Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 13.5s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(n \cdot 2\right) \cdot \pi\\ \frac{1}{\sqrt{k}} \cdot \frac{\sqrt{t_0}}{{t_0}^{\left(k \cdot 0.5\right)}} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(k, n):
	t_0 = (n * 2.0) * math.pi
	return (1.0 / math.sqrt(k)) * (math.sqrt(t_0) / math.pow(t_0, (k * 0.5)))

Julia

function code(k, n)
	t_0 = Float64(Float64(n * 2.0) * pi)
	return Float64(Float64(1.0 / sqrt(k)) * Float64(sqrt(t_0) / (t_0 ^ Float64(k * 0.5))))
end

MATLAB

function tmp = code(k, n)
	t_0 = (n * 2.0) * pi;
	tmp = (1.0 / sqrt(k)) * (sqrt(t_0) / (t_0 ^ (k * 0.5)));
end

Wolfram

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

Derivation

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. div-sub 99.3%

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

   2. metadata-eval 99.3%

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

   3. pow-sub 99.6%

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

   4. pow1/2 99.6%

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

   5. associate-*l* 99.6%

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

   6. associate-*l* 99.6%

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

   7. div-inv 99.6%

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

   8. metadata-eval 99.6%

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

3. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. *-commutative 99.6%

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

   4. *-commutative 99.6%

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

   5. associate-*r* 99.6%

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

   6. *-commutative 99.6%

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

   7. *-commutative 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. expm1-log1p-u 95.9%

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

   2. expm1-udef 76.0%

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

   3. pow1/2 76.0%

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

   4. pow-flip 76.0%

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

   5. metadata-eval 76.0%

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

3. Applied egg-rr 76.0%

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

   1. expm1-def 95.9%

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

   2. expm1-log1p 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.3 \cdot 10^{-100}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.3e-100)
   (/ (sqrt (* n (* 2.0 PI))) (sqrt k))
   (sqrt (/ (pow (* (* n 2.0) PI) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.3e-100) {
		tmp = sqrt((n * (2.0 * ((double) M_PI)))) / sqrt(k);
	} else {
		tmp = sqrt((pow(((n * 2.0) * ((double) M_PI)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.3e-100) {
		tmp = Math.sqrt((n * (2.0 * Math.PI))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow(((n * 2.0) * Math.PI), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 4.3e-100:
		tmp = math.sqrt((n * (2.0 * math.pi))) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow(((n * 2.0) * math.pi), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.3e-100)
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * pi))) / sqrt(k));
	else
		tmp = sqrt(Float64((Float64(Float64(n * 2.0) * pi) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 4.3e-100)
		tmp = sqrt((n * (2.0 * pi))) / sqrt(k);
	else
		tmp = sqrt(((((n * 2.0) * pi) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.3e-100], N[(N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(N[(n * 2.0), $MachinePrecision] * Pi), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.29999999999999998e-100

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-*r* 99.2%

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

      4. div-inv 99.4%

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

      5. expm1-log1p-u 93.1%

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

      6. expm1-udef 74.2%

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

   3. Applied egg-rr 43.7%

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

      1. expm1-def 62.6%

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

      2. expm1-log1p 65.4%

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

      3. *-commutative 65.4%

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

      4. associate-*r* 65.4%

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

      5. *-commutative 65.4%

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

   5. Simplified 65.4%

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

      1. add-exp-log 60.9%

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

      2. associate-*l* 60.9%

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

      3. *-commutative 60.9%

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

      4. associate-*r* 60.9%

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

      5. add-exp-log 60.9%

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

      6. log-pow 60.9%

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

      7. add-log-exp 60.9%

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

      8. associate-*r* 60.9%

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

      9. *-commutative 60.9%

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

      10. associate-*l* 60.9%

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

      11. *-commutative 60.9%

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

      12. *-commutative 60.9%

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

      13. *-commutative 60.9%

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

   7. Applied egg-rr 60.9%

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

   8. Taylor expanded in k around 0 65.4%

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

      1. *-commutative 65.4%

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

   10. Simplified 65.4%

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

       1. sqrt-div 99.4%

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

   12. Applied egg-rr 99.4%

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

       1. associate-*r* 99.4%

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

   14. Simplified 99.4%

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

   if 4.29999999999999998e-100 < k

   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. *-commutative 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-*r* 99.4%

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

      4. div-inv 99.3%

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

      5. expm1-log1p-u 98.6%

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

      6. expm1-udef 84.5%

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

   3. Applied egg-rr 84.5%

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

      1. expm1-def 98.6%

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

      2. expm1-log1p 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*r* 99.5%

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

      5. *-commutative 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.3 \cdot 10^{-100}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Final simplification 99.4%

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

Alternative 5: 50.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. expm1-log1p-u 95.9%

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

   2. expm1-udef 76.0%

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

   3. pow1/2 76.0%

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

   4. pow-flip 76.0%

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

   5. metadata-eval 76.0%

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

3. Applied egg-rr 76.0%

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

   1. expm1-def 95.9%

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

   2. expm1-log1p 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in k around 0 52.1%

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

   1. *-commutative 52.1%

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

   2. sqrt-prod 52.1%

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

   3. expm1-log1p-u 50.5%

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

   4. expm1-udef 25.1%

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

8. Applied egg-rr 25.1%

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

   1. expm1-def 50.5%

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

   2. expm1-log1p 52.1%

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

   3. associate-*r* 52.1%

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

10. Simplified 52.1%

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

11. Final simplification 52.1%

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

Alternative 6: 50.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. *-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*r* 99.3%

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

   4. div-inv 99.4%

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

   5. expm1-log1p-u 96.8%

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

   6. expm1-udef 81.1%

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

3. Applied egg-rr 70.9%

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

   1. expm1-def 86.7%

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

   2. expm1-log1p 88.1%

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

   3. *-commutative 88.1%

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

   4. associate-*r* 88.1%

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

   5. *-commutative 88.1%

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

5. Simplified 88.1%

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

   1. add-exp-log 85.5%

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

   2. associate-*l* 85.5%

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

   3. *-commutative 85.5%

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

   4. associate-*r* 85.5%

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

   5. add-exp-log 85.5%

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

   6. log-pow 85.5%

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

   7. add-log-exp 85.5%

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

   8. associate-*r* 85.5%

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

   9. *-commutative 85.5%

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

   10. associate-*l* 85.5%

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

   11. *-commutative 85.5%

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

   12. *-commutative 85.5%

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

   13. *-commutative 85.5%

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

7. Applied egg-rr 85.5%

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

8. Taylor expanded in k around 0 40.8%

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

   1. *-commutative 40.8%

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

10. Simplified 40.8%

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

    1. sqrt-div 52.1%

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

12. Applied egg-rr 52.1%

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

    1. associate-*r* 52.1%

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

14. Simplified 52.1%

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

15. Final simplification 52.1%

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

Alternative 7: 38.5% 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 99.3%

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

   1. *-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*r* 99.3%

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

   4. div-inv 99.4%

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

   5. expm1-log1p-u 96.8%

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

   6. expm1-udef 81.1%

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

3. Applied egg-rr 70.9%

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

   1. expm1-def 86.7%

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

   2. expm1-log1p 88.1%

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

   3. *-commutative 88.1%

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

   4. associate-*r* 88.1%

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

   5. *-commutative 88.1%

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

5. Simplified 88.1%

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

   1. add-exp-log 85.5%

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

   2. associate-*l* 85.5%

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

   3. *-commutative 85.5%

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

   4. associate-*r* 85.5%

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

   5. add-exp-log 85.5%

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

   6. log-pow 85.5%

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

   7. add-log-exp 85.5%

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

   8. associate-*r* 85.5%

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

   9. *-commutative 85.5%

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

   10. associate-*l* 85.5%

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

   11. *-commutative 85.5%

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

   12. *-commutative 85.5%

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

   13. *-commutative 85.5%

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

7. Applied egg-rr 85.5%

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

8. Taylor expanded in k around 0 40.8%

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

   1. *-commutative 40.8%

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

10. Simplified 40.8%

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

11. Taylor expanded in n around 0 40.8%

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

    1. associate-/l* 40.7%

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

    2. associate-/r/ 40.8%

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

13. Simplified 40.8%

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

14. Final simplification 40.8%

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

Reproduce

herbie shell --seed 2023192 
(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))))