Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 11.9s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. k = 1.3863971241004152e+217
  2. n = -7.170880586110564e-35

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(n * 2.0), $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 99.6%

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

   1. associate-*l/ 99.6%

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

   2. *-un-lft-identity 99.6%

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

   3. associate-*r* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

   6. pow-div 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. div-inv 99.7%

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

   2. associate-*r* 99.7%

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

   3. pow1/2 99.7%

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

   4. pow-unpow 99.7%

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

   5. pow-prod-down 99.7%

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

   6. associate-*r* 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

   3. associate-*l* 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*l* 99.7%

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

   6. unpow1/2 99.7%

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

   7. associate-*l* 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-*l* 99.7%

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

8. Simplified 99.7%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 8.8e-75)
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
   (/ 1.0 (sqrt (/ k (pow (* PI (* n 2.0)) (- 1.0 k)))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 8.8e-75) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = 1.0 / sqrt((k / pow((((double) M_PI) * (n * 2.0)), (1.0 - k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 8.8e-75) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	} else {
		tmp = 1.0 / Math.sqrt((k / Math.pow((Math.PI * (n * 2.0)), (1.0 - k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 8.8e-75:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (math.pi / k)))
	else:
		tmp = 1.0 / math.sqrt((k / math.pow((math.pi * (n * 2.0)), (1.0 - k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 8.8e-75)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = Float64(1.0 / sqrt(Float64(k / (Float64(pi * Float64(n * 2.0)) ^ Float64(1.0 - k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 8.8e-75)
		tmp = sqrt(n) * sqrt((2.0 * (pi / k)));
	else
		tmp = 1.0 / sqrt((k / ((pi * (n * 2.0)) ^ (1.0 - k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 8.8e-75], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(k / N[Power[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.80000000000000022e-75

   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 74.4%

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

      1. associate-/l* 74.4%

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

   5. Simplified 74.4%

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

      1. sqrt-unprod 73.6%

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

   7. Applied egg-rr 73.6%

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

      1. associate-*l* 73.6%

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

      2. sqrt-prod 99.5%

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

   9. Applied egg-rr 99.5%

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

   if 8.80000000000000022e-75 < k

   1. Initial program 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.7%

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

      6. metadata-eval 99.7%

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

      7. neg-mul-1 99.7%

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

      8. sub-neg 99.7%

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

   6. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. sqrt-div 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   10. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot n\right)\\ \mathbf{if}\;k \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{t\_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t\_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* 2.0 (* PI n))))
   (if (<= k 5.6e-17)
     (/ (sqrt t_0) (sqrt k))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = 2.0 * (((double) M_PI) * n);
	double tmp;
	if (k <= 5.6e-17) {
		tmp = sqrt(t_0) / sqrt(k);
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = 2.0 * (Math.PI * n);
	double tmp;
	if (k <= 5.6e-17) {
		tmp = Math.sqrt(t_0) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = 2.0 * (math.pi * n)
	tmp = 0
	if k <= 5.6e-17:
		tmp = math.sqrt(t_0) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(2.0 * Float64(pi * n))
	tmp = 0.0
	if (k <= 5.6e-17)
		tmp = Float64(sqrt(t_0) / sqrt(k));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = 2.0 * (pi * n);
	tmp = 0.0;
	if (k <= 5.6e-17)
		tmp = sqrt(t_0) / sqrt(k);
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 5.6e-17], N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $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 < 5.5999999999999998e-17

   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 79.8%

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

      1. associate-/l* 79.8%

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

   5. Simplified 79.8%

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

      1. sqrt-unprod 79.1%

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

   7. Applied egg-rr 79.1%

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

   8. Taylor expanded in n around 0 79.2%

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

      1. associate-*l/ 79.2%

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

      2. associate-/r/ 79.1%

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

   10. Simplified 79.1%

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

       1. associate-*r/ 79.1%

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

       2. *-commutative 79.1%

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

       3. div-inv 79.2%

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

       4. clear-num 79.1%

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

       5. associate-/l* 79.2%

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

       6. sqrt-div 99.5%

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

       7. associate-*l* 99.5%

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

   12. Applied egg-rr 99.5%

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

       1. associate-*r* 99.5%

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

       2. *-commutative 99.5%

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

       3. associate-*r* 99.5%

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

       4. *-commutative 99.5%

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

   14. Simplified 99.5%

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

   if 5.5999999999999998e-17 < k

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-lft-in 99.8%

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

      3. metadata-eval 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*r* 99.8%

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

      6. metadata-eval 99.8%

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

      7. neg-mul-1 99.8%

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

      8. sub-neg 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
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 99.6%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. associate-*l* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

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

Alternative 5: 50.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in k around 0 42.5%

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

   1. associate-/l* 42.5%

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

5. Simplified 42.5%

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

   1. sqrt-unprod 42.2%

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

7. Applied egg-rr 42.2%

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

8. Taylor expanded in n around 0 42.2%

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

   1. associate-*l/ 42.2%

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

   2. associate-/r/ 42.1%

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

10. Simplified 42.1%

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

    1. associate-*r/ 42.1%

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

    2. *-commutative 42.1%

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

    3. div-inv 42.2%

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

    4. clear-num 42.2%

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

    5. associate-/l* 42.2%

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

    6. sqrt-div 52.4%

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

    7. associate-*l* 52.4%

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

12. Applied egg-rr 52.4%

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

    1. associate-*r* 52.4%

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

    2. *-commutative 52.4%

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

    3. associate-*r* 52.4%

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

    4. *-commutative 52.4%

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

14. Simplified 52.4%

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

15. Final simplification 52.4%

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

Alternative 6: 50.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in k around 0 42.5%

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

   1. associate-/l* 42.5%

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

5. Simplified 42.5%

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

   1. sqrt-unprod 42.2%

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

7. Applied egg-rr 42.2%

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

   1. associate-*l* 42.2%

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

   2. sqrt-prod 52.4%

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

9. Applied egg-rr 52.4%

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

10. Final simplification 52.4%

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

Alternative 7: 50.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in k around 0 42.5%

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

   1. associate-/l* 42.5%

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

5. Simplified 42.5%

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

   1. sqrt-unprod 42.2%

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

7. Applied egg-rr 42.2%

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

   1. pow1/2 42.2%

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

   2. associate-*l* 42.2%

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

   3. unpow-prod-down 52.4%

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

   4. pow1/2 52.4%

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

9. Applied egg-rr 52.4%

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

    1. unpow1/2 52.4%

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

    2. associate-*l/ 52.4%

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

    3. associate-/l* 52.4%

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

11. Simplified 52.4%

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

Alternative 8: 39.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

4. Applied egg-rr 89.4%

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

   1. *-commutative 89.4%

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

   2. distribute-lft-in 89.4%

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

   3. metadata-eval 89.4%

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

   4. *-commutative 89.4%

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

   5. associate-*r* 89.4%

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

   6. metadata-eval 89.4%

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

   7. neg-mul-1 89.4%

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

   8. sub-neg 89.4%

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

6. Simplified 89.4%

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

   1. clear-num 89.4%

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

   2. sqrt-div 89.9%

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

   3. metadata-eval 89.9%

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

   4. *-commutative 89.9%

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

8. Applied egg-rr 89.9%

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

   1. *-commutative 89.9%

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

   2. associate-*r* 89.9%

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

   3. *-commutative 89.9%

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

10. Simplified 89.9%

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

11. Taylor expanded in k around 0 42.6%

    \[\leadsto \frac{1}{\sqrt{\color{blue}{0.5 \cdot \frac{k}{n \cdot \pi}}}} \]
12. Step-by-step derivation

    1. associate-*r/ 42.6%

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

    2. *-commutative 42.6%

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

    3. *-commutative 42.6%

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

    4. times-frac 42.6%

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

13. Simplified 42.6%

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

Alternative 9: 38.6% accurate, 2.0× speedup

Math

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

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(Float64(pi * 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[(N[(Pi * 2.0), $MachinePrecision] * N[(n / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

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. Add Preprocessing

3. Taylor expanded in k around 0 42.5%

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

   1. associate-/l* 42.5%

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

5. Simplified 42.5%

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

   1. pow1 42.5%

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

   2. sqrt-unprod 42.2%

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

7. Applied egg-rr 42.2%

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

   1. unpow1 42.2%

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

   2. *-commutative 42.2%

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

   3. associate-*r/ 42.2%

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

   4. associate-/l* 42.2%

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

   5. *-commutative 42.2%

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

   6. associate-*l* 42.2%

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

   7. associate-/l* 42.2%

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

   8. *-commutative 42.2%

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

9. Simplified 42.2%

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

Alternative 10: 38.6% 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 99.6%

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

3. Taylor expanded in k around 0 42.5%

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

   1. associate-/l* 42.5%

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

5. Simplified 42.5%

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

   1. sqrt-unprod 42.2%

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

7. Applied egg-rr 42.2%

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

8. Final simplification 42.2%

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

Alternative 11: 38.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 42.5%

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

   1. associate-/l* 42.5%

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

5. Simplified 42.5%

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

   1. sqrt-unprod 42.2%

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

7. Applied egg-rr 42.2%

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

8. Taylor expanded in n around 0 42.2%

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

   1. associate-*l/ 42.2%

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

   2. associate-/r/ 42.1%

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

10. Simplified 42.1%

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

Reproduce

herbie shell --seed 2024191 
(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))))