Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 14.4s

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: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Power[k, -0.5], $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 99.5%

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

   1. div-sub 99.5%

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

   2. metadata-eval 99.5%

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

   3. pow-sub 99.7%

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

   4. pow1/2 99.7%

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

   5. associate-*l* 99.7%

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

   6. associate-*l* 99.7%

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

   7. div-inv 99.7%

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

   8. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*r* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*r* 99.7%

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

   5. *-commutative 99.7%

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

5. Simplified 99.7%

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

   1. expm1-log1p-u 96.5%

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

   2. expm1-udef 73.0%

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

   3. pow1/2 73.0%

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

   4. pow-flip 73.0%

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

   5. metadata-eval 73.0%

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

7. Applied egg-rr 73.0%

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

   1. expm1-def 96.5%

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

   2. expm1-log1p 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot n\right) \cdot \pi\\ \mathbf{if}\;k \leq 3 \cdot 10^{-39}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{k}}{\sqrt{t_0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{t_0}^{\left(1 - k\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* 2.0 n) PI)))
   (if (<= k 3e-39)
     (/ 1.0 (/ (sqrt k) (sqrt t_0)))
     (/ 1.0 (sqrt (/ k (pow t_0 (- 1.0 k))))))))

C

double code(double k, double n) {
	double t_0 = (2.0 * n) * ((double) M_PI);
	double tmp;
	if (k <= 3e-39) {
		tmp = 1.0 / (sqrt(k) / sqrt(t_0));
	} else {
		tmp = 1.0 / sqrt((k / pow(t_0, (1.0 - k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = (2.0 * n) * Math.PI;
	double tmp;
	if (k <= 3e-39) {
		tmp = 1.0 / (Math.sqrt(k) / Math.sqrt(t_0));
	} else {
		tmp = 1.0 / Math.sqrt((k / Math.pow(t_0, (1.0 - k))));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = (2.0 * n) * math.pi
	tmp = 0
	if k <= 3e-39:
		tmp = 1.0 / (math.sqrt(k) / math.sqrt(t_0))
	else:
		tmp = 1.0 / math.sqrt((k / math.pow(t_0, (1.0 - k))))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(Float64(2.0 * n) * pi)
	tmp = 0.0
	if (k <= 3e-39)
		tmp = Float64(1.0 / Float64(sqrt(k) / sqrt(t_0)));
	else
		tmp = Float64(1.0 / sqrt(Float64(k / (t_0 ^ Float64(1.0 - k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = (2.0 * n) * pi;
	tmp = 0.0;
	if (k <= 3e-39)
		tmp = 1.0 / (sqrt(k) / sqrt(t_0));
	else
		tmp = 1.0 / sqrt((k / (t_0 ^ (1.0 - k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[k, 3e-39], N[(1.0 / N[(N[Sqrt[k], $MachinePrecision] / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(k / N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.00000000000000028e-39

   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%

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

      2. associate-*r* 99.3%

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

      3. associate-/r/ 99.4%

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

      4. add-sqr-sqrt 99.0%

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

      5. sqrt-unprod 99.4%

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

      6. associate-*r* 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-*r* 99.4%

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

      9. *-commutative 99.4%

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

      10. pow-prod-up 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. sqrt-undiv 77.0%

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

      2. *-commutative 77.0%

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

      3. associate-*l* 77.0%

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

      4. *-commutative 77.0%

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

   5. Applied egg-rr 77.0%

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

   6. Taylor expanded in k around 0 77.0%

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

      1. *-commutative 77.0%

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

      2. *-commutative 77.0%

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

      3. associate-*l* 77.0%

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

   8. Simplified 77.0%

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

      1. sqrt-div 99.4%

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

      2. *-commutative 99.4%

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

   10. Applied egg-rr 99.4%

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

   if 3.00000000000000028e-39 < k

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*r* 99.6%

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

      3. associate-/r/ 99.6%

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

      4. add-sqr-sqrt 99.5%

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

      5. sqrt-unprod 99.6%

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

      6. associate-*r* 99.6%

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

      7. *-commutative 99.6%

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

      8. associate-*r* 99.6%

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

      9. *-commutative 99.6%

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

      10. pow-prod-up 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. sqrt-undiv 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l* 99.6%

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

      4. *-commutative 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-39}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{k}}{\sqrt{\left(2 \cdot n\right) \cdot \pi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[k, -0.5], $MachinePrecision] / N[Power[N[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision], N[(N[(k + -1.0), $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. div-sub 99.5%

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

   2. metadata-eval 99.5%

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

   3. pow-sub 99.7%

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

   4. pow1/2 99.7%

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

   5. associate-*l* 99.7%

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

   6. associate-*l* 99.7%

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

   7. div-inv 99.7%

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

   8. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*r* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*r* 99.7%

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

   5. *-commutative 99.7%

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

5. Simplified 99.7%

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

   1. expm1-log1p-u 96.5%

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

   2. expm1-udef 73.0%

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

   3. pow1/2 73.0%

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

   4. pow-flip 73.0%

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

   5. metadata-eval 73.0%

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

7. Applied egg-rr 73.0%

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

   1. expm1-def 96.5%

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

   2. expm1-log1p 99.7%

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

9. Simplified 99.7%

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

    1. clear-num 99.7%

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

    2. un-div-inv 99.7%

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

    3. pow-unpow 99.7%

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

    4. pow1/2 99.7%

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

    5. pow1 99.7%

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

    6. pow-div 99.5%

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

    7. associate-*l* 99.5%

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

11. Applied egg-rr 99.5%

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

    1. *-commutative 99.5%

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

    2. *-commutative 99.5%

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

    3. associate-*l* 99.5%

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

    4. sub-neg 99.5%

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

    5. metadata-eval 99.5%

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

13. Simplified 99.5%

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

    1. sqrt-pow2 99.6%

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

    2. associate-*r* 99.6%

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

    3. *-commutative 99.6%

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

    4. *-commutative 99.6%

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

    5. associate-*r* 99.6%

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

    6. *-commutative 99.6%

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

    7. pow-to-exp 96.1%

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

15. Applied egg-rr 96.1%

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

    1. exp-to-pow 99.6%

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

17. Simplified 99.6%

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

18. Final simplification 99.6%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot n\right) \cdot \pi\\ \mathbf{if}\;k \leq 1.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{k}}{\sqrt{t_0}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* 2.0 n) PI)))
   (if (<= k 1.8e-45)
     (/ 1.0 (/ (sqrt k) (sqrt t_0)))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = (2.0 * n) * ((double) M_PI);
	double tmp;
	if (k <= 1.8e-45) {
		tmp = 1.0 / (sqrt(k) / sqrt(t_0));
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = (2.0 * n) * Math.PI;
	double tmp;
	if (k <= 1.8e-45) {
		tmp = 1.0 / (Math.sqrt(k) / Math.sqrt(t_0));
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = (2.0 * n) * math.pi
	tmp = 0
	if k <= 1.8e-45:
		tmp = 1.0 / (math.sqrt(k) / math.sqrt(t_0))
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(Float64(2.0 * n) * pi)
	tmp = 0.0
	if (k <= 1.8e-45)
		tmp = Float64(1.0 / Float64(sqrt(k) / sqrt(t_0)));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = (2.0 * n) * pi;
	tmp = 0.0;
	if (k <= 1.8e-45)
		tmp = 1.0 / (sqrt(k) / sqrt(t_0));
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[k, 1.8e-45], N[(1.0 / N[(N[Sqrt[k], $MachinePrecision] / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.8e-45

   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%

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

      2. associate-*r* 99.3%

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

      3. associate-/r/ 99.4%

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

      4. add-sqr-sqrt 99.0%

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

      5. sqrt-unprod 99.4%

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

      6. associate-*r* 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-*r* 99.4%

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

      9. *-commutative 99.4%

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

      10. pow-prod-up 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. sqrt-undiv 76.2%

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

      2. *-commutative 76.2%

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

      3. associate-*l* 76.2%

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

      4. *-commutative 76.2%

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

   5. Applied egg-rr 76.2%

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

   6. Taylor expanded in k around 0 76.2%

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

      1. *-commutative 76.2%

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

      2. *-commutative 76.2%

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

      3. associate-*l* 76.2%

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

   8. Simplified 76.2%

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

      1. sqrt-div 99.4%

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

      2. *-commutative 99.4%

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

   10. Applied egg-rr 99.4%

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

   if 1.8e-45 < k

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*r* 99.6%

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

      4. div-inv 99.6%

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

      5. expm1-log1p-u 99.1%

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

      6. expm1-udef 94.2%

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

   3. Applied egg-rr 94.2%

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

      1. expm1-def 99.1%

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

      2. expm1-log1p 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*r* 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{k}}{\sqrt{\left(2 \cdot n\right) \cdot \pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 5: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(N[(2.0 * n), $MachinePrecision] * Pi), $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/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 6: 50.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{k} \cdot \sqrt{\frac{0.5}{n \cdot \pi}}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(1.0 / N[(N[Sqrt[k], $MachinePrecision] * N[Sqrt[N[(0.5 / N[(n * Pi), $MachinePrecision]), $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. *-commutative 99.5%

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

   2. associate-*r* 99.5%

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

   3. associate-/r/ 99.5%

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

   4. add-sqr-sqrt 99.3%

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

   5. sqrt-unprod 99.5%

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

   6. associate-*r* 99.5%

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

   7. *-commutative 99.5%

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

   8. associate-*r* 99.5%

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

   9. *-commutative 99.5%

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

   10. pow-prod-up 99.5%

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

3. Applied egg-rr 99.5%

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

   1. sqrt-undiv 89.9%

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

   2. *-commutative 89.9%

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

   3. associate-*l* 89.9%

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

   4. *-commutative 89.9%

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

5. Applied egg-rr 89.9%

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

6. Taylor expanded in k around 0 39.2%

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

   1. *-commutative 39.2%

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

   2. *-commutative 39.2%

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

   3. associate-*l* 39.2%

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

8. Simplified 39.2%

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

   1. pow1/2 39.2%

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

   2. div-inv 39.2%

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

   3. unpow-prod-down 48.9%

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

   4. pow1/2 48.9%

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

   5. *-commutative 48.9%

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

10. Applied egg-rr 48.9%

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

    1. unpow1/2 48.9%

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

    2. *-commutative 48.9%

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

    3. associate-*r* 48.9%

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

    4. *-commutative 48.9%

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

    5. associate-/r* 48.9%

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

    6. metadata-eval 48.9%

       \[\leadsto \frac{1}{\sqrt{k} \cdot \sqrt{\frac{\color{blue}{0.5}}{\pi \cdot n}}} \]

12. Simplified 48.9%

    \[\leadsto \frac{1}{\color{blue}{\sqrt{k} \cdot \sqrt{\frac{0.5}{\pi \cdot n}}}} \]

13. Final simplification 48.9%

    \[\leadsto \frac{1}{\sqrt{k} \cdot \sqrt{\frac{0.5}{n \cdot \pi}}} \]

Alternative 7: 50.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(1.0 / N[(N[Sqrt[k], $MachinePrecision] / N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * 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. *-commutative 99.5%

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

   2. associate-*r* 99.5%

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

   3. associate-/r/ 99.5%

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

   4. add-sqr-sqrt 99.3%

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

   5. sqrt-unprod 99.5%

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

   6. associate-*r* 99.5%

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

   7. *-commutative 99.5%

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

   8. associate-*r* 99.5%

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

   9. *-commutative 99.5%

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

   10. pow-prod-up 99.5%

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

3. Applied egg-rr 99.5%

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

   1. sqrt-undiv 89.9%

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

   2. *-commutative 89.9%

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

   3. associate-*l* 89.9%

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

   4. *-commutative 89.9%

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

5. Applied egg-rr 89.9%

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

6. Taylor expanded in k around 0 39.2%

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

   1. *-commutative 39.2%

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

   2. *-commutative 39.2%

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

   3. associate-*l* 39.2%

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

8. Simplified 39.2%

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

   1. sqrt-div 48.9%

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

   2. *-commutative 48.9%

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

10. Applied egg-rr 48.9%

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

11. Final simplification 48.9%

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

Alternative 8: 39.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Power[N[(N[(k / 2.0), $MachinePrecision] / N[(n * Pi), $MachinePrecision]), $MachinePrecision], -0.5], $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. *-commutative 99.5%

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

   2. associate-*r* 99.5%

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

   3. associate-/r/ 99.5%

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

   4. add-sqr-sqrt 99.3%

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

   5. sqrt-unprod 99.5%

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

   6. associate-*r* 99.5%

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

   7. *-commutative 99.5%

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

   8. associate-*r* 99.5%

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

   9. *-commutative 99.5%

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

   10. pow-prod-up 99.5%

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

3. Applied egg-rr 99.5%

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

   1. sqrt-undiv 89.9%

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

   2. *-commutative 89.9%

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

   3. associate-*l* 89.9%

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

   4. *-commutative 89.9%

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

5. Applied egg-rr 89.9%

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

6. Taylor expanded in k around 0 39.2%

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

   1. *-commutative 39.2%

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

   2. *-commutative 39.2%

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

   3. associate-*l* 39.2%

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

8. Simplified 39.2%

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

   1. expm1-log1p-u 37.6%

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

   2. expm1-udef 33.6%

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

   3. pow1/2 33.6%

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

   4. pow-flip 33.6%

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

   5. *-commutative 33.6%

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

   6. metadata-eval 33.6%

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

10. Applied egg-rr 33.6%

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

    1. expm1-def 37.6%

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

    2. expm1-log1p 39.3%

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

    3. *-commutative 39.3%

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

    4. associate-*r* 39.3%

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

    5. *-commutative 39.3%

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

    6. associate-/r* 39.3%

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

12. Simplified 39.3%

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

13. Final simplification 39.3%

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

Alternative 9: 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 99.5%

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

   1. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*r* 99.5%

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

   4. div-inv 99.5%

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

   5. expm1-log1p-u 96.9%

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

   6. expm1-udef 84.9%

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

3. Applied egg-rr 76.0%

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

   1. expm1-def 88.0%

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

   2. expm1-log1p 89.8%

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

   3. *-commutative 89.8%

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

   4. associate-*r* 89.8%

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

5. Simplified 89.8%

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

   1. pow-sub 90.0%

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

   2. pow1 90.0%

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

   3. *-commutative 90.0%

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

   4. *-commutative 90.0%

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

7. Applied egg-rr 90.0%

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

8. Taylor expanded in k around 0 39.1%

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

   1. associate-/l* 39.1%

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

   2. associate-/r/ 39.1%

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

10. Simplified 39.1%

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

11. Final simplification 39.1%

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

Alternative 10: 38.5% 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. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*r* 99.5%

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

   4. div-inv 99.5%

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

   5. expm1-log1p-u 96.9%

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

   6. expm1-udef 84.9%

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

3. Applied egg-rr 76.0%

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

   1. expm1-def 88.0%

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

   2. expm1-log1p 89.8%

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

   3. *-commutative 89.8%

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

   4. associate-*r* 89.8%

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

5. Simplified 89.8%

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

   1. pow-sub 90.0%

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

   2. pow1 90.0%

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

   3. *-commutative 90.0%

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

   4. *-commutative 90.0%

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

7. Applied egg-rr 90.0%

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

8. Taylor expanded in k around 0 39.1%

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

   1. associate-/l* 39.1%

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

10. Simplified 39.1%

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

11. Final simplification 39.1%

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

Reproduce

herbie shell --seed 2023174 
(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))))