Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 6.9s

Alternatives: 7

Speedup: 1.2×

• Could not converge on a ground truth

  1. k = 4.243911146424012e+136
  2. n = -9.70775719155593e+209

Specification

Math

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

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.2× speedup

Math

\[\frac{{\left(n \cdot 6.283185307179586\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}{\sqrt{k}} \]

FPCore

(FPCore (k n)
  :precision binary64
  (/ (pow (* n 6.283185307179586) (fma k -0.5 0.5)) (sqrt k)))

C

double code(double k, double n) {
	return pow((n * 6.283185307179586), fma(k, -0.5, 0.5)) / sqrt(k);
}

Julia

function code(k, n)
	return Float64((Float64(n * 6.283185307179586) ^ fma(k, -0.5, 0.5)) / sqrt(k))
end

Wolfram

code[k_, n_] := N[(N[Power[N[(n * 6.283185307179586), $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 99.4%

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

3. Applied rewrites 99.4%

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

4. Evaluated real constant 99.4%

   \[\leadsto \frac{{\left(n \cdot \color{blue}{6.283185307179586}\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}{\sqrt{k}} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;k \leq 0.38:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(n \cdot 6.283185307179586\right)}^{\left(-0.5 \cdot k\right)}}{\sqrt{k}}\\ \end{array} \]

FPCore

(FPCore (k n)
  :precision binary64
  (if (<= k 0.38)
  (* (sqrt (+ n n)) (sqrt (/ PI k)))
  (/ (pow (* n 6.283185307179586) (* -0.5 k)) (sqrt k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 0.38) {
		tmp = sqrt((n + n)) * sqrt((((double) M_PI) / k));
	} else {
		tmp = pow((n * 6.283185307179586), (-0.5 * k)) / sqrt(k);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 0.38) {
		tmp = Math.sqrt((n + n)) * Math.sqrt((Math.PI / k));
	} else {
		tmp = Math.pow((n * 6.283185307179586), (-0.5 * k)) / Math.sqrt(k);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 0.38:
		tmp = math.sqrt((n + n)) * math.sqrt((math.pi / k))
	else:
		tmp = math.pow((n * 6.283185307179586), (-0.5 * k)) / math.sqrt(k)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 0.38)
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(pi / k)));
	else
		tmp = Float64((Float64(n * 6.283185307179586) ^ Float64(-0.5 * k)) / sqrt(k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 0.38)
		tmp = sqrt((n + n)) * sqrt((pi / k));
	else
		tmp = ((n * 6.283185307179586) ^ (-0.5 * k)) / sqrt(k);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 0.38], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(n * 6.283185307179586), $MachinePrecision], N[(-0.5 * k), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.38

   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

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
   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 49.6%

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

   4. Applied rewrites 49.6%

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

      1. lift-/.f64 N/A

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

   6. Applied rewrites 37.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 37.6%

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

   8. Applied rewrites 37.6%

      \[\leadsto \sqrt{n \cdot \frac{\pi + \pi}{k}} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ 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. count-2-rev N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. sqrt-prod N/A

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

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

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

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

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

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

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

      16. lower-/.f64 49.6%

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

   10. Applied rewrites 49.6%

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

   if 0.38 < 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 99.4%

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

   3. Applied rewrites 99.4%

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

   4. Evaluated real constant 99.4%

      \[\leadsto \frac{{\left(n \cdot \color{blue}{6.283185307179586}\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}{\sqrt{k}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{{\left(n \cdot 6.283185307179586\right)}^{\color{blue}{\left(\frac{-1}{2} \cdot k\right)}}}{\sqrt{k}} \]
   6. Step-by-step derivation

      1. lower-*.f64 53.2%

         \[\leadsto \frac{{\left(n \cdot 6.283185307179586\right)}^{\left(-0.5 \cdot \color{blue}{k}\right)}}{\sqrt{k}} \]

   7. Applied rewrites 53.2%

      \[\leadsto \frac{{\left(n \cdot 6.283185307179586\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}}}{\sqrt{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 74.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;n \leq 10^{-12}:\\ \;\;\;\;\frac{n \cdot \sqrt{2 \cdot \frac{k \cdot \pi}{n}}}{k}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{\pi}{k \cdot n}}\\ \end{array} \]

FPCore

(FPCore (k n)
  :precision binary64
  (if (<= n 1e-12)
  (/ (* n (sqrt (* 2.0 (/ (* k PI) n)))) k)
  (* n (sqrt (* 2.0 (/ PI (* k n)))))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 1e-12) {
		tmp = (n * sqrt((2.0 * ((k * ((double) M_PI)) / n)))) / 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 <= 1e-12) {
		tmp = (n * Math.sqrt((2.0 * ((k * Math.PI) / n)))) / k;
	} else {
		tmp = n * Math.sqrt((2.0 * (Math.PI / (k * n))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 1e-12:
		tmp = (n * math.sqrt((2.0 * ((k * math.pi) / n)))) / k
	else:
		tmp = n * math.sqrt((2.0 * (math.pi / (k * n))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 1e-12)
		tmp = Float64(Float64(n * sqrt(Float64(2.0 * Float64(Float64(k * pi) / n)))) / 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 <= 1e-12)
		tmp = (n * sqrt((2.0 * ((k * pi) / n)))) / k;
	else
		tmp = n * sqrt((2.0 * (pi / (k * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 1e-12], N[(N[(n * N[Sqrt[N[(2.0 * N[(N[(k * Pi), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k), $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 < 9.9999999999999998e-13

   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

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
   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 49.6%

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

   4. Applied rewrites 49.6%

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

      1. lift-/.f64 N/A

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

   6. Applied rewrites 37.7%

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

   7. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 38.2%

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

   9. Applied rewrites 38.2%

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

   10. Taylor expanded in n around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 50.2%

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

   12. Applied rewrites 50.2%

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

   if 9.9999999999999998e-13 < 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

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
   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 49.6%

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

   4. Applied rewrites 49.6%

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

      1. lift-/.f64 N/A

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

   6. Applied rewrites 37.7%

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

   7. Taylor expanded in n around inf

      \[\leadsto n \cdot \color{blue}{\sqrt{2 \cdot \frac{\pi}{k \cdot n}}} \]
   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 4: 61.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;n \leq 1.85 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{n \cdot \frac{\pi + \pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \sqrt{2 \cdot \frac{\pi}{k \cdot n}}\\ \end{array} \]

FPCore

(FPCore (k n)
  :precision binary64
  (if (<= n 1.85e-50)
  (sqrt (* n (/ (+ PI PI) k)))
  (* n (sqrt (* 2.0 (/ PI (* k n)))))))

C

double code(double k, double n) {
	double tmp;
	if (n <= 1.85e-50) {
		tmp = sqrt((n * ((((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 <= 1.85e-50) {
		tmp = Math.sqrt((n * ((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 <= 1.85e-50:
		tmp = math.sqrt((n * ((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 <= 1.85e-50)
		tmp = sqrt(Float64(n * 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 <= 1.85e-50)
		tmp = sqrt((n * ((pi + pi) / k)));
	else
		tmp = n * sqrt((2.0 * (pi / (k * n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 1.85e-50], N[Sqrt[N[(n * 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 < 1.85e-50

   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

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
   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 49.6%

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

   4. Applied rewrites 49.6%

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

      1. lift-/.f64 N/A

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

   6. Applied rewrites 37.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 37.6%

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

   8. Applied rewrites 37.6%

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

   if 1.85e-50 < 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

      \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
   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 49.6%

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

   4. Applied rewrites 49.6%

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

      1. lift-/.f64 N/A

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

   6. Applied rewrites 37.7%

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

   7. Taylor expanded in n around inf

      \[\leadsto n \cdot \color{blue}{\sqrt{2 \cdot \frac{\pi}{k \cdot n}}} \]
   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 5: 49.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
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 49.6%

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

4. Applied rewrites 49.6%

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

   1. lift-/.f64 N/A

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

6. Applied rewrites 37.7%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. mult-flip-rev N/A

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

   10. lower-/.f64 37.6%

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

8. Applied rewrites 37.6%

   \[\leadsto \sqrt{n \cdot \frac{\pi + \pi}{k}} \]
9. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ 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. count-2-rev N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. sqrt-prod N/A

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

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

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

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

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

   13. count-2-rev N/A

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

   14. lower-+.f64 N/A

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

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

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

   16. lower-/.f64 49.6%

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

10. Applied rewrites 49.6%

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

Alternative 6: 49.6% accurate, 2.7× speedup

Math

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

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

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
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 49.6%

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

4. Applied rewrites 49.6%

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

   1. lift-/.f64 N/A

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

6. Applied rewrites 37.7%

   \[\leadsto \sqrt{\frac{\left(\pi + \pi\right) \cdot n}{k}} \]
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 49.6%

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

8. Applied rewrites 49.6%

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

Alternative 7: 37.6% accurate, 3.2× speedup

Math

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

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

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\frac{1}{2}}}{\sqrt{k}}} \]
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 49.6%

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

4. Applied rewrites 49.6%

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

   1. lift-/.f64 N/A

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

6. Applied rewrites 37.7%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. mult-flip-rev N/A

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

   10. lower-/.f64 37.6%

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

8. Applied rewrites 37.6%

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

Reproduce

herbie shell --seed 2025212 
(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))))