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in k around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 52.8

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

   6. Applied rewrites 52.8%

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

Alternative 4: 61.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 5 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{\frac{n \cdot \left(\pi + \pi\right)}{\sqrt{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\frac{\pi + \pi}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))) 5e-62)
   (sqrt (/ (* n (+ PI PI)) (sqrt (* k k))))
   (* (sqrt n) (sqrt (/ (+ PI PI) k)))))

C

double code(double k, double n) {
	double tmp;
	if (((1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0))) <= 5e-62) {
		tmp = sqrt(((n * (((double) M_PI) + ((double) M_PI))) / sqrt((k * k))));
	} else {
		tmp = sqrt(n) * sqrt(((((double) M_PI) + ((double) M_PI)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (((1.0 / Math.sqrt(k)) * Math.pow(((2.0 * Math.PI) * n), ((1.0 - k) / 2.0))) <= 5e-62) {
		tmp = Math.sqrt(((n * (Math.PI + Math.PI)) / Math.sqrt((k * k))));
	} else {
		tmp = Math.sqrt(n) * Math.sqrt(((Math.PI + Math.PI) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if ((1.0 / math.sqrt(k)) * math.pow(((2.0 * math.pi) * n), ((1.0 - k) / 2.0))) <= 5e-62:
		tmp = math.sqrt(((n * (math.pi + math.pi)) / math.sqrt((k * k))))
	else:
		tmp = math.sqrt(n) * math.sqrt(((math.pi + math.pi) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0))) <= 5e-62)
		tmp = sqrt(Float64(Float64(n * Float64(pi + pi)) / sqrt(Float64(k * k))));
	else
		tmp = Float64(sqrt(n) * sqrt(Float64(Float64(pi + pi) / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (((1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0))) <= 5e-62)
		tmp = sqrt(((n * (pi + pi)) / sqrt((k * k))));
	else
		tmp = sqrt(n) * sqrt(((pi + pi) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[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], 5e-62], N[Sqrt[N[(N[(n * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(k * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 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)))) < 5.0000000000000002e-62

   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

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

      1. sqrt-undiv N/A

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

      2. rem-square-sqrt N/A

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

      3. sqrt-undiv N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. sqrt-undiv N/A

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

      8. rem-square-sqrt N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 38.5%

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

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 38.5

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

   6. Applied rewrites 38.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 38.5

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

      7. rem-square-sqrt N/A

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

      8. sqrt-prod N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-+.f64 N/A

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

   8. Applied rewrites 35.1%

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

   if 5.0000000000000002e-62 < (*.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

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

      1. sqrt-undiv N/A

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

      2. rem-square-sqrt N/A

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

      3. sqrt-undiv N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. sqrt-undiv N/A

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

      8. rem-square-sqrt N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 38.5%

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

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 38.5

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

   6. Applied rewrites 38.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. sqrt-prod N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 50.1

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

   8. Applied rewrites 50.1%

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

Alternative 5: 61.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{n \cdot \frac{\pi + \pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 2e-7)
   (sqrt (* n (/ (+ PI PI) k)))
   (* (sqrt (/ (+ PI PI) (* n k))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 2e-7) {
		tmp = sqrt((n * ((((double) M_PI) + ((double) M_PI)) / 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-7) {
		tmp = Math.sqrt((n * ((Math.PI + Math.PI) / k)));
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (n * k))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 2e-7:
		tmp = math.sqrt((n * ((math.pi + math.pi) / 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-7)
		tmp = sqrt(Float64(n * Float64(Float64(pi + pi) / 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-7)
		tmp = sqrt((n * ((pi + pi) / k)));
	else
		tmp = sqrt(((pi + pi) / (n * k))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 2e-7], N[Sqrt[N[(n * N[(N[(Pi + Pi), $MachinePrecision] / 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 < 1.9999999999999999e-7

   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

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

      1. sqrt-undiv N/A

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

      2. rem-square-sqrt N/A

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

      3. sqrt-undiv N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. sqrt-undiv N/A

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

      8. rem-square-sqrt N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 38.5%

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

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 38.5

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

   6. Applied rewrites 38.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 38.4

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

   8. Applied rewrites 38.4%

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

   if 1.9999999999999999e-7 < 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

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

      1. sqrt-undiv N/A

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

      2. rem-square-sqrt N/A

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

      3. sqrt-undiv N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. sqrt-undiv N/A

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

      8. rem-square-sqrt N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 38.5%

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

   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. count-2-rev N/A

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

      6. lift-+.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.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 48.8

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

   7. Applied rewrites 48.8%

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

Alternative 6: 50.1% 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

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

   1. sqrt-undiv N/A

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

   2. rem-square-sqrt N/A

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

   3. sqrt-undiv N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. sqrt-undiv N/A

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

   7. sqrt-undiv N/A

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

   8. rem-square-sqrt N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 38.5%

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

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 38.5

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

6. Applied rewrites 38.5%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. associate-*r/ N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. sqrt-prod N/A

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

   14. lower-*.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. lower-sqrt.f64 N/A

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

   17. lower-/.f64 50.1

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

8. Applied rewrites 50.1%

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

Alternative 7: 38.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(n * 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

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

   1. sqrt-undiv N/A

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

   2. rem-square-sqrt N/A

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

   3. sqrt-undiv N/A

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

   4. sqrt-undiv N/A

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

   5. lower-sqrt.f64 N/A

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

   6. sqrt-undiv N/A

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

   7. sqrt-undiv N/A

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

   8. rem-square-sqrt N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 38.5%

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

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 38.5

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

6. Applied rewrites 38.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 38.4

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

8. Applied rewrites 38.4%

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

Reproduce

herbie shell --seed 2025127 
(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))))