Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.5%

Time: 13.8s

Alternatives: 8

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: 99.5% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot n\right)\\ \mathbf{if}\;k \leq 3.2 \cdot 10^{-52}:\\ \;\;\;\;{k}^{-0.5} \cdot \frac{1}{{t_0}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* 2.0 n))))
   (if (<= k 3.2e-52)
     (* (pow k -0.5) (/ 1.0 (pow t_0 -0.5)))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (2.0 * n);
	double tmp;
	if (k <= 3.2e-52) {
		tmp = pow(k, -0.5) * (1.0 / pow(t_0, -0.5));
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (2.0 * n);
	double tmp;
	if (k <= 3.2e-52) {
		tmp = Math.pow(k, -0.5) * (1.0 / Math.pow(t_0, -0.5));
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.pi * (2.0 * n)
	tmp = 0
	if k <= 3.2e-52:
		tmp = math.pow(k, -0.5) * (1.0 / math.pow(t_0, -0.5))
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(2.0 * n))
	tmp = 0.0
	if (k <= 3.2e-52)
		tmp = Float64((k ^ -0.5) * Float64(1.0 / (t_0 ^ -0.5)));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = pi * (2.0 * n);
	tmp = 0.0;
	if (k <= 3.2e-52)
		tmp = (k ^ -0.5) * (1.0 / (t_0 ^ -0.5));
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 3.2e-52], N[(N[Power[k, -0.5], $MachinePrecision] * N[(1.0 / N[Power[t$95$0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.2000000000000001e-52

   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/ 99.4%

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

      2. *-lft-identity 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-*l* 99.4%

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

   3. Simplified 99.4%

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

      1. div-inv 99.1%

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

      2. *-commutative 99.1%

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

      3. unpow-prod-down 99.1%

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

      4. associate-*r* 99.0%

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

      5. pow1/2 99.0%

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

      6. pow-flip 99.1%

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

      7. metadata-eval 99.1%

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

      8. div-sub 99.1%

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

      9. metadata-eval 99.1%

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

      10. div-inv 99.1%

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

      11. metadata-eval 99.1%

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

      12. div-sub 99.1%

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

      13. metadata-eval 99.1%

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

      14. div-inv 99.1%

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

      15. metadata-eval 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. add-sqr-sqrt 98.7%

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

      2. sqrt-unprod 61.9%

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

      3. associate-*l* 61.8%

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

      4. associate-*l* 61.8%

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

      5. swap-sqr 61.7%

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

      6. pow-prod-up 61.8%

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

      7. metadata-eval 61.8%

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

      8. inv-pow 61.8%

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

   7. Applied egg-rr 62.1%

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

      1. sqrt-prod 99.3%

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

      2. sqrt-div 99.1%

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

      3. metadata-eval 99.1%

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

      4. associate-/r/ 99.3%

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

      5. inv-pow 99.3%

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

      6. div-inv 99.2%

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

      7. unpow-prod-down 99.1%

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

      8. inv-pow 99.1%

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

      9. pow1/2 99.1%

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

      10. pow-flip 99.3%

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

      11. metadata-eval 99.3%

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

   9. Applied egg-rr 99.3%

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

       1. unpow-1 99.3%

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

   11. Simplified 99.3%

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

   12. Taylor expanded in k around 0 99.3%

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

       1. associate-*r* 99.3%

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

       2. *-commutative 99.3%

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

       3. *-commutative 99.3%

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

   14. Simplified 99.3%

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

   if 3.2000000000000001e-52 < k

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*r* 99.6%

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

      4. div-inv 99.7%

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

      5. expm1-log1p-u 99.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

      6. expm1-udef 92.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)} - 1} \]

   3. Applied egg-rr 92.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 99.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

      2. expm1-log1p 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*r* 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-52}:\\ \;\;\;\;{k}^{-0.5} \cdot \frac{1}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot n\right)\\ \mathbf{if}\;k \leq 5.8 \cdot 10^{-52}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* 2.0 n))))
   (if (<= k 5.8e-52)
     (* (pow k -0.5) (sqrt t_0))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (2.0 * n);
	double tmp;
	if (k <= 5.8e-52) {
		tmp = pow(k, -0.5) * sqrt(t_0);
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (2.0 * n);
	double tmp;
	if (k <= 5.8e-52) {
		tmp = Math.pow(k, -0.5) * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.pi * (2.0 * n)
	tmp = 0
	if k <= 5.8e-52:
		tmp = math.pow(k, -0.5) * math.sqrt(t_0)
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(2.0 * n))
	tmp = 0.0
	if (k <= 5.8e-52)
		tmp = Float64((k ^ -0.5) * sqrt(t_0));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = pi * (2.0 * n);
	tmp = 0.0;
	if (k <= 5.8e-52)
		tmp = (k ^ -0.5) * sqrt(t_0);
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 5.8e-52], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.8000000000000003e-52

   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. expm1-log1p-u 91.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{k}}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

      2. expm1-udef 91.6%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{k}}\right)} - 1\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

      3. pow1/2 91.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{k}^{0.5}}}\right)} - 1\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

      4. pow-flip 91.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{k}^{\left(-0.5\right)}}\right)} - 1\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

      5. metadata-eval 91.6%

         \[\leadsto \left(e^{\mathsf{log1p}\left({k}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   3. Applied egg-rr 91.6%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-0.5}\right)} - 1\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   4. Step-by-step derivation

      1. expm1-def 91.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

      2. expm1-log1p 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in k around 0 99.2%

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

      1. expm1-log1p-u 95.8%

         \[\leadsto {k}^{-0.5} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{n \cdot \pi}\right)\right)} \]

      2. expm1-udef 54.1%

         \[\leadsto {k}^{-0.5} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{n \cdot \pi}\right)} - 1\right)} \]

      3. sqrt-unprod 54.1%

         \[\leadsto {k}^{-0.5} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{2 \cdot \left(n \cdot \pi\right)}}\right)} - 1\right) \]

      4. *-commutative 54.1%

         \[\leadsto {k}^{-0.5} \cdot \left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \color{blue}{\left(\pi \cdot n\right)}}\right)} - 1\right) \]

   8. Applied egg-rr 54.1%

      \[\leadsto {k}^{-0.5} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\pi \cdot n\right)}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 95.8%

         \[\leadsto {k}^{-0.5} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(\pi \cdot n\right)}\right)\right)} \]

      2. expm1-log1p 99.3%

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

      3. *-commutative 99.3%

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

      4. associate-*r* 99.3%

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

   10. Simplified 99.3%

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

   if 5.8000000000000003e-52 < k

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*r* 99.6%

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

      4. div-inv 99.7%

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

      5. expm1-log1p-u 99.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

      6. expm1-udef 93.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)} - 1} \]

   3. Applied egg-rr 93.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 99.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

      2. expm1-log1p 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*r* 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-52}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 4: 49.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. expm1-log1p-u 96.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{k}}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. expm1-udef 73.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{k}}\right)} - 1\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   3. pow1/2 73.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{k}^{0.5}}}\right)} - 1\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   4. pow-flip 73.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{k}^{\left(-0.5\right)}}\right)} - 1\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   5. metadata-eval 73.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left({k}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

3. Applied egg-rr 73.0%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-0.5}\right)} - 1\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
4. Step-by-step derivation

   1. expm1-def 96.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. expm1-log1p 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in k around 0 50.0%

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

   1. expm1-log1p-u 48.5%

      \[\leadsto {k}^{-0.5} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{n \cdot \pi}\right)\right)} \]

   2. expm1-udef 27.3%

      \[\leadsto {k}^{-0.5} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2} \cdot \sqrt{n \cdot \pi}\right)} - 1\right)} \]

   3. sqrt-unprod 27.3%

      \[\leadsto {k}^{-0.5} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{2 \cdot \left(n \cdot \pi\right)}}\right)} - 1\right) \]

   4. *-commutative 27.3%

      \[\leadsto {k}^{-0.5} \cdot \left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \color{blue}{\left(\pi \cdot n\right)}}\right)} - 1\right) \]

8. Applied egg-rr 27.3%

   \[\leadsto {k}^{-0.5} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\pi \cdot n\right)}\right)} - 1\right)} \]
9. Step-by-step derivation

   1. expm1-def 48.5%

      \[\leadsto {k}^{-0.5} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(\pi \cdot n\right)}\right)\right)} \]

   2. expm1-log1p 50.1%

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

   3. *-commutative 50.1%

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

   4. associate-*r* 50.1%

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

10. Simplified 50.1%

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

11. Final simplification 50.1%

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

Alternative 5: 49.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.4%

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

   2. *-commutative 99.4%

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

   3. unpow-prod-down 74.0%

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

   4. associate-*r* 74.0%

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

   5. pow1/2 74.0%

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

   6. pow-flip 74.0%

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

   7. metadata-eval 74.0%

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

   8. div-sub 74.0%

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

   9. metadata-eval 74.0%

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

   10. div-inv 74.0%

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

   11. metadata-eval 74.0%

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

   12. div-sub 74.0%

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

   13. metadata-eval 74.0%

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

   14. div-inv 74.0%

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

   15. metadata-eval 74.0%

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

5. Applied egg-rr 74.0%

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

   1. add-sqr-sqrt 73.8%

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

   2. sqrt-unprod 58.7%

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

   3. associate-*l* 58.7%

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

   4. associate-*l* 58.7%

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

   5. swap-sqr 58.7%

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

   6. pow-prod-up 58.7%

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

   7. metadata-eval 58.7%

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

   8. inv-pow 58.7%

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

7. Applied egg-rr 84.2%

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

8. Taylor expanded in k around 0 34.8%

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

   1. *-commutative 34.8%

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

10. Simplified 34.8%

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

    1. pow1/2 34.8%

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

    2. associate-*r* 34.8%

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

    3. unpow-prod-down 49.8%

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

    4. pow1/2 49.8%

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

12. Applied egg-rr 49.8%

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

    1. *-commutative 49.8%

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

    2. *-commutative 49.8%

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

    3. unpow1/2 49.8%

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

    4. associate-*l/ 49.8%

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

    5. metadata-eval 49.8%

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

14. Simplified 49.8%

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

15. Final simplification 49.8%

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

Alternative 6: 37.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.4%

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

   2. *-commutative 99.4%

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

   3. unpow-prod-down 74.0%

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

   4. associate-*r* 74.0%

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

   5. pow1/2 74.0%

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

   6. pow-flip 74.0%

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

   7. metadata-eval 74.0%

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

   8. div-sub 74.0%

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

   9. metadata-eval 74.0%

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

   10. div-inv 74.0%

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

   11. metadata-eval 74.0%

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

   12. div-sub 74.0%

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

   13. metadata-eval 74.0%

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

   14. div-inv 74.0%

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

   15. metadata-eval 74.0%

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

5. Applied egg-rr 74.0%

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

   1. add-sqr-sqrt 73.8%

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

   2. sqrt-unprod 58.7%

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

   3. associate-*l* 58.7%

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

   4. associate-*l* 58.7%

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

   5. swap-sqr 58.7%

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

   6. pow-prod-up 58.7%

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

   7. metadata-eval 58.7%

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

   8. inv-pow 58.7%

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

7. Applied egg-rr 84.2%

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

8. Taylor expanded in k around 0 34.8%

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

   1. *-commutative 34.8%

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

   2. associate-/l* 34.8%

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

10. Simplified 34.8%

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

11. Taylor expanded in k around 0 34.8%

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

    1. associate-/l* 34.8%

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

    2. associate-/r/ 34.8%

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

13. Simplified 34.8%

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

14. Final simplification 34.8%

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

Alternative 7: 37.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.4%

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

   2. *-commutative 99.4%

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

   3. unpow-prod-down 74.0%

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

   4. associate-*r* 74.0%

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

   5. pow1/2 74.0%

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

   6. pow-flip 74.0%

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

   7. metadata-eval 74.0%

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

   8. div-sub 74.0%

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

   9. metadata-eval 74.0%

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

   10. div-inv 74.0%

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

   11. metadata-eval 74.0%

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

   12. div-sub 74.0%

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

   13. metadata-eval 74.0%

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

   14. div-inv 74.0%

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

   15. metadata-eval 74.0%

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

5. Applied egg-rr 74.0%

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

   1. add-sqr-sqrt 73.8%

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

   2. sqrt-unprod 58.7%

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

   3. associate-*l* 58.7%

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

   4. associate-*l* 58.7%

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

   5. swap-sqr 58.7%

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

   6. pow-prod-up 58.7%

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

   7. metadata-eval 58.7%

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

   8. inv-pow 58.7%

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

7. Applied egg-rr 84.2%

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

8. Taylor expanded in k around 0 34.8%

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

   1. *-commutative 34.8%

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

   2. associate-/l* 34.8%

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

10. Simplified 34.8%

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

    1. associate-/r/ 34.8%

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

12. Applied egg-rr 34.8%

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

13. Final simplification 34.8%

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

Alternative 8: 37.6% accurate, 1.5× 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 99.4%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.4%

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

   2. *-commutative 99.4%

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

   3. unpow-prod-down 74.0%

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

   4. associate-*r* 74.0%

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

   5. pow1/2 74.0%

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

   6. pow-flip 74.0%

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

   7. metadata-eval 74.0%

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

   8. div-sub 74.0%

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

   9. metadata-eval 74.0%

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

   10. div-inv 74.0%

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

   11. metadata-eval 74.0%

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

   12. div-sub 74.0%

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

   13. metadata-eval 74.0%

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

   14. div-inv 74.0%

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

   15. metadata-eval 74.0%

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

5. Applied egg-rr 74.0%

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

   1. add-sqr-sqrt 73.8%

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

   2. sqrt-unprod 58.7%

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

   3. associate-*l* 58.7%

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

   4. associate-*l* 58.7%

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

   5. swap-sqr 58.7%

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

   6. pow-prod-up 58.7%

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

   7. metadata-eval 58.7%

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

   8. inv-pow 58.7%

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

7. Applied egg-rr 84.2%

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

8. Taylor expanded in k around 0 34.8%

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

   1. *-commutative 34.8%

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

   2. associate-/l* 34.8%

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

10. Simplified 34.8%

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

11. Final simplification 34.8%

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

Reproduce

herbie shell --seed 2023230 
(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))))