Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 17.3s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(k, n):
	return 1.0 / (math.sqrt(k) * math.pow((math.pi * (2.0 * n)), ((k * 0.5) - 0.5)))

Julia

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

MATLAB

function tmp = code(k, n)
	tmp = 1.0 / (sqrt(k) * ((pi * (2.0 * n)) ^ ((k * 0.5) - 0.5)));
end

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 99.2%

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

   1. *-commutative 99.2%

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

   2. associate-*r* 99.2%

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

   3. div-sub 99.2%

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

   4. metadata-eval 99.2%

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

   5. sqrt-div 99.1%

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

   6. metadata-eval 99.1%

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

   7. pow1/2 99.1%

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

   8. metadata-eval 99.1%

      \[\leadsto {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)} \cdot \frac{1}{{k}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}} \]

   9. pow-pow 74.5%

      \[\leadsto {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)} \cdot \frac{1}{\color{blue}{{\left({k}^{1.5}\right)}^{0.3333333333333333}}} \]

   10. pow1/3 76.6%

       \[\leadsto {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt[3]{{k}^{1.5}}}} \]

   11. div-inv 76.6%

       \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt[3]{{k}^{1.5}}}} \]

   12. clear-num 76.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{{k}^{1.5}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}}} \]

   13. pow1/3 74.5%

       \[\leadsto \frac{1}{\frac{\color{blue}{{\left({k}^{1.5}\right)}^{0.3333333333333333}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}} \]

   14. pow-pow 99.1%

       \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}} \]

   15. metadata-eval 99.1%

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

   16. pow1/2 99.1%

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

   17. associate-*r* 99.1%

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

   18. *-commutative 99.1%

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

   19. div-inv 99.1%

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

5. Applied egg-rr 99.1%

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

   1. div-inv 99.0%

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

   2. pow-flip 99.4%

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

   3. associate-*r* 99.4%

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

   4. *-commutative 99.4%

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

   5. *-commutative 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

   \[\leadsto \frac{1}{\sqrt{k} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(k \cdot 0.5 - 0.5\right)}} \]
9. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;k \leq 2 \cdot 10^{-21}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{k} \cdot {t\_0}^{\left(1 - k\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* n (* PI 2.0))))
   (if (<= k 2e-21)
     (* (pow k -0.5) (sqrt t_0))
     (sqrt (* (/ 1.0 k) (pow t_0 (- 1.0 k)))))))

C

double code(double k, double n) {
	double t_0 = n * (((double) M_PI) * 2.0);
	double tmp;
	if (k <= 2e-21) {
		tmp = pow(k, -0.5) * sqrt(t_0);
	} else {
		tmp = sqrt(((1.0 / k) * pow(t_0, (1.0 - k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = n * (Math.PI * 2.0);
	double tmp;
	if (k <= 2e-21) {
		tmp = Math.pow(k, -0.5) * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt(((1.0 / k) * Math.pow(t_0, (1.0 - k))));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = n * (math.pi * 2.0)
	tmp = 0
	if k <= 2e-21:
		tmp = math.pow(k, -0.5) * math.sqrt(t_0)
	else:
		tmp = math.sqrt(((1.0 / k) * math.pow(t_0, (1.0 - k))))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(n * Float64(pi * 2.0))
	tmp = 0.0
	if (k <= 2e-21)
		tmp = Float64((k ^ -0.5) * sqrt(t_0));
	else
		tmp = sqrt(Float64(Float64(1.0 / k) * (t_0 ^ Float64(1.0 - k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = n * (pi * 2.0);
	tmp = 0.0;
	if (k <= 2e-21)
		tmp = (k ^ -0.5) * sqrt(t_0);
	else
		tmp = sqrt(((1.0 / k) * (t_0 ^ (1.0 - 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, 2e-21], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(1.0 / k), $MachinePrecision] * N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999982e-21

   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 72.4%

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

      1. associate-/l* 72.4%

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

   5. Simplified 72.4%

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

      1. pow1 72.4%

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

      2. sqrt-unprod 72.7%

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

   7. Applied egg-rr 72.7%

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

      1. unpow1 72.7%

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

      2. associate-*l* 72.7%

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

   9. Simplified 72.7%

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

   10. Taylor expanded in n around 0 72.7%

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

       1. associate-*r/ 72.7%

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

       2. associate-*l* 72.7%

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

       3. *-commutative 72.7%

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

       4. sqrt-undiv 99.5%

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

       5. div-inv 99.4%

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

       6. associate-*r* 99.4%

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

       7. pow1/2 99.4%

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

       8. pow-flip 99.5%

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

       9. metadata-eval 99.5%

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

   12. Applied egg-rr 99.5%

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

   if 1.99999999999999982e-21 < k

   1. Initial program 98.9%

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

      1. associate-*l/ 98.9%

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

      2. *-lft-identity 98.9%

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

      3. associate-*l* 98.9%

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

      4. div-sub 98.9%

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

      5. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

      1. div-inv 98.9%

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

      2. metadata-eval 98.9%

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

      3. div-sub 98.9%

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

      4. sqr-pow 98.9%

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

      5. associate-*l* 99.4%

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

      6. div-inv 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate-/l* 99.4%

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

      9. metadata-eval 99.4%

         \[\leadsto {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\left(1 - k\right) \cdot \color{blue}{0.25}\right)} \cdot \left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot \frac{1}{\sqrt{k}}\right) \]

   6. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\left(1 - k\right) \cdot 0.25\right)} \cdot \left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\left(1 - k\right) \cdot 0.25\right)} \cdot {k}^{-0.5}\right)} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 99.4%

         \[\leadsto \color{blue}{\sqrt{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\left(1 - k\right) \cdot 0.25\right)} \cdot \left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\left(1 - k\right) \cdot 0.25\right)} \cdot {k}^{-0.5}\right)} \cdot \sqrt{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\left(1 - k\right) \cdot 0.25\right)} \cdot \left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\left(1 - k\right) \cdot 0.25\right)} \cdot {k}^{-0.5}\right)}} \]

      2. sqrt-unprod 98.9%

         \[\leadsto \color{blue}{\sqrt{\left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\left(1 - k\right) \cdot 0.25\right)} \cdot \left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\left(1 - k\right) \cdot 0.25\right)} \cdot {k}^{-0.5}\right)\right) \cdot \left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\left(1 - k\right) \cdot 0.25\right)} \cdot \left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\left(1 - k\right) \cdot 0.25\right)} \cdot {k}^{-0.5}\right)\right)}} \]

      3. pow2 98.9%

         \[\leadsto \sqrt{\color{blue}{{\left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\left(1 - k\right) \cdot 0.25\right)} \cdot \left({\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\left(1 - k\right) \cdot 0.25\right)} \cdot {k}^{-0.5}\right)\right)}^{2}}} \]

   8. Applied egg-rr 98.9%

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

      1. unpow2 98.9%

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

      2. unpow1/2 98.9%

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

      3. *-commutative 98.9%

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

      4. unpow1/2 98.9%

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

      5. *-commutative 98.9%

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

      6. swap-sqr 98.9%

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

      7. pow-sqr 98.9%

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

      8. metadata-eval 98.9%

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

      9. unpow-1 98.9%

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

      10. rem-square-sqrt 98.9%

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

      11. *-commutative 98.9%

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

      12. *-commutative 98.9%

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

      13. associate-*l* 98.9%

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

   10. Simplified 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-21}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{k} \cdot {\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(k, n)
	tmp = sqrt((1.0 / k)) * ((n * (pi * 2.0)) ^ ((1.0 - k) / 2.0));
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[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 99.2%

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

4. Final simplification 99.2%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 + k \cdot -0.5\right)} \cdot {k}^{-0.5} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (pow (* PI (* 2.0 n)) (+ 0.5 (* k -0.5))) (pow k -0.5)))

C

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

Java

public static double code(double k, double n) {
	return Math.pow((Math.PI * (2.0 * n)), (0.5 + (k * -0.5))) * Math.pow(k, -0.5);
}

Python

def code(k, n):
	return math.pow((math.pi * (2.0 * n)), (0.5 + (k * -0.5))) * math.pow(k, -0.5)

Julia

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

MATLAB

function tmp = code(k, n)
	tmp = ((pi * (2.0 * n)) ^ (0.5 + (k * -0.5))) * (k ^ -0.5);
end

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. associate-*l/ 99.2%

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

   2. *-lft-identity 99.2%

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

   3. associate-*l* 99.2%

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

   4. div-sub 99.2%

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

   5. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. pow-div 99.0%

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

   2. pow1/2 99.0%

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

   3. associate-/l/ 99.2%

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

   4. *-un-lft-identity 99.2%

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

   5. frac-times 99.0%

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

   6. pow1/2 99.0%

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

   7. pow-flip 99.0%

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

   8. metadata-eval 99.0%

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

   9. pow1/2 99.0%

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

   10. pow-div 99.2%

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

   11. div-inv 99.2%

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

   12. metadata-eval 99.2%

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

6. Applied egg-rr 99.2%

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

   1. *-commutative 99.2%

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

   2. *-commutative 99.2%

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

   3. *-commutative 99.2%

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

   4. cancel-sign-sub-inv 99.2%

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

   5. metadata-eval 99.2%

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

   6. associate-*r* 99.2%

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

   7. *-commutative 99.2%

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

8. Simplified 99.2%

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

9. Final simplification 99.2%

   \[\leadsto {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 + k \cdot -0.5\right)} \cdot {k}^{-0.5} \]
10. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(k, n):
	return math.pow((2.0 * (math.pi * n)), (0.5 - (k / 2.0))) / math.sqrt(k)

Julia

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

MATLAB

function tmp = code(k, n)
	tmp = ((2.0 * (pi * n)) ^ (0.5 - (k / 2.0))) / sqrt(k);
end

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. associate-*l/ 99.2%

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

   2. *-lft-identity 99.2%

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

   3. associate-*l* 99.2%

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

   4. div-sub 99.2%

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

   5. metadata-eval 99.2%

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

3. Simplified 99.2%

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

Alternative 6: 49.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double k, double n) {
	return Math.pow(k, -0.5) * Math.sqrt((n * (Math.PI * 2.0)));
}

Python

def code(k, n):
	return math.pow(k, -0.5) * math.sqrt((n * (math.pi * 2.0)))

Julia

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

MATLAB

function tmp = code(k, n)
	tmp = (k ^ -0.5) * sqrt((n * (pi * 2.0)));
end

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 38.6%

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

   1. associate-/l* 38.6%

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

5. Simplified 38.6%

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

   1. pow1 38.6%

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

   2. sqrt-unprod 38.8%

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

7. Applied egg-rr 38.8%

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

   1. unpow1 38.8%

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

   2. associate-*l* 38.8%

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

9. Simplified 38.8%

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

10. Taylor expanded in n around 0 38.8%

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

    1. associate-*r/ 38.8%

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

    2. associate-*l* 38.8%

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

    3. *-commutative 38.8%

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

    4. sqrt-undiv 51.6%

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

    5. div-inv 51.5%

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

    6. associate-*r* 51.5%

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

    7. pow1/2 51.5%

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

    8. pow-flip 51.6%

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

    9. metadata-eval 51.6%

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

12. Applied egg-rr 51.6%

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

13. Final simplification 51.6%

    \[\leadsto {k}^{-0.5} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)} \]
14. Add Preprocessing

Alternative 7: 49.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 38.6%

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

   1. associate-/l* 38.6%

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

5. Simplified 38.6%

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

   1. associate-*r/ 38.6%

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

   2. *-commutative 38.6%

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

7. Applied egg-rr 38.6%

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

   1. *-commutative 38.6%

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

   2. sqrt-unprod 38.8%

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

   3. metadata-eval 38.8%

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

   4. times-frac 38.8%

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

   5. *-commutative 38.8%

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

   6. associate-*r* 38.8%

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

   7. *-commutative 38.8%

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

   8. associate-*r* 38.8%

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

   9. *-un-lft-identity 38.8%

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

   10. sqrt-div 51.6%

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

   11. associate-*r* 51.6%

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

   12. *-commutative 51.6%

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

   13. *-commutative 51.6%

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

9. Applied egg-rr 51.6%

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

Alternative 8: 49.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 38.6%

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

   1. associate-/l* 38.6%

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

5. Simplified 38.6%

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

   1. pow1 38.6%

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

   2. sqrt-unprod 38.8%

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

7. Applied egg-rr 38.8%

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

   1. unpow1 38.8%

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

   2. associate-*l* 38.8%

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

9. Simplified 38.8%

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

    1. *-commutative 38.8%

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

    2. sqrt-prod 51.6%

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

    3. *-commutative 51.6%

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

11. Applied egg-rr 51.6%

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

Alternative 9: 38.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 38.6%

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

   1. associate-/l* 38.6%

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

5. Simplified 38.6%

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

   1. pow1 38.6%

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

   2. sqrt-unprod 38.8%

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

7. Applied egg-rr 38.8%

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

   1. unpow1 38.8%

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

   2. associate-*l* 38.8%

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

9. Simplified 38.8%

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

10. Taylor expanded in n around 0 38.8%

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

    1. clear-num 38.8%

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

    2. *-commutative 38.8%

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

    3. associate-/l/ 38.8%

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

    4. un-div-inv 38.8%

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

    5. sqrt-undiv 38.8%

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

    6. clear-num 38.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\frac{\frac{k}{n}}{\pi}}}{\sqrt{2}}}} \]

    7. inv-pow 38.8%

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

    8. sqrt-undiv 38.9%

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

    9. sqrt-pow2 39.0%

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

    10. div-inv 39.0%

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

    11. associate-/l/ 39.0%

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

    12. metadata-eval 39.0%

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

    13. metadata-eval 39.0%

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

12. Applied egg-rr 39.0%

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

    1. associate-*l/ 39.0%

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

    2. *-commutative 39.0%

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

    3. times-frac 39.0%

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

14. Simplified 39.0%

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

Alternative 10: 37.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 38.6%

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

   1. associate-/l* 38.6%

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

5. Simplified 38.6%

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

   1. pow1 38.6%

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

   2. sqrt-unprod 38.8%

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

7. Applied egg-rr 38.8%

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

   1. unpow1 38.8%

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

   2. associate-*r/ 38.8%

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

   3. *-commutative 38.8%

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

   4. associate-/l* 38.8%

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

   5. associate-*l* 38.8%

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

9. Simplified 38.8%

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

10. Final simplification 38.8%

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

Alternative 11: 37.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 38.6%

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

   1. associate-/l* 38.6%

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

5. Simplified 38.6%

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

   1. pow1 38.6%

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

   2. sqrt-unprod 38.8%

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

7. Applied egg-rr 38.8%

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

   1. unpow1 38.8%

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

   2. associate-*l* 38.8%

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

9. Simplified 38.8%

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

10. Final simplification 38.8%

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

Alternative 12: 37.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0 38.6%

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

   1. associate-/l* 38.6%

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

5. Simplified 38.6%

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

   1. pow1 38.6%

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

   2. sqrt-unprod 38.8%

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

7. Applied egg-rr 38.8%

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

   1. unpow1 38.8%

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

   2. associate-*l* 38.8%

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

9. Simplified 38.8%

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

10. Taylor expanded in n around 0 38.8%

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

    1. clear-num 38.8%

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

    2. *-commutative 38.8%

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

    3. associate-/l/ 38.8%

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

    4. clear-num 38.8%

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

    5. div-inv 38.8%

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

    6. clear-num 38.8%

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

12. Applied egg-rr 38.8%

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

    1. associate-*r/ 38.8%

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

    2. associate-*l/ 38.8%

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

    3. associate-/r/ 38.8%

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

14. Simplified 38.8%

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

Reproduce

herbie shell --seed 2024112 
(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))))