Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 11.2s

Alternatives: 9

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 1.0696938381760762e+161
  2. n = -1.6360501313250228e+90

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, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}}{\sqrt{k}} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (/ (pow (* 2.0 (* PI n)) (fma -0.5 k 0.5)) (sqrt k)))

C

double code(double k, double n) {
	return pow((2.0 * (((double) M_PI) * n)), fma(-0.5, k, 0.5)) / sqrt(k);
}

Julia

function code(k, n)
	return Float64((Float64(2.0 * Float64(pi * n)) ^ fma(-0.5, k, 0.5)) / sqrt(k))
end

Wolfram

code[k_, n_] := N[(N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(-0.5 * k + 0.5), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in n around -inf

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. exp-prod N/A

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

   4. *-commutative N/A

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

   5. lower-pow.f64 N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. exp-diff N/A

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

   9. lower-/.f64 N/A

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

   10. rem-exp-log N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-PI.f64 N/A

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

   14. rem-exp-log N/A

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

   15. lower-/.f64 N/A

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

5. Applied rewrites 99.1%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{{\left(\frac{\pi \cdot -2}{\frac{-1}{n}}\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}} \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. associate-*l/ N/A

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

   9. *-lft-identity N/A

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

   10. lower-/.f64 99.2

       \[\leadsto \color{blue}{\frac{{\left(\frac{\pi \cdot -2}{\frac{-1}{n}}\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}}{\sqrt{k}}} \]

7. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}}{\sqrt{k}}} \]
8. Add Preprocessing

Alternative 2: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(-0.5 \cdot k\right)}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.0)
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
   (/ (pow (* 2.0 (* PI n)) (* -0.5 k)) (sqrt k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.0) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = pow((2.0 * (((double) M_PI) * n)), (-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((2.0 * (Math.PI / k)));
	} else {
		tmp = Math.pow((2.0 * (Math.PI * n)), (-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((2.0 * (math.pi / k)))
	else:
		tmp = math.pow((2.0 * (math.pi * n)), (-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(2.0 * Float64(pi / k))));
	else
		tmp = Float64((Float64(2.0 * Float64(pi * n)) ^ 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((2.0 * (pi / k)));
	else
		tmp = ((2.0 * (pi * n)) ^ (-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[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 * N[(Pi * n), $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 98.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 82.0

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

   5. Applied rewrites 82.0%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 98.0

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

   7. Applied rewrites 98.0%

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

   if 1 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around inf

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

      1. lower-*.f64 99.2

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. lower-/.f64 99.2

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 99.2

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

   7. Applied rewrites 99.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(-0.5 \cdot k\right)}}{\sqrt{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.5:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(k \cdot k\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 0.5)
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
   (sqrt (pow (* k k) -0.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 0.5) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = sqrt(pow((k * k), -0.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 0.5) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	} else {
		tmp = Math.sqrt(Math.pow((k * k), -0.5));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 0.5:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (math.pi / k)))
	else:
		tmp = math.sqrt(math.pow((k * k), -0.5))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 0.5)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = sqrt((Float64(k * k) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 0.5)
		tmp = sqrt(n) * sqrt((2.0 * (pi / k)));
	else
		tmp = sqrt(((k * k) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 0.5], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Power[N[(k * k), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.5

   1. Initial program 98.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 82.0

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

   5. Applied rewrites 82.0%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 98.0

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

   7. Applied rewrites 98.0%

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

   if 0.5 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around inf

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

      1. lower-*.f64 99.2

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]

      2. lower-/.f64 3.2

         \[\leadsto \sqrt{\color{blue}{\frac{1}{k}}} \]

   8. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   9. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto \sqrt{\color{blue}{{k}^{-1}}} \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

         \[\leadsto \sqrt{\color{blue}{{k}^{\frac{-1}{2}} \cdot {k}^{\frac{-1}{2}}}} \]

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 28.0

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

   10. Applied rewrites 28.0%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.5:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(k \cdot k\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.5:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k} \cdot \frac{1}{k}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 0.5)
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
   (sqrt (sqrt (* (/ 1.0 k) (/ 1.0 k))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 0.5) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = sqrt(sqrt(((1.0 / k) * (1.0 / k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 0.5) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	} else {
		tmp = Math.sqrt(Math.sqrt(((1.0 / k) * (1.0 / k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 0.5:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (math.pi / k)))
	else:
		tmp = math.sqrt(math.sqrt(((1.0 / k) * (1.0 / k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 0.5)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = sqrt(sqrt(Float64(Float64(1.0 / k) * Float64(1.0 / k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 0.5)
		tmp = sqrt(n) * sqrt((2.0 * (pi / k)));
	else
		tmp = sqrt(sqrt(((1.0 / k) * (1.0 / k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 0.5], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Sqrt[N[(N[(1.0 / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.5

   1. Initial program 98.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 82.0

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

   5. Applied rewrites 82.0%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*l* N/A

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

      9. sqrt-prod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 98.0

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

   7. Applied rewrites 98.0%

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

   if 0.5 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around inf

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

      1. lower-*.f64 99.2

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]

      2. lower-/.f64 3.2

         \[\leadsto \sqrt{\color{blue}{\frac{1}{k}}} \]

   8. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 3.2

         \[\leadsto \sqrt{\color{blue}{\frac{1}{k}}} \]

      2. rem-square-sqrt N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k}} \cdot \sqrt{\frac{1}{k}}}} \]

      3. sqrt-unprod N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k} \cdot \frac{1}{k}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k} \cdot \frac{1}{k}}}} \]

      5. lower-*.f64 25.7

         \[\leadsto \sqrt{\sqrt{\color{blue}{\frac{1}{k} \cdot \frac{1}{k}}}} \]

   10. Applied rewrites 25.7%

       \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{k} \cdot \frac{1}{k}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.5:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k} \cdot \frac{1}{k}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 42.6

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

5. Applied rewrites 42.6%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sqrt-unprod N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. sqrt-prod N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 50.7

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

7. Applied rewrites 50.7%

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

8. Final simplification 50.7%

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

Alternative 6: 38.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{\pi \cdot n}{k \cdot 0.5}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (/ (* PI n) (* k 0.5))))

C

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

Java

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

Python

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

Julia

function code(k, n)
	return sqrt(Float64(Float64(pi * n) / Float64(k * 0.5)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 42.6

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

5. Applied rewrites 42.6%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sqrt-unprod N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. sqrt-div N/A

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

7. Applied rewrites 50.3%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. sqrt-undiv N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 43.1

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

9. Applied rewrites 43.1%

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

    1. lift-PI.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. div-inv N/A

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

    5. div-inv N/A

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

    6. lift-*.f64 N/A

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

    7. associate-/r* N/A

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

    8. div-inv N/A

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

    9. metadata-eval N/A

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

    10. lift-*.f64 N/A

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

    11. sqrt-div N/A

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

    12. lift-sqrt.f64 N/A

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

    13. lift-*.f64 N/A

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

    14. sqrt-unprod N/A

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

    15. lift-sqrt.f64 N/A

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

    16. lift-sqrt.f64 N/A

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

    17. associate-/l/ N/A

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

11. Applied rewrites 42.7%

    \[\leadsto \color{blue}{\sqrt{\frac{\pi \cdot n}{k \cdot 0.5}}} \]
12. Add Preprocessing

Alternative 7: 38.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 42.6

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

5. Applied rewrites 42.6%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sqrt-unprod N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*l/ N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-/.f64 42.7

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

7. Applied rewrites 42.7%

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

Alternative 8: 38.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 42.6

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

5. Applied rewrites 42.6%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sqrt-unprod N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*l/ N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. sqrt-div N/A

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

7. Applied rewrites 50.3%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. sqrt-undiv N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 43.1

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

9. Applied rewrites 43.1%

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

    1. lift-PI.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. div-inv N/A

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

    5. div-inv N/A

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

    6. lift-*.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. *-commutative N/A

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

    9. associate-*r* N/A

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

    10. lift-*.f64 N/A

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

    11. sqrt-undiv N/A

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

    12. lift-sqrt.f64 N/A

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

    13. sqrt-unprod N/A

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

    14. lift-sqrt.f64 N/A

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

    15. lift-sqrt.f64 N/A

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

    16. lift-*.f64 N/A

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

11. Applied rewrites 42.7%

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

Alternative 9: 5.1% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{k}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (/ 1.0 k)))

C

double code(double k, double n) {
	return sqrt((1.0 / k));
}

Fortran

real(8) function code(k, n)
    real(8), intent (in) :: k
    real(8), intent (in) :: n
    code = sqrt((1.0d0 / k))
end function

Java

public static double code(double k, double n) {
	return Math.sqrt((1.0 / k));
}

Python

def code(k, n):
	return math.sqrt((1.0 / k))

Julia

function code(k, n)
	return sqrt(Float64(1.0 / k))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt((1.0 / k));
end

Wolfram

code[k_, n_] := N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around inf

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

   1. lower-*.f64 53.0

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

5. Applied rewrites 53.0%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \]

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
7. Step-by-step derivation

   1. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]

   2. lower-/.f64 5.4

      \[\leadsto \sqrt{\color{blue}{\frac{1}{k}}} \]

8. Applied rewrites 5.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{k}}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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))))