Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 18.7s

Alternatives: 9

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot n\right)\\ \frac{\sqrt{t\_0}}{\sqrt{k \cdot {t\_0}^{k}}} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. associate-*l* 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. div-inv 99.4%

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

   2. div-inv 99.4%

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

   3. metadata-eval 99.4%

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

   4. inv-pow 99.4%

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

   5. sqrt-pow2 99.4%

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

   6. metadata-eval 99.4%

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

6. Applied egg-rr 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*r* 99.4%

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

   3. *-commutative 99.4%

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

   4. cancel-sign-sub-inv 99.4%

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

   5. metadata-eval 99.4%

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

   6. *-commutative 99.4%

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

8. Simplified 99.4%

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

   1. *-commutative 99.4%

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

   2. metadata-eval 99.4%

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

   3. cancel-sign-sub-inv 99.4%

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

   4. associate-*r* 99.4%

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

   5. exp-to-pow 96.5%

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

   6. metadata-eval 96.5%

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

   7. pow-flip 96.5%

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

   8. pow1/2 96.5%

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

   9. div-inv 96.5%

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

   10. exp-to-pow 99.4%

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

   11. associate-*r* 99.4%

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

   12. pow-sub 99.7%

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

   13. pow1/2 99.7%

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

   14. associate-/r* 99.7%

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

10. Applied egg-rr 99.7%

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

11. Final simplification 99.7%

    \[\leadsto \frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}}} \]
12. Add Preprocessing

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. associate-*l* 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in k around inf 96.5%

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

   1. exp-to-pow 99.4%

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

   2. associate-*r* 99.4%

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

   3. pow-sub 99.7%

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

   4. pow1/2 99.7%

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

   5. pow1 99.7%

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

   6. pow-unpow 99.7%

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

   7. pow1/2 99.7%

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

   8. pow-div 99.4%

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

   9. *-commutative 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3e-24)
   (* (sqrt (* 2.0 (/ PI k))) (sqrt n))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3e-24) {
		tmp = sqrt((2.0 * (((double) M_PI) / k))) * sqrt(n);
	} else {
		tmp = sqrt((pow((((double) M_PI) * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3e-24) {
		tmp = Math.sqrt((2.0 * (Math.PI / k))) * Math.sqrt(n);
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3e-24:
		tmp = math.sqrt((2.0 * (math.pi / k))) * math.sqrt(n)
	else:
		tmp = math.sqrt((math.pow((math.pi * (2.0 * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3e-24)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi / k))) * sqrt(n));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(2.0 * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3e-24)
		tmp = sqrt((2.0 * (pi / k))) * sqrt(n);
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3e-24], N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.99999999999999995e-24

   1. Initial program 99.2%

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

   3. Taylor expanded in k around 0 71.8%

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

      1. *-commutative 71.8%

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

      2. associate-/l* 71.7%

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

   5. Simplified 71.7%

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

   7. Applied egg-rr 71.9%

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

      1. unpow1 71.9%

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

      2. associate-/r/ 71.9%

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

      3. associate-*l/ 71.8%

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

      4. associate-/l* 71.9%

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

   9. Simplified 71.9%

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

       1. sqrt-prod 71.7%

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

       2. div-inv 71.7%

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

       3. add-cbrt-cube 50.6%

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

       4. pow1/3 47.7%

          \[\leadsto \color{blue}{{\left(\left(\left(\sqrt{2} \cdot \sqrt{n \cdot \left(\pi \cdot \frac{1}{k}\right)}\right) \cdot \left(\sqrt{2} \cdot \sqrt{n \cdot \left(\pi \cdot \frac{1}{k}\right)}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{n \cdot \left(\pi \cdot \frac{1}{k}\right)}\right)\right)}^{0.3333333333333333}} \]

   11. Applied egg-rr 47.7%

       \[\leadsto \color{blue}{{\left({\left(n \cdot \frac{2}{\frac{k}{\pi}}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   12. Step-by-step derivation

       1. unpow1/3 50.6%

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

   13. Simplified 50.6%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(n \cdot \frac{2}{\frac{k}{\pi}}\right)}^{1.5}}} \]
   14. Step-by-step derivation

       1. pow1/3 47.7%

          \[\leadsto \color{blue}{{\left({\left(n \cdot \frac{2}{\frac{k}{\pi}}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       2. pow-pow 71.8%

          \[\leadsto \color{blue}{{\left(n \cdot \frac{2}{\frac{k}{\pi}}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       3. metadata-eval 71.8%

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

       4. pow1/2 71.8%

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

       5. *-commutative 71.8%

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

       6. sqrt-prod 99.4%

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

       7. div-inv 99.4%

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

       8. clear-num 99.5%

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

   15. Applied egg-rr 99.5%

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

   if 2.99999999999999995e-24 < k

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 99.4%

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

      2. sqrt-unprod 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*r* 99.5%

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

      5. div-sub 99.5%

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

      6. metadata-eval 99.5%

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

      7. div-inv 99.5%

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

      8. *-commutative 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. associate-*l* 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

3. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 5: 50.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 33.8%

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

   1. *-commutative 33.8%

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

   2. associate-/l* 33.8%

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

5. Simplified 33.8%

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

7. Applied egg-rr 33.8%

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

   1. unpow1 33.8%

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

   2. associate-/r/ 33.9%

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

   3. associate-*l/ 33.8%

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

   4. associate-/l* 33.8%

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

9. Simplified 33.8%

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

    1. associate-*r* 33.8%

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

    2. sqrt-prod 45.2%

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

11. Applied egg-rr 45.2%

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

12. Final simplification 45.2%

    \[\leadsto \sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}} \]
13. Add Preprocessing

Alternative 6: 50.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 33.8%

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

   1. *-commutative 33.8%

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

   2. associate-/l* 33.8%

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

5. Simplified 33.8%

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

7. Applied egg-rr 33.8%

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

   1. unpow1 33.8%

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

   2. associate-/r/ 33.9%

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

   3. associate-*l/ 33.8%

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

   4. associate-/l* 33.8%

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

9. Simplified 33.8%

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

    1. sqrt-prod 33.8%

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

    2. div-inv 33.8%

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

    3. add-cbrt-cube 23.9%

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

    4. pow1/3 22.6%

       \[\leadsto \color{blue}{{\left(\left(\left(\sqrt{2} \cdot \sqrt{n \cdot \left(\pi \cdot \frac{1}{k}\right)}\right) \cdot \left(\sqrt{2} \cdot \sqrt{n \cdot \left(\pi \cdot \frac{1}{k}\right)}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{n \cdot \left(\pi \cdot \frac{1}{k}\right)}\right)\right)}^{0.3333333333333333}} \]

11. Applied egg-rr 22.6%

    \[\leadsto \color{blue}{{\left({\left(n \cdot \frac{2}{\frac{k}{\pi}}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
12. Step-by-step derivation

    1. unpow1/3 23.9%

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

13. Simplified 23.9%

    \[\leadsto \color{blue}{\sqrt[3]{{\left(n \cdot \frac{2}{\frac{k}{\pi}}\right)}^{1.5}}} \]
14. Step-by-step derivation

    1. pow1/3 22.6%

       \[\leadsto \color{blue}{{\left({\left(n \cdot \frac{2}{\frac{k}{\pi}}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

    2. pow-pow 33.8%

       \[\leadsto \color{blue}{{\left(n \cdot \frac{2}{\frac{k}{\pi}}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

    3. metadata-eval 33.8%

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

    4. pow1/2 33.8%

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

    5. *-commutative 33.8%

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

    6. sqrt-prod 45.2%

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

    7. div-inv 45.2%

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

    8. clear-num 45.2%

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

15. Applied egg-rr 45.2%

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

16. Final simplification 45.2%

    \[\leadsto \sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n} \]
17. Add Preprocessing

Alternative 7: 37.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 33.8%

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

   1. *-commutative 33.8%

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

   2. associate-/l* 33.8%

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

5. Simplified 33.8%

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

7. Applied egg-rr 33.8%

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

   1. unpow1 33.8%

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

   2. associate-/r/ 33.9%

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

9. Simplified 33.9%

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

    1. pow1/2 33.9%

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

    2. *-commutative 33.9%

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

    3. associate-*r* 33.9%

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

    4. *-commutative 33.9%

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

11. Applied egg-rr 33.9%

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

12. Final simplification 33.9%

    \[\leadsto {\left(\left(\pi \cdot 2\right) \cdot \frac{n}{k}\right)}^{0.5} \]
13. Add Preprocessing

Alternative 8: 37.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 33.8%

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

   1. *-commutative 33.8%

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

   2. associate-/l* 33.8%

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

5. Simplified 33.8%

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

7. Applied egg-rr 33.8%

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

   1. unpow1 33.8%

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

   2. associate-/r/ 33.9%

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

   3. associate-*l/ 33.8%

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

   4. associate-/l* 33.8%

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

9. Simplified 33.8%

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

10. Final simplification 33.8%

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

Alternative 9: 37.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 33.8%

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

   1. *-commutative 33.8%

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

   2. associate-/l* 33.8%

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

5. Simplified 33.8%

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

7. Applied egg-rr 33.8%

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

   1. unpow1 33.8%

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

   2. associate-/r/ 33.9%

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

9. Simplified 33.9%

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

10. Final simplification 33.9%

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

Reproduce

herbie shell --seed 2024073 
(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))))