Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.2%

Time: 16.4s

Alternatives: 10

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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot n\right)\\ \mathbf{if}\;k \leq 2 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{t_0} \cdot {k}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* 2.0 n))))
   (if (<= k 2e-38)
     (* (sqrt t_0) (pow k -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 <= 2e-38) {
		tmp = sqrt(t_0) * pow(k, -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 <= 2e-38) {
		tmp = Math.sqrt(t_0) * Math.pow(k, -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 <= 2e-38:
		tmp = math.sqrt(t_0) * math.pow(k, -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 <= 2e-38)
		tmp = Float64(sqrt(t_0) * (k ^ -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 <= 2e-38)
		tmp = sqrt(t_0) * (k ^ -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, 2e-38], N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Power[k, -0.5], $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 < 1.9999999999999999e-38

   1. Initial program 99.3%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 99.3%

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

      2. *-commutative 99.3%

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

      3. pow1/2 99.3%

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

      4. pow-flip 99.5%

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

      5. metadata-eval 99.5%

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

      6. add-sqr-sqrt 99.1%

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

      7. sqrt-unprod 99.5%

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

      8. associate-*r* 99.5%

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

      9. *-commutative 99.5%

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

      10. associate-*r* 99.5%

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

      11. *-commutative 99.5%

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

      12. pow-prod-up 99.5%

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

      13. associate-*l* 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in k around 0 99.5%

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

      1. associate-*r* 99.5%

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

      2. *-commutative 99.5%

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

      3. *-commutative 99.5%

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

   8. Simplified 99.5%

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

   if 1.9999999999999999e-38 < k

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*r* 99.7%

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

      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 99.3%

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

      6. expm1-udef 95.8%

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

   3. Applied egg-rr 95.8%

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

      1. expm1-def 99.3%

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

      2. expm1-log1p 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r* 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{\pi \cdot \left(2 \cdot n\right)} \cdot {k}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. inv-pow 99.5%

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

   2. pow1/2 99.5%

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

   3. pow-pow 99.6%

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

   4. add-exp-log 95.8%

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

   5. pow-exp 95.8%

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

   6. metadata-eval 95.8%

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

3. Applied egg-rr 95.8%

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

   1. pow-to-exp 99.6%

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

   2. metadata-eval 99.6%

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

   3. pow-div 99.5%

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

5. Applied egg-rr 99.5%

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

   1. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*r* 99.5%

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

   4. sqrt-pow1 99.5%

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

   5. add-sqr-sqrt 99.4%

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

   6. sqrt-unprod 99.1%

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

   7. sqrt-prod 88.8%

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

   8. pow-div 88.9%

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

   9. metadata-eval 88.9%

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

   10. pow-div 88.9%

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

   11. metadata-eval 88.9%

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

   12. pow-prod-up 89.0%

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

   13. metadata-eval 89.0%

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

   14. inv-pow 89.0%

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

   15. div-inv 89.4%

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

7. Applied egg-rr 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 3: 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 99.5%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 4: 37.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. *-commutative 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 99.4%

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

   7. sqrt-unprod 99.6%

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

   8. associate-*r* 99.6%

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

   9. *-commutative 99.6%

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

   10. associate-*r* 99.6%

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

   11. *-commutative 99.6%

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

   12. pow-prod-up 99.6%

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

   13. associate-*l* 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in k around 0 55.3%

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

   1. associate-*r* 55.3%

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

   2. *-commutative 55.3%

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

   3. *-commutative 55.3%

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

8. Simplified 55.3%

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

   1. expm1-log1p-u 52.1%

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

   2. expm1-udef 51.6%

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

10. Applied egg-rr 42.5%

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

    1. expm1-def 42.9%

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

    2. expm1-log1p 45.1%

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

    3. associate-/l* 45.0%

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

    4. *-commutative 45.0%

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

    5. associate-/r* 45.1%

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

12. Simplified 45.1%

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

    1. sqrt-div 45.9%

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

    2. associate-/l/ 45.9%

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

14. Applied egg-rr 45.9%

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

15. Final simplification 45.9%

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

Alternative 5: 37.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. *-commutative 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 99.4%

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

   7. sqrt-unprod 99.6%

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

   8. associate-*r* 99.6%

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

   9. *-commutative 99.6%

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

   10. associate-*r* 99.6%

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

   11. *-commutative 99.6%

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

   12. pow-prod-up 99.6%

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

   13. associate-*l* 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in k around 0 55.3%

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

   1. associate-*r* 55.3%

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

   2. *-commutative 55.3%

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

   3. *-commutative 55.3%

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

8. Simplified 55.3%

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

   1. expm1-log1p-u 52.1%

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

   2. expm1-udef 51.6%

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

10. Applied egg-rr 42.5%

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

    1. expm1-def 42.9%

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

    2. expm1-log1p 45.1%

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

    3. associate-/l* 45.0%

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

    4. *-commutative 45.0%

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

    5. associate-/r* 45.1%

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

12. Simplified 45.1%

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

    1. sqrt-div 45.9%

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

    2. associate-/l/ 45.9%

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

14. Applied egg-rr 45.9%

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

    1. associate-/l/ 45.9%

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

16. Simplified 45.9%

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

17. Final simplification 45.9%

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

Alternative 6: 49.0% 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 99.5%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. *-commutative 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 99.4%

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

   7. sqrt-unprod 99.6%

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

   8. associate-*r* 99.6%

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

   9. *-commutative 99.6%

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

   10. associate-*r* 99.6%

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

   11. *-commutative 99.6%

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

   12. pow-prod-up 99.6%

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

   13. associate-*l* 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in k around 0 55.3%

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

   1. associate-*r* 55.3%

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

   2. *-commutative 55.3%

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

   3. *-commutative 55.3%

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

8. Simplified 55.3%

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

   1. metadata-eval 55.3%

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

   2. sqrt-pow2 55.2%

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

   3. inv-pow 55.2%

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

   4. associate-*r* 55.2%

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

   5. *-commutative 55.2%

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

   6. sqrt-prod 55.2%

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

   7. *-commutative 55.2%

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

   8. associate-*l/ 55.2%

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

   9. *-un-lft-identity 55.2%

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

   10. sqrt-unprod 55.3%

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

   11. *-commutative 55.3%

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

   12. associate-*l* 55.3%

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

   13. *-commutative 55.3%

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

   14. associate-*r* 55.3%

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

   15. associate-*r* 55.3%

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

   16. *-commutative 55.3%

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

   17. associate-*l* 55.3%

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

10. Applied egg-rr 55.3%

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

    1. associate-*r* 55.3%

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

    2. *-commutative 55.3%

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

    3. associate-*r* 55.3%

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

12. Simplified 55.3%

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

13. Final simplification 55.3%

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

Alternative 7: 37.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\frac{\frac{k}{\pi \cdot n}}{2}}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. *-commutative 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 99.4%

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

   7. sqrt-unprod 99.6%

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

   8. associate-*r* 99.6%

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

   9. *-commutative 99.6%

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

   10. associate-*r* 99.6%

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

   11. *-commutative 99.6%

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

   12. pow-prod-up 99.6%

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

   13. associate-*l* 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in k around 0 55.3%

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

   1. associate-*r* 55.3%

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

   2. *-commutative 55.3%

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

   3. *-commutative 55.3%

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

8. Simplified 55.3%

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

   1. expm1-log1p-u 52.1%

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

   2. expm1-udef 51.6%

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

10. Applied egg-rr 42.5%

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

    1. expm1-def 42.9%

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

    2. expm1-log1p 45.1%

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

    3. associate-/l* 45.0%

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

    4. *-commutative 45.0%

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

    5. associate-/r* 45.1%

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

12. Simplified 45.1%

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

    1. clear-num 45.1%

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

    2. sqrt-div 45.9%

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

    3. metadata-eval 45.9%

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

    4. associate-/l/ 45.9%

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

14. Applied egg-rr 45.9%

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

15. Final simplification 45.9%

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

Alternative 8: 36.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. *-commutative 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 99.4%

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

   7. sqrt-unprod 99.6%

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

   8. associate-*r* 99.6%

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

   9. *-commutative 99.6%

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

   10. associate-*r* 99.6%

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

   11. *-commutative 99.6%

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

   12. pow-prod-up 99.6%

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

   13. associate-*l* 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in k around 0 55.3%

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

   1. associate-*r* 55.3%

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

   2. *-commutative 55.3%

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

   3. *-commutative 55.3%

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

8. Simplified 55.3%

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

   1. expm1-log1p-u 52.1%

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

   2. expm1-udef 51.6%

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

10. Applied egg-rr 42.5%

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

    1. expm1-def 42.9%

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

    2. expm1-log1p 45.1%

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

    3. associate-/l* 45.0%

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

    4. *-commutative 45.0%

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

    5. associate-/r* 45.1%

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

12. Simplified 45.1%

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

13. Taylor expanded in k around 0 45.1%

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

    1. associate-/l* 45.0%

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

15. Simplified 45.0%

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

16. Final simplification 45.0%

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

Alternative 9: 36.6% 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 99.5%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. *-commutative 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 99.4%

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

   7. sqrt-unprod 99.6%

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

   8. associate-*r* 99.6%

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

   9. *-commutative 99.6%

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

   10. associate-*r* 99.6%

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

   11. *-commutative 99.6%

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

   12. pow-prod-up 99.6%

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

   13. associate-*l* 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in k around 0 55.3%

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

   1. associate-*r* 55.3%

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

   2. *-commutative 55.3%

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

   3. *-commutative 55.3%

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

8. Simplified 55.3%

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

   1. expm1-log1p-u 52.1%

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

   2. expm1-udef 51.6%

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

10. Applied egg-rr 42.5%

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

    1. expm1-def 42.9%

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

    2. expm1-log1p 45.1%

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

    3. associate-/l* 45.0%

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

    4. *-commutative 45.0%

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

    5. associate-/r* 45.1%

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

12. Simplified 45.1%

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

13. Taylor expanded in k around 0 45.1%

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

    1. *-commutative 45.1%

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

    2. associate-/l* 45.1%

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

15. Simplified 45.1%

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

16. Final simplification 45.1%

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

Alternative 10: 36.6% accurate, 1.5× speedup

Math

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

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(Float64(pi * Float64(2.0 * n)) / k))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. *-commutative 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

   6. add-sqr-sqrt 99.4%

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

   7. sqrt-unprod 99.6%

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

   8. associate-*r* 99.6%

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

   9. *-commutative 99.6%

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

   10. associate-*r* 99.6%

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

   11. *-commutative 99.6%

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

   12. pow-prod-up 99.6%

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

   13. associate-*l* 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in k around 0 55.3%

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

   1. associate-*r* 55.3%

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

   2. *-commutative 55.3%

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

   3. *-commutative 55.3%

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

8. Simplified 55.3%

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

   1. expm1-log1p-u 52.1%

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

   2. expm1-udef 51.6%

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

10. Applied egg-rr 42.5%

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

    1. expm1-def 42.9%

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

    2. expm1-log1p 45.1%

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

    3. associate-*r* 45.1%

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

    4. *-commutative 45.1%

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

    5. associate-*r* 45.1%

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

12. Simplified 45.1%

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

13. Final simplification 45.1%

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

Reproduce

herbie shell --seed 2023256 
(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))))