Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 20.2s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot n\right)\\ {k}^{-0.5} \cdot \frac{\sqrt{t\_0}}{{t\_0}^{\left(k \cdot 0.5\right)}} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*r* 99.5%

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

   2. div-sub 99.5%

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

   3. metadata-eval 99.5%

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

   4. pow-div 99.7%

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

   5. pow1/2 99.7%

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

   6. associate-*r/ 99.7%

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

   7. inv-pow 99.7%

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

   8. sqrt-pow2 99.7%

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

   9. metadata-eval 99.7%

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

   10. div-inv 99.7%

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

   11. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-/l* 99.7%

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

   2. *-rgt-identity 99.7%

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

   3. associate-*l* 99.7%

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

   4. *-lft-identity 99.7%

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

   5. associate-*r* 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.7%

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

   8. associate-*r* 99.7%

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

   9. *-commutative 99.7%

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

   10. associate-*l* 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot n\right)\\ \frac{\sqrt{t\_0}}{\sqrt{k} \cdot {t\_0}^{\left(k \cdot 0.5\right)}} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. associate-*r* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

   6. pow-div 99.6%

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

   7. pow1/2 99.6%

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

   8. associate-/l/ 99.6%

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

   9. div-inv 99.6%

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

   10. metadata-eval 99.6%

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 39.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{+205}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\pi \cdot \frac{n}{k}\right)}^{2} \cdot 4\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.55e+205)
   (sqrt (* 2.0 (* n (/ PI k))))
   (pow (* (pow (* PI (/ n k)) 2.0) 4.0) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.55e+205) {
		tmp = sqrt((2.0 * (n * (((double) M_PI) / k))));
	} else {
		tmp = pow((pow((((double) M_PI) * (n / k)), 2.0) * 4.0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.55e+205) {
		tmp = Math.sqrt((2.0 * (n * (Math.PI / k))));
	} else {
		tmp = Math.pow((Math.pow((Math.PI * (n / k)), 2.0) * 4.0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.55e+205:
		tmp = math.sqrt((2.0 * (n * (math.pi / k))))
	else:
		tmp = math.pow((math.pow((math.pi * (n / k)), 2.0) * 4.0), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.55e+205)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(pi / k))));
	else
		tmp = Float64((Float64(pi * Float64(n / k)) ^ 2.0) * 4.0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.55e+205)
		tmp = sqrt((2.0 * (n * (pi / k))));
	else
		tmp = (((pi * (n / k)) ^ 2.0) * 4.0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.55e+205], N[Sqrt[N[(2.0 * N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[Power[N[(Pi * N[(n / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 4.0), $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.55000000000000009e205

   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 48.5%

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

      1. associate-/l* 48.5%

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

   5. Simplified 48.5%

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

      1. sqrt-unprod 48.7%

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

   7. Applied egg-rr 48.7%

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

   if 1.55000000000000009e205 < 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. Add Preprocessing

   3. Taylor expanded in k around 0 2.9%

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

      1. associate-/l* 2.9%

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

   5. Simplified 2.9%

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

      1. pow1 2.9%

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

      2. sqrt-unprod 2.9%

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

   7. Applied egg-rr 2.9%

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

      1. unpow1 2.9%

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

      2. *-commutative 2.9%

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

      3. associate-*r/ 2.9%

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

      4. *-commutative 2.9%

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

      5. associate-/l* 2.9%

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

   9. Simplified 2.9%

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

       1. pow1/2 2.9%

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

       2. metadata-eval 2.9%

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

       3. pow-prod-up 2.9%

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

       4. pow-prod-down 22.9%

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

       5. *-commutative 22.9%

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

       6. *-commutative 22.9%

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

       7. swap-sqr 22.9%

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

       8. pow2 22.9%

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

       9. associate-*r/ 22.9%

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

       10. *-commutative 22.9%

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

       11. associate-/l* 22.9%

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

       12. metadata-eval 22.9%

           \[\leadsto {\left({\left(n \cdot \frac{\pi}{k}\right)}^{2} \cdot \color{blue}{4}\right)}^{0.25} \]

   11. Applied egg-rr 22.9%

       \[\leadsto \color{blue}{{\left({\left(n \cdot \frac{\pi}{k}\right)}^{2} \cdot 4\right)}^{0.25}} \]
   12. Step-by-step derivation

       1. associate-*r/ 22.9%

          \[\leadsto {\left({\color{blue}{\left(\frac{n \cdot \pi}{k}\right)}}^{2} \cdot 4\right)}^{0.25} \]

       2. *-commutative 22.9%

          \[\leadsto {\left({\left(\frac{\color{blue}{\pi \cdot n}}{k}\right)}^{2} \cdot 4\right)}^{0.25} \]

       3. associate-/l* 22.9%

          \[\leadsto {\left({\color{blue}{\left(\pi \cdot \frac{n}{k}\right)}}^{2} \cdot 4\right)}^{0.25} \]

   13. Simplified 22.9%

       \[\leadsto \color{blue}{{\left({\left(\pi \cdot \frac{n}{k}\right)}^{2} \cdot 4\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{+205}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\pi \cdot \frac{n}{k}\right)}^{2} \cdot 4\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 37.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{+218}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\pi \cdot \left(2 \cdot \frac{n}{k}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.7e+218)
   (sqrt (* 2.0 (* n (/ PI k))))
   (cbrt (pow (* PI (* 2.0 (/ n k))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.7e+218) {
		tmp = sqrt((2.0 * (n * (((double) M_PI) / k))));
	} else {
		tmp = cbrt(pow((((double) M_PI) * (2.0 * (n / k))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.7e+218) {
		tmp = Math.sqrt((2.0 * (n * (Math.PI / k))));
	} else {
		tmp = Math.cbrt(Math.pow((Math.PI * (2.0 * (n / k))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.7e+218)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(pi / k))));
	else
		tmp = cbrt((Float64(pi * Float64(2.0 * Float64(n / k))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.7e+218], N[Sqrt[N[(2.0 * N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[Power[N[(Pi * N[(2.0 * N[(n / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.70000000000000004e218

   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 46.9%

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

      1. associate-/l* 46.9%

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

   5. Simplified 46.9%

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

      1. sqrt-unprod 47.0%

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

   7. Applied egg-rr 47.0%

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

   if 1.70000000000000004e218 < 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. Add Preprocessing

   3. Taylor expanded in k around 0 2.9%

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

      1. associate-/l* 2.9%

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

   5. Simplified 2.9%

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

      1. pow1 2.9%

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

      2. sqrt-unprod 2.9%

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

   7. Applied egg-rr 2.9%

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

      1. unpow1 2.9%

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

      2. *-commutative 2.9%

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

      3. associate-*r/ 2.9%

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

      4. *-commutative 2.9%

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

      5. associate-/l* 2.9%

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

   9. Simplified 2.9%

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

       1. add-cbrt-cube 16.5%

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

       2. pow1/3 16.5%

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

       3. add-sqr-sqrt 16.5%

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

       4. pow1 16.5%

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

       5. pow1/2 16.5%

          \[\leadsto {\left({\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

       6. pow-prod-up 16.5%

          \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

       7. associate-*r* 16.5%

          \[\leadsto {\left({\color{blue}{\left(\left(2 \cdot \pi\right) \cdot \frac{n}{k}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

       8. *-commutative 16.5%

          \[\leadsto {\left({\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot \frac{n}{k}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

       9. associate-*l* 16.5%

          \[\leadsto {\left({\color{blue}{\left(\pi \cdot \left(2 \cdot \frac{n}{k}\right)\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

       10. metadata-eval 16.5%

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

   11. Applied egg-rr 16.5%

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

       1. unpow1/3 16.5%

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

   13. Simplified 16.5%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{+218}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\pi \cdot \left(2 \cdot \frac{n}{k}\right)\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. associate-*l* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.5%

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

   2. div-inv 99.5%

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

   3. metadata-eval 99.5%

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

   4. inv-pow 99.5%

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

   5. sqrt-pow2 99.5%

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

   6. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 6: 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 99.5%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. associate-*l* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 7: 88.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 88.9%

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

   3. *-commutative 88.9%

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

   4. associate-*r* 88.9%

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

   5. div-sub 88.9%

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

   6. metadata-eval 88.9%

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

   7. div-inv 88.9%

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

   8. *-commutative 88.9%

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

4. Applied egg-rr 89.0%

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

6. Simplified 89.0%

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

7. Final simplification 89.0%

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

Alternative 8: 36.8% 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 99.5%

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

3. Taylor expanded in k around 0 40.8%

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

   1. associate-/l* 40.8%

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

5. Simplified 40.8%

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

   1. pow1 40.8%

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

   2. sqrt-unprod 41.0%

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

7. Applied egg-rr 41.0%

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

   1. unpow1 41.0%

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

   2. *-commutative 41.0%

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

   3. associate-*r/ 40.9%

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

   4. *-commutative 40.9%

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

   5. associate-/l* 40.9%

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

9. Simplified 40.9%

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

10. Final simplification 40.9%

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

Alternative 9: 36.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 40.8%

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

   1. associate-/l* 40.8%

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

5. Simplified 40.8%

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

   1. pow1 40.8%

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

   2. sqrt-unprod 41.0%

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

7. Applied egg-rr 41.0%

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

   1. unpow1 41.0%

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

   2. *-commutative 41.0%

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

   3. associate-*r/ 40.9%

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

   4. *-commutative 40.9%

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

   5. associate-/l* 40.9%

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

9. Simplified 40.9%

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

    1. *-un-lft-identity 40.9%

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

    2. *-commutative 40.9%

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

    3. associate-*r* 40.9%

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

    4. *-commutative 40.9%

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

    5. associate-*l* 40.9%

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

11. Applied egg-rr 40.9%

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

    1. *-rgt-identity 40.9%

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

13. Simplified 40.9%

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

14. Final simplification 40.9%

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

Alternative 10: 36.8% 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 99.5%

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

3. Taylor expanded in k around 0 40.8%

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

   1. associate-/l* 40.8%

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

5. Simplified 40.8%

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

   1. sqrt-unprod 41.0%

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

7. Applied egg-rr 41.0%

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

8. Final simplification 41.0%

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

Reproduce

herbie shell --seed 2024095 
(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))))