Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.5%

Time: 11.2s

Alternatives: 11

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 4.3149894729957014e+182
  2. n = -1.0938641443995715e+21

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: 99.5% 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

\[\begin{array}{l} \\ \sqrt{\frac{1}{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)} \end{array} \]

FPCore

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

C

double code(double k, double n) {
	return sqrt((1.0 / k)) * pow((n * (((double) M_PI) * 2.0)), fma(-0.5, k, 0.5));
}

Julia

function code(k, n)
	return Float64(sqrt(Float64(1.0 / k)) * (Float64(n * Float64(pi * 2.0)) ^ fma(-0.5, k, 0.5)))
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. exp-prod N/A

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

   7. *-commutative N/A

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

   8. lower-pow.f64 N/A

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

   9. rem-exp-log N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. rem-exp-log N/A

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

   14. lower-*.f64 N/A

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

   15. rem-exp-log N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-PI.f64 N/A

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

   19. sub-neg N/A

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

   20. mul-1-neg N/A

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

5. Simplified 99.6%

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

Alternative 2: 69.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {t\_0}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;{\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}^{-0.125}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{k}} \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* n (* PI 2.0))))
   (if (<= (* (/ 1.0 (sqrt k)) (pow t_0 (/ (- 1.0 k) 2.0))) 0.0)
     (pow (* k (* k (* k k))) -0.125)
     (* (sqrt (/ 1.0 k)) (sqrt t_0)))))

C

double code(double k, double n) {
	double t_0 = n * (((double) M_PI) * 2.0);
	double tmp;
	if (((1.0 / sqrt(k)) * pow(t_0, ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = pow((k * (k * (k * k))), -0.125);
	} else {
		tmp = sqrt((1.0 / k)) * sqrt(t_0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = n * (Math.PI * 2.0);
	double tmp;
	if (((1.0 / Math.sqrt(k)) * Math.pow(t_0, ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = Math.pow((k * (k * (k * k))), -0.125);
	} else {
		tmp = Math.sqrt((1.0 / k)) * Math.sqrt(t_0);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = n * (math.pi * 2.0)
	tmp = 0
	if ((1.0 / math.sqrt(k)) * math.pow(t_0, ((1.0 - k) / 2.0))) <= 0.0:
		tmp = math.pow((k * (k * (k * k))), -0.125)
	else:
		tmp = math.sqrt((1.0 / k)) * math.sqrt(t_0)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(n * Float64(pi * 2.0))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(k)) * (t_0 ^ Float64(Float64(1.0 - k) / 2.0))) <= 0.0)
		tmp = Float64(k * Float64(k * Float64(k * k))) ^ -0.125;
	else
		tmp = Float64(sqrt(Float64(1.0 / k)) * sqrt(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = n * (pi * 2.0);
	tmp = 0.0;
	if (((1.0 / sqrt(k)) * (t_0 ^ ((1.0 - k) / 2.0))) <= 0.0)
		tmp = (k * (k * (k * k))) ^ -0.125;
	else
		tmp = sqrt((1.0 / k)) * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[Power[N[(k * N[(k * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.125], $MachinePrecision], N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   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 98.3

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

   5. Simplified 98.3%

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

   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 3.8

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

   8. Simplified 3.8%

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lower-/.f64 3.8

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

   10. Applied egg-rr 3.8%

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

       1. pow1/2 N/A

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

       2. pow-flip N/A

          \[\leadsto \color{blue}{{k}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

       3. metadata-eval N/A

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

       4. sqr-pow N/A

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

       5. unpow-prod-down N/A

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

       6. lift-*.f64 N/A

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

       7. metadata-eval N/A

          \[\leadsto {\left(k \cdot k\right)}^{\color{blue}{\frac{-1}{4}}} \]

       8. metadata-eval N/A

          \[\leadsto {\left(k \cdot k\right)}^{\color{blue}{\left(2 \cdot \frac{-1}{8}\right)}} \]

       9. metadata-eval N/A

          \[\leadsto {\left(k \cdot k\right)}^{\left(2 \cdot \color{blue}{{\frac{-1}{2}}^{3}}\right)} \]

       10. pow-sqr N/A

           \[\leadsto \color{blue}{{\left(k \cdot k\right)}^{\left({\frac{-1}{2}}^{3}\right)} \cdot {\left(k \cdot k\right)}^{\left({\frac{-1}{2}}^{3}\right)}} \]

       11. pow-prod-down N/A

           \[\leadsto \color{blue}{{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}^{\left({\frac{-1}{2}}^{3}\right)}} \]

       12. lower-pow.f64 N/A

           \[\leadsto \color{blue}{{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}^{\left({\frac{-1}{2}}^{3}\right)}} \]

       13. lift-*.f64 N/A

           \[\leadsto {\left(\color{blue}{\left(k \cdot k\right)} \cdot \left(k \cdot k\right)\right)}^{\left({\frac{-1}{2}}^{3}\right)} \]

       14. associate-*l* N/A

           \[\leadsto {\color{blue}{\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}}^{\left({\frac{-1}{2}}^{3}\right)} \]

       15. lift-*.f64 N/A

           \[\leadsto {\left(k \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)}^{\left({\frac{-1}{2}}^{3}\right)} \]

       16. cube-mult N/A

           \[\leadsto {\left(k \cdot \color{blue}{{k}^{3}}\right)}^{\left({\frac{-1}{2}}^{3}\right)} \]

       17. lower-*.f64 N/A

           \[\leadsto {\color{blue}{\left(k \cdot {k}^{3}\right)}}^{\left({\frac{-1}{2}}^{3}\right)} \]

       18. cube-mult N/A

           \[\leadsto {\left(k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}\right)}^{\left({\frac{-1}{2}}^{3}\right)} \]

       19. lift-*.f64 N/A

           \[\leadsto {\left(k \cdot \left(k \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)}^{\left({\frac{-1}{2}}^{3}\right)} \]

       20. lower-*.f64 N/A

           \[\leadsto {\left(k \cdot \color{blue}{\left(k \cdot \left(k \cdot k\right)\right)}\right)}^{\left({\frac{-1}{2}}^{3}\right)} \]

       21. metadata-eval 73.3

           \[\leadsto {\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}^{\color{blue}{-0.125}} \]

   12. Applied egg-rr 73.3%

       \[\leadsto \color{blue}{{\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}^{-0.125}} \]

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

   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. 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 50.1

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

   5. Simplified 50.1%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   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. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sqrt-div N/A

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

      14. unpow1/2 N/A

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

      15. lift-pow.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. *-lft-identity N/A

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

      18. associate-*l/ N/A

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

   7. Applied egg-rr 65.8%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;{\left(k \cdot \left(k \cdot \left(k \cdot k\right)\right)\right)}^{-0.125}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{k}} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   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 98.3

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

   5. Simplified 98.3%

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

   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 3.8

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

   8. Simplified 3.8%

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

      1. lift-/.f64 3.8

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

      2. rem-square-sqrt N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 3.8

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

   10. Applied egg-rr 3.8%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 3.8

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

       5. rem-square-sqrt N/A

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

       6. sqrt-unprod N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. rem-square-sqrt N/A

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

       13. lift-/.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lift-sqrt.f64 N/A

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

       16. lift-sqrt.f64 N/A

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

       17. rem-square-sqrt N/A

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

       18. frac-times N/A

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

       19. metadata-eval N/A

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

       20. lift-*.f64 N/A

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

       21. lower-/.f64 45.1

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

   12. Applied egg-rr 45.1%

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

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

   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. 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 50.1

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

   5. Simplified 50.1%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   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. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sqrt-div N/A

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

      14. unpow1/2 N/A

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

      15. lift-pow.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. *-lft-identity N/A

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

      18. associate-*l/ N/A

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

   7. Applied egg-rr 65.8%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{k}} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{k}} \cdot \sqrt{n \cdot \pi}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* (/ 1.0 (sqrt k)) (pow (* n (* PI 2.0)) (/ (- 1.0 k) 2.0))) 0.0)
   (sqrt (sqrt (/ 1.0 (* k k))))
   (* (sqrt (/ 2.0 k)) (sqrt (* n PI)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[Sqrt[N[Sqrt[N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(2.0 / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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 98.3

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

   5. Simplified 98.3%

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

   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 3.8

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

   8. Simplified 3.8%

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

      1. lift-/.f64 3.8

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

      2. rem-square-sqrt N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 3.8

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

   10. Applied egg-rr 3.8%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 3.8

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

       5. rem-square-sqrt N/A

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

       6. sqrt-unprod N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. rem-square-sqrt N/A

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

       13. lift-/.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lift-sqrt.f64 N/A

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

       16. lift-sqrt.f64 N/A

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

       17. rem-square-sqrt N/A

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

       18. frac-times N/A

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

       19. metadata-eval N/A

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

       20. lift-*.f64 N/A

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

       21. lower-/.f64 45.1

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

   12. Applied egg-rr 45.1%

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

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

   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. 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 50.1

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

   5. Simplified 50.1%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   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. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sqrt-div N/A

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

      14. unpow1/2 N/A

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

      15. lift-pow.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. un-div-inv N/A

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

      18. lift-/.f64 N/A

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

   7. Applied egg-rr 49.9%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. div-inv N/A

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

      15. lift-/.f64 N/A

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

      16. associate-/r/ N/A

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

      17. sqrt-prod N/A

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

      18. lower-*.f64 N/A

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

   9. Applied egg-rr 65.8%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{k}} \cdot \sqrt{n \cdot \pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* (/ 1.0 (sqrt k)) (pow (* n (* PI 2.0)) (/ (- 1.0 k) 2.0))) 0.0)
   (sqrt (sqrt (/ 1.0 (* k k))))
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[Sqrt[N[Sqrt[N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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 98.3

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

   5. Simplified 98.3%

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

   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 3.8

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

   8. Simplified 3.8%

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

      1. lift-/.f64 3.8

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

      2. rem-square-sqrt N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 3.8

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

   10. Applied egg-rr 3.8%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 3.8

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

       5. rem-square-sqrt N/A

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

       6. sqrt-unprod N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. rem-square-sqrt N/A

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

       13. lift-/.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lift-sqrt.f64 N/A

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

       16. lift-sqrt.f64 N/A

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

       17. rem-square-sqrt N/A

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

       18. frac-times N/A

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

       19. metadata-eval N/A

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

       20. lift-*.f64 N/A

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

       21. lower-/.f64 45.1

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

   12. Applied egg-rr 45.1%

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

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

   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. 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 50.1

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

   5. Simplified 50.1%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   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. unpow1/2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 65.8

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

   7. Applied egg-rr 65.8%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;k \leq 1:\\ \;\;\;\;\sqrt{\frac{1}{k}} \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_0}^{\left(k \cdot -0.5\right)}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* n (* PI 2.0))))
   (if (<= k 1.0)
     (* (sqrt (/ 1.0 k)) (sqrt t_0))
     (/ (pow t_0 (* k -0.5)) (sqrt k)))))

C

double code(double k, double n) {
	double t_0 = n * (((double) M_PI) * 2.0);
	double tmp;
	if (k <= 1.0) {
		tmp = sqrt((1.0 / k)) * sqrt(t_0);
	} else {
		tmp = pow(t_0, (k * -0.5)) / sqrt(k);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = n * (Math.PI * 2.0);
	double tmp;
	if (k <= 1.0) {
		tmp = Math.sqrt((1.0 / k)) * Math.sqrt(t_0);
	} else {
		tmp = Math.pow(t_0, (k * -0.5)) / Math.sqrt(k);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = n * (math.pi * 2.0)
	tmp = 0
	if k <= 1.0:
		tmp = math.sqrt((1.0 / k)) * math.sqrt(t_0)
	else:
		tmp = math.pow(t_0, (k * -0.5)) / math.sqrt(k)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(n * Float64(pi * 2.0))
	tmp = 0.0
	if (k <= 1.0)
		tmp = Float64(sqrt(Float64(1.0 / k)) * sqrt(t_0));
	else
		tmp = Float64((t_0 ^ Float64(k * -0.5)) / sqrt(k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = n * (pi * 2.0);
	tmp = 0.0;
	if (k <= 1.0)
		tmp = sqrt((1.0 / k)) * sqrt(t_0);
	else
		tmp = (t_0 ^ (k * -0.5)) / sqrt(k);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.0], N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[Power[t$95$0, N[(k * -0.5), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1

   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 73.1

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

   5. Simplified 73.1%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   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. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sqrt-div N/A

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

      14. unpow1/2 N/A

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

      15. lift-pow.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. *-lft-identity N/A

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

      18. associate-*l/ N/A

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

   7. Applied egg-rr 96.2%

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

   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. Simplified 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. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 99.2

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 99.2

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

   7. Applied egg-rr 99.2%

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

Alternative 7: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. exp-prod N/A

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

   7. *-commutative N/A

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

   8. lower-pow.f64 N/A

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

   9. rem-exp-log N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. rem-exp-log N/A

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

   14. lower-*.f64 N/A

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

   15. rem-exp-log N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-PI.f64 N/A

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

   19. sub-neg N/A

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

   20. mul-1-neg N/A

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

5. Simplified 99.6%

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. associate-*l/ N/A

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

   10. *-lft-identity N/A

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

   11. lower-/.f64 99.6

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

   12. lift-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 99.6

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

7. Applied egg-rr 99.6%

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

Alternative 8: 51.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.49:\\ \;\;\;\;\sqrt{\frac{n \cdot \left(\pi \cdot 2\right)}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{k \cdot k}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 0.49) (sqrt (/ (* n (* PI 2.0)) k)) (sqrt (sqrt (/ 1.0 (* k k))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 0.49) {
		tmp = sqrt(((n * (((double) M_PI) * 2.0)) / k));
	} else {
		tmp = sqrt(sqrt((1.0 / (k * k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 0.49) {
		tmp = Math.sqrt(((n * (Math.PI * 2.0)) / k));
	} else {
		tmp = Math.sqrt(Math.sqrt((1.0 / (k * k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 0.49:
		tmp = math.sqrt(((n * (math.pi * 2.0)) / k))
	else:
		tmp = math.sqrt(math.sqrt((1.0 / (k * k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 0.49)
		tmp = sqrt(Float64(Float64(n * Float64(pi * 2.0)) / k));
	else
		tmp = sqrt(sqrt(Float64(1.0 / Float64(k * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 0.49)
		tmp = sqrt(((n * (pi * 2.0)) / k));
	else
		tmp = sqrt(sqrt((1.0 / (k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 0.49], N[Sqrt[N[(N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Sqrt[N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.48999999999999999

   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 73.6

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

   5. Simplified 73.6%

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
   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. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 73.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 73.7

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 73.7

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

   7. Applied egg-rr 73.7%

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

   if 0.48999999999999999 < 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 98.4

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

   5. Simplified 98.4%

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

   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. Simplified 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. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. metadata-eval N/A

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

      10. lift-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 3.2

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

   10. Applied egg-rr 3.2%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 3.2

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

       5. rem-square-sqrt N/A

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

       6. sqrt-unprod N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. rem-square-sqrt N/A

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

       13. lift-/.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lift-sqrt.f64 N/A

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

       16. lift-sqrt.f64 N/A

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

       17. rem-square-sqrt N/A

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

       18. frac-times N/A

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

       19. metadata-eval N/A

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

       20. lift-*.f64 N/A

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

       21. lower-/.f64 22.4

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

   12. Applied egg-rr 22.4%

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

Alternative 9: 38.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 39.5

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

5. Simplified 39.5%

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \pi}{k}} \cdot \sqrt{2}} \]
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. associate-*r* N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lower-/.f64 39.6

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 39.6

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lift-*.f64 39.6

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

7. Applied egg-rr 39.6%

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

Alternative 10: 38.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. exp-prod N/A

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

   7. *-commutative N/A

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

   8. lower-pow.f64 N/A

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

   9. rem-exp-log N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. rem-exp-log N/A

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

   14. lower-*.f64 N/A

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

   15. rem-exp-log N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-PI.f64 N/A

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

   19. sub-neg N/A

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

   20. mul-1-neg N/A

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

5. Simplified 99.6%

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. associate-*l/ N/A

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

   10. *-lft-identity N/A

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

   11. lower-/.f64 99.6

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

   12. lift-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 99.6

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

7. Applied egg-rr 99.6%

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

8. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 51.5

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

10. Simplified 51.5%

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

    1. *-rgt-identity N/A

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

    2. lift-PI.f64 N/A

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

    3. sqrt-prod N/A

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

    4. *-rgt-identity N/A

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

    5. sqrt-prod N/A

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

    6. lift-*.f64 N/A

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

    7. sqrt-unprod N/A

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

    8. sqrt-undiv N/A

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

    9. lower-sqrt.f64 N/A

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

    10. *-commutative N/A

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

    11. lift-*.f64 N/A

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

    12. associate-*r* N/A

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

    13. lift-*.f64 N/A

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

    14. associate-/l* N/A

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

    15. lower-*.f64 N/A

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

    16. lower-/.f64 39.6

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

12. Applied egg-rr 39.6%

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

Alternative 11: 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 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 51.1

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

5. Simplified 51.1%

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

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 5.3

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

8. Simplified 5.3%

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

Reproduce

herbie shell --seed 2024207 
(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))))