Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 14.8s

Alternatives: 11

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: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

def code(k, n):
	t_0 = math.pi * (n * 2.0)
	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(pi * Float64(n * 2.0))
	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 = pi * (n * 2.0);
	tmp = (1.0 / sqrt(k)) * (sqrt(t_0) / (t_0 ^ (k * 0.5)));
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(n * 2.0), $MachinePrecision]), $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 99.1%

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

   1. div-sub 99.1%

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

   2. metadata-eval 99.1%

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

   3. pow-sub 99.3%

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

   4. pow1/2 99.3%

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

   5. associate-*l* 99.3%

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

   6. associate-*l* 99.3%

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

   7. div-inv 99.3%

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

   8. metadata-eval 99.3%

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

3. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. *-commutative 99.3%

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

   7. *-commutative 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-75}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{\left(\pi \cdot \left(n + n\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.35e-75)
   (* (pow k -0.5) (sqrt (* PI (* n 2.0))))
   (pow (/ k (pow (* PI (+ n n)) (- 1.0 k))) -0.5)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.35e-75) {
		tmp = pow(k, -0.5) * sqrt((((double) M_PI) * (n * 2.0)));
	} else {
		tmp = pow((k / pow((((double) M_PI) * (n + n)), (1.0 - k))), -0.5);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.35e-75) {
		tmp = Math.pow(k, -0.5) * Math.sqrt((Math.PI * (n * 2.0)));
	} else {
		tmp = Math.pow((k / Math.pow((Math.PI * (n + n)), (1.0 - k))), -0.5);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.35e-75:
		tmp = math.pow(k, -0.5) * math.sqrt((math.pi * (n * 2.0)))
	else:
		tmp = math.pow((k / math.pow((math.pi * (n + n)), (1.0 - k))), -0.5)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.35e-75)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(pi * Float64(n * 2.0))));
	else
		tmp = Float64(k / (Float64(pi * Float64(n + n)) ^ Float64(1.0 - k))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.35e-75)
		tmp = (k ^ -0.5) * sqrt((pi * (n * 2.0)));
	else
		tmp = (k / ((pi * (n + n)) ^ (1.0 - k))) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.35e-75], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(k / N[Power[N[(Pi * N[(n + n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3499999999999999e-75

   1. Initial program 98.2%

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

      1. *-commutative 98.2%

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

      2. *-commutative 98.2%

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

      3. associate-*r* 98.2%

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

      4. div-inv 98.2%

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

      5. expm1-log1p-u 92.0%

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

      6. expm1-udef 77.4%

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

   3. Applied egg-rr 49.1%

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

      1. expm1-def 63.8%

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

      2. expm1-log1p 67.0%

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

      3. *-commutative 67.0%

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

      4. associate-*r* 67.0%

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

      5. *-commutative 67.0%

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

   5. Simplified 67.0%

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

   6. Taylor expanded in k around 0 67.0%

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

      1. div-inv 67.0%

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

      2. sqrt-prod 98.3%

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

      3. *-commutative 98.3%

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

      4. *-commutative 98.3%

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

      5. associate-*r* 98.3%

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

      6. *-commutative 98.3%

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

      7. inv-pow 98.3%

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

      8. metadata-eval 98.3%

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

      9. pow-prod-up 98.3%

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

      10. sqrt-unprod 97.8%

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

      11. add-sqr-sqrt 98.3%

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

   8. Applied egg-rr 98.3%

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

   if 1.3499999999999999e-75 < 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. expm1-log1p-u 98.8%

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

      2. expm1-udef 65.8%

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

      3. pow1/2 65.8%

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

      4. pow-flip 65.8%

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

      5. metadata-eval 65.8%

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

   3. Applied egg-rr 65.8%

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

      1. expm1-def 98.8%

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

      2. expm1-log1p 99.6%

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

   5. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. *-commutative 99.6%

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

      4. sqrt-pow1 99.6%

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

      5. *-commutative 99.6%

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

      6. *-commutative 99.6%

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

      7. associate-*r* 99.6%

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

      8. add-sqr-sqrt 99.5%

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

      9. sqrt-unprod 99.6%

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

      10. pow-prod-up 99.5%

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

      11. metadata-eval 99.5%

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

      12. inv-pow 99.5%

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

      13. sqrt-prod 99.6%

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

      14. div-inv 99.6%

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

      15. clear-num 99.6%

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

      16. sqrt-div 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. expm1-log1p-u 98.8%

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

      2. expm1-udef 90.7%

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

      3. pow1/2 90.7%

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

      4. pow-flip 90.7%

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

      5. metadata-eval 90.7%

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

   9. Applied egg-rr 90.7%

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

   11. Simplified 99.6%

       \[\leadsto \color{blue}{{\left(\frac{k}{{\left(\pi \cdot \left(n + n\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-75}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{\left(\pi \cdot \left(n + n\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \]

Alternative 3: 99.5% 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 99.1%

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

   1. expm1-log1p-u 96.1%

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

   2. expm1-udef 74.3%

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

   3. pow1/2 74.3%

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

   4. pow-flip 74.3%

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

   5. metadata-eval 74.3%

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

3. Applied egg-rr 74.3%

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

   1. expm1-def 96.1%

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

   2. expm1-log1p 99.2%

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;k \leq 1.35 \cdot 10^{-75}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* n 2.0))))
   (if (<= k 1.35e-75)
     (* (pow k -0.5) (sqrt t_0))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (n * 2.0);
	double tmp;
	if (k <= 1.35e-75) {
		tmp = pow(k, -0.5) * sqrt(t_0);
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (n * 2.0);
	double tmp;
	if (k <= 1.35e-75) {
		tmp = Math.pow(k, -0.5) * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.pi * (n * 2.0)
	tmp = 0
	if k <= 1.35e-75:
		tmp = math.pow(k, -0.5) * math.sqrt(t_0)
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(n * 2.0))
	tmp = 0.0
	if (k <= 1.35e-75)
		tmp = Float64((k ^ -0.5) * sqrt(t_0));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = pi * (n * 2.0);
	tmp = 0.0;
	if (k <= 1.35e-75)
		tmp = (k ^ -0.5) * sqrt(t_0);
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.35e-75], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3499999999999999e-75

   1. Initial program 98.2%

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

      1. *-commutative 98.2%

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

      2. *-commutative 98.2%

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

      3. associate-*r* 98.2%

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

      4. div-inv 98.2%

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

      5. expm1-log1p-u 92.0%

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

      6. expm1-udef 77.4%

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

   3. Applied egg-rr 49.1%

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

      1. expm1-def 63.8%

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

      2. expm1-log1p 67.0%

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

      3. *-commutative 67.0%

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

      4. associate-*r* 67.0%

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

      5. *-commutative 67.0%

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

   5. Simplified 67.0%

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

   6. Taylor expanded in k around 0 67.0%

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

      1. div-inv 67.0%

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

      2. sqrt-prod 98.3%

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

      3. *-commutative 98.3%

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

      4. *-commutative 98.3%

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

      5. associate-*r* 98.3%

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

      6. *-commutative 98.3%

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

      7. inv-pow 98.3%

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

      8. metadata-eval 98.3%

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

      9. pow-prod-up 98.3%

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

      10. sqrt-unprod 97.8%

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

      11. add-sqr-sqrt 98.3%

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

   8. Applied egg-rr 98.3%

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

   if 1.3499999999999999e-75 < 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 98.8%

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

      6. expm1-udef 90.7%

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

   3. Applied egg-rr 90.7%

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

      1. expm1-def 98.8%

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

      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. *-commutative 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-75}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. associate-*l/ 99.1%

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

   2. *-lft-identity 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-*l* 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 6: 49.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{+228}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \frac{\pi}{\frac{k}{n}}\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3e+228)
   (/ (sqrt (* PI (* n 2.0))) (sqrt k))
   (cbrt (pow (* 2.0 (/ PI (/ k n))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3e+228) {
		tmp = sqrt((((double) M_PI) * (n * 2.0))) / sqrt(k);
	} else {
		tmp = cbrt(pow((2.0 * (((double) M_PI) / (k / n))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3e+228) {
		tmp = Math.sqrt((Math.PI * (n * 2.0))) / Math.sqrt(k);
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * (Math.PI / (k / n))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3e+228)
		tmp = Float64(sqrt(Float64(pi * Float64(n * 2.0))) / sqrt(k));
	else
		tmp = cbrt((Float64(2.0 * Float64(pi / Float64(k / n))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3e+228], N[(N[Sqrt[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(Pi / N[(k / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.0000000000000001e228

   1. Initial program 99.0%

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

      1. *-commutative 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*r* 99.0%

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

      4. div-inv 99.0%

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

      5. expm1-log1p-u 96.0%

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

      6. expm1-udef 84.2%

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

   3. Applied egg-rr 73.3%

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

      1. expm1-def 85.1%

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

      2. expm1-log1p 87.0%

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

      3. *-commutative 87.0%

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

      4. associate-*r* 87.0%

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

      5. *-commutative 87.0%

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

   5. Simplified 87.0%

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

   6. Taylor expanded in k around 0 41.4%

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

      1. sqrt-div 53.4%

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

      2. *-commutative 53.4%

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

      3. *-commutative 53.4%

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

      4. associate-*r* 53.4%

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

      5. *-commutative 53.4%

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

   8. Applied egg-rr 53.4%

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

   if 3.0000000000000001e228 < k

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. div-inv 100.0%

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

      5. expm1-log1p-u 100.0%

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

      6. expm1-udef 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in k around 0 2.8%

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

      1. *-un-lft-identity 2.8%

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

      2. times-frac 2.8%

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

      3. metadata-eval 2.8%

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

      4. *-commutative 2.8%

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

   8. Applied egg-rr 2.8%

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

      1. add-cbrt-cube 23.0%

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

      2. add-sqr-sqrt 23.0%

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

      3. pow1 23.0%

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

      4. pow1/2 23.0%

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

      5. pow-prod-up 23.0%

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

      6. associate-/l* 23.0%

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

      7. metadata-eval 23.0%

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

   10. Applied egg-rr 23.0%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(2 \cdot \frac{\pi}{\frac{k}{n}}\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{+228}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \frac{\pi}{\frac{k}{n}}\right)}^{1.5}}\\ \end{array} \]

Alternative 7: 49.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{+229}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \frac{\pi}{\frac{k}{n}}\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 8.2e+229)
   (* (pow k -0.5) (sqrt (* PI (* n 2.0))))
   (cbrt (pow (* 2.0 (/ PI (/ k n))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 8.2e+229) {
		tmp = pow(k, -0.5) * sqrt((((double) M_PI) * (n * 2.0)));
	} else {
		tmp = cbrt(pow((2.0 * (((double) M_PI) / (k / n))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 8.2e+229) {
		tmp = Math.pow(k, -0.5) * Math.sqrt((Math.PI * (n * 2.0)));
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * (Math.PI / (k / n))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 8.2e+229)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(pi * Float64(n * 2.0))));
	else
		tmp = cbrt((Float64(2.0 * Float64(pi / Float64(k / n))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 8.2e+229], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(Pi / N[(k / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.2000000000000003e229

   1. Initial program 99.0%

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

      1. *-commutative 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*r* 99.0%

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

      4. div-inv 99.0%

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

      5. expm1-log1p-u 96.0%

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

      6. expm1-udef 84.2%

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

   3. Applied egg-rr 73.3%

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

      1. expm1-def 85.1%

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

      2. expm1-log1p 87.0%

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

      3. *-commutative 87.0%

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

      4. associate-*r* 87.0%

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

      5. *-commutative 87.0%

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

   5. Simplified 87.0%

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

   6. Taylor expanded in k around 0 41.4%

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

      1. div-inv 41.3%

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

      2. sqrt-prod 53.4%

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

      3. *-commutative 53.4%

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

      4. *-commutative 53.4%

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

      5. associate-*r* 53.4%

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

      6. *-commutative 53.4%

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

      7. inv-pow 53.4%

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

      8. metadata-eval 53.4%

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

      9. pow-prod-up 53.5%

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

      10. sqrt-unprod 53.2%

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

      11. add-sqr-sqrt 53.5%

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

   8. Applied egg-rr 53.5%

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

   if 8.2000000000000003e229 < k

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. div-inv 100.0%

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

      5. expm1-log1p-u 100.0%

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

      6. expm1-udef 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in k around 0 2.8%

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

      1. *-un-lft-identity 2.8%

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

      2. times-frac 2.8%

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

      3. metadata-eval 2.8%

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

      4. *-commutative 2.8%

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

   8. Applied egg-rr 2.8%

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

      1. add-cbrt-cube 23.0%

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

      2. add-sqr-sqrt 23.0%

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

      3. pow1 23.0%

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

      4. pow1/2 23.0%

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

      5. pow-prod-up 23.0%

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

      6. associate-/l* 23.0%

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

      7. metadata-eval 23.0%

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

   10. Applied egg-rr 23.0%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(2 \cdot \frac{\pi}{\frac{k}{n}}\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{+229}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \frac{\pi}{\frac{k}{n}}\right)}^{1.5}}\\ \end{array} \]

Alternative 8: 48.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. *-commutative 99.1%

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

   2. *-commutative 99.1%

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

   3. associate-*r* 99.1%

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

   4. div-inv 99.1%

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

   5. expm1-log1p-u 96.5%

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

   6. expm1-udef 86.2%

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

3. Applied egg-rr 76.7%

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

   1. expm1-def 87.1%

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

   2. expm1-log1p 88.7%

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

   3. *-commutative 88.7%

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

   4. associate-*r* 88.7%

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

   5. *-commutative 88.7%

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

5. Simplified 88.7%

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

6. Taylor expanded in k around 0 36.4%

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

   1. sqrt-div 46.9%

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

   2. *-commutative 46.9%

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

   3. *-commutative 46.9%

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

   4. associate-*r* 46.9%

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

   5. *-commutative 46.9%

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

8. Applied egg-rr 46.9%

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

9. Final simplification 46.9%

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

Alternative 9: 38.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. *-commutative 99.1%

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

   2. *-commutative 99.1%

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

   3. associate-*r* 99.1%

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

   4. div-inv 99.1%

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

   5. expm1-log1p-u 96.5%

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

   6. expm1-udef 86.2%

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

3. Applied egg-rr 76.7%

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

   1. expm1-def 87.1%

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

   2. expm1-log1p 88.7%

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

   3. *-commutative 88.7%

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

   4. associate-*r* 88.7%

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

   5. *-commutative 88.7%

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

5. Simplified 88.7%

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

6. Taylor expanded in k around 0 36.4%

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

   1. clear-num 36.4%

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

   2. sqrt-div 37.0%

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

   3. metadata-eval 37.0%

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

   4. *-commutative 37.0%

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

   5. *-commutative 37.0%

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

   6. associate-*r* 37.0%

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

   7. *-commutative 37.0%

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

8. Applied egg-rr 37.0%

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

   1. associate-/r* 37.0%

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

   2. *-commutative 37.0%

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

   3. associate-/r* 37.0%

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

   4. associate-/r* 37.0%

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

10. Simplified 37.0%

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

    1. expm1-log1p-u 35.4%

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

    2. expm1-udef 36.6%

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

    3. pow1/2 36.6%

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

    4. pow-flip 36.6%

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

    5. div-inv 36.6%

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

    6. metadata-eval 36.6%

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

    7. metadata-eval 36.6%

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

12. Applied egg-rr 36.6%

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

    1. expm1-def 35.4%

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

    2. expm1-log1p 37.1%

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

    3. associate-*l/ 37.1%

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

    4. *-commutative 37.1%

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

    5. times-frac 37.0%

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

14. Simplified 37.0%

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

15. Final simplification 37.0%

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

Alternative 10: 38.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. *-commutative 99.1%

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

   2. *-commutative 99.1%

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

   3. associate-*r* 99.1%

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

   4. div-inv 99.1%

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

   5. expm1-log1p-u 96.5%

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

   6. expm1-udef 86.2%

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

3. Applied egg-rr 76.7%

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

   1. expm1-def 87.1%

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

   2. expm1-log1p 88.7%

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

   3. *-commutative 88.7%

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

   4. associate-*r* 88.7%

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

   5. *-commutative 88.7%

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

5. Simplified 88.7%

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

6. Taylor expanded in k around 0 36.4%

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

   1. clear-num 36.4%

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

   2. sqrt-div 37.0%

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

   3. metadata-eval 37.0%

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

   4. *-commutative 37.0%

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

   5. *-commutative 37.0%

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

   6. associate-*r* 37.0%

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

   7. *-commutative 37.0%

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

8. Applied egg-rr 37.0%

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

   1. associate-/r* 37.0%

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

   2. *-commutative 37.0%

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

   3. associate-/r* 37.0%

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

   4. associate-/r* 37.0%

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

10. Simplified 37.0%

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

    1. pow1/2 37.0%

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

    2. pow-flip 37.1%

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

    3. div-inv 37.1%

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

    4. metadata-eval 37.1%

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

    5. metadata-eval 37.1%

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

12. Applied egg-rr 37.1%

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

13. Final simplification 37.1%

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

Alternative 11: 37.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. *-commutative 99.1%

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

   2. *-commutative 99.1%

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

   3. associate-*r* 99.1%

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

   4. div-inv 99.1%

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

   5. expm1-log1p-u 96.5%

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

   6. expm1-udef 86.2%

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

3. Applied egg-rr 76.7%

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

   1. expm1-def 87.1%

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

   2. expm1-log1p 88.7%

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

   3. *-commutative 88.7%

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

   4. associate-*r* 88.7%

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

   5. *-commutative 88.7%

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

5. Simplified 88.7%

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

6. Taylor expanded in k around 0 36.4%

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

   1. *-un-lft-identity 36.4%

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

   2. times-frac 36.4%

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

   3. metadata-eval 36.4%

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

   4. *-commutative 36.4%

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

8. Applied egg-rr 36.4%

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

9. Final simplification 36.4%

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

Reproduce

herbie shell --seed 2023181 
(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))))