Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 32.8s

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

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

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. 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. Step-by-step derivation

   1. div-inv 99.5%

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

   2. associate-*r* 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-unpow 99.5%

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

   5. pow-prod-down 99.5%

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

   6. associate-*r* 99.5%

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

6. Applied egg-rr 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-*l/ 99.6%

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

   3. *-lft-identity 99.6%

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

   4. *-commutative 99.6%

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

   5. associate-*l* 99.6%

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

   6. unpow1/2 99.6%

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

   7. *-commutative 99.6%

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

   8. associate-*l* 99.6%

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

8. Simplified 99.6%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 8e-23)
   (* (sqrt (* 2.0 (/ PI k))) (sqrt n))
   (sqrt (/ (pow (* 2.0 (* PI n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 8e-23) {
		tmp = sqrt((2.0 * (((double) M_PI) / k))) * sqrt(n);
	} else {
		tmp = sqrt((pow((2.0 * (((double) M_PI) * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 8e-23) {
		tmp = Math.sqrt((2.0 * (Math.PI / k))) * Math.sqrt(n);
	} else {
		tmp = Math.sqrt((Math.pow((2.0 * (Math.PI * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 8e-23:
		tmp = math.sqrt((2.0 * (math.pi / k))) * math.sqrt(n)
	else:
		tmp = math.sqrt((math.pow((2.0 * (math.pi * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 8e-23)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi / k))) * sqrt(n));
	else
		tmp = sqrt(Float64((Float64(2.0 * Float64(pi * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 8e-23)
		tmp = sqrt((2.0 * (pi / k))) * sqrt(n);
	else
		tmp = sqrt((((2.0 * (pi * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 8e-23], N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.99999999999999968e-23

   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 71.0%

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

      1. associate-/l* 71.0%

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

   5. Simplified 71.0%

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

      1. pow1 71.0%

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

      2. sqrt-unprod 71.2%

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

   7. Applied egg-rr 71.2%

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

      1. unpow1 71.2%

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

      2. associate-*l* 71.2%

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

   9. Simplified 71.2%

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

       1. sqrt-prod 99.4%

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

       2. *-commutative 99.4%

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

       3. *-commutative 99.4%

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

   11. Applied egg-rr 99.4%

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

   if 7.99999999999999968e-23 < 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. distribute-lft-in 99.7%

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

      2. metadata-eval 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r* 99.7%

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

      5. metadata-eval 99.7%

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

      6. neg-mul-1 99.7%

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

      7. sub-neg 99.7%

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

      8. *-commutative 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{+219}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(8 \cdot {\left(n \cdot \frac{\pi}{k}\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.3e+219)
   (* (sqrt (* 2.0 (/ PI k))) (sqrt n))
   (pow (* 8.0 (pow (* n (/ PI k)) 3.0)) 0.16666666666666666)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.3e+219) {
		tmp = sqrt((2.0 * (((double) M_PI) / k))) * sqrt(n);
	} else {
		tmp = pow((8.0 * pow((n * (((double) M_PI) / k)), 3.0)), 0.16666666666666666);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.3e+219) {
		tmp = Math.sqrt((2.0 * (Math.PI / k))) * Math.sqrt(n);
	} else {
		tmp = Math.pow((8.0 * Math.pow((n * (Math.PI / k)), 3.0)), 0.16666666666666666);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.3e+219:
		tmp = math.sqrt((2.0 * (math.pi / k))) * math.sqrt(n)
	else:
		tmp = math.pow((8.0 * math.pow((n * (math.pi / k)), 3.0)), 0.16666666666666666)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.3e+219)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi / k))) * sqrt(n));
	else
		tmp = Float64(8.0 * (Float64(n * Float64(pi / k)) ^ 3.0)) ^ 0.16666666666666666;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.3e+219)
		tmp = sqrt((2.0 * (pi / k))) * sqrt(n);
	else
		tmp = (8.0 * ((n * (pi / k)) ^ 3.0)) ^ 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.3e+219], N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Power[N[(8.0 * N[Power[N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 0.16666666666666666], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3e219

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 48.3%

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

      1. associate-/l* 48.3%

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

   5. Simplified 48.3%

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

      1. pow1 48.3%

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

      2. sqrt-unprod 48.4%

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

   7. Applied egg-rr 48.4%

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

      1. unpow1 48.4%

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

      2. associate-*l* 48.4%

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

   9. Simplified 48.4%

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

       1. sqrt-prod 66.2%

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

       2. *-commutative 66.2%

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

       3. *-commutative 66.2%

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

   11. Applied egg-rr 66.2%

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

   if 1.3e219 < 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 2.6%

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

      1. associate-/l* 2.6%

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

   5. Simplified 2.6%

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

      1. pow1 2.6%

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

      2. sqrt-unprod 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. unpow1 2.6%

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

      2. associate-*l* 2.6%

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

   9. Simplified 2.6%

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

       1. associate-*r* 2.6%

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

       2. rem-cbrt-cube 11.6%

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

       3. unpow1/3 11.6%

          \[\leadsto \color{blue}{{\left({\left(\sqrt{\left(n \cdot \frac{\pi}{k}\right) \cdot 2}\right)}^{3}\right)}^{0.3333333333333333}} \]

       4. sqr-pow 11.6%

          \[\leadsto \color{blue}{{\left({\left(\sqrt{\left(n \cdot \frac{\pi}{k}\right) \cdot 2}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left({\left(\sqrt{\left(n \cdot \frac{\pi}{k}\right) \cdot 2}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

       5. pow-prod-down 20.8%

          \[\leadsto \color{blue}{{\left({\left(\sqrt{\left(n \cdot \frac{\pi}{k}\right) \cdot 2}\right)}^{3} \cdot {\left(\sqrt{\left(n \cdot \frac{\pi}{k}\right) \cdot 2}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

       6. pow-prod-down 20.8%

          \[\leadsto {\color{blue}{\left({\left(\sqrt{\left(n \cdot \frac{\pi}{k}\right) \cdot 2} \cdot \sqrt{\left(n \cdot \frac{\pi}{k}\right) \cdot 2}\right)}^{3}\right)}}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       7. add-sqr-sqrt 20.8%

          \[\leadsto {\left({\color{blue}{\left(\left(n \cdot \frac{\pi}{k}\right) \cdot 2\right)}}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       8. *-commutative 20.8%

          \[\leadsto {\left({\color{blue}{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       9. clear-num 20.8%

          \[\leadsto {\left({\left(2 \cdot \left(n \cdot \color{blue}{\frac{1}{\frac{k}{\pi}}}\right)\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       10. un-div-inv 20.8%

           \[\leadsto {\left({\left(2 \cdot \color{blue}{\frac{n}{\frac{k}{\pi}}}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       11. metadata-eval 20.8%

           \[\leadsto {\left({\left(2 \cdot \frac{n}{\frac{k}{\pi}}\right)}^{3}\right)}^{\color{blue}{0.16666666666666666}} \]

   11. Applied egg-rr 20.8%

       \[\leadsto \color{blue}{{\left({\left(2 \cdot \frac{n}{\frac{k}{\pi}}\right)}^{3}\right)}^{0.16666666666666666}} \]
   12. Step-by-step derivation

       1. cube-prod 25.4%

          \[\leadsto {\color{blue}{\left({2}^{3} \cdot {\left(\frac{n}{\frac{k}{\pi}}\right)}^{3}\right)}}^{0.16666666666666666} \]

       2. metadata-eval 25.4%

          \[\leadsto {\left(\color{blue}{8} \cdot {\left(\frac{n}{\frac{k}{\pi}}\right)}^{3}\right)}^{0.16666666666666666} \]

       3. associate-/r/ 25.4%

          \[\leadsto {\left(8 \cdot {\color{blue}{\left(\frac{n}{k} \cdot \pi\right)}}^{3}\right)}^{0.16666666666666666} \]

       4. associate-*l/ 25.4%

          \[\leadsto {\left(8 \cdot {\color{blue}{\left(\frac{n \cdot \pi}{k}\right)}}^{3}\right)}^{0.16666666666666666} \]

       5. associate-/l* 25.4%

          \[\leadsto {\left(8 \cdot {\color{blue}{\left(n \cdot \frac{\pi}{k}\right)}}^{3}\right)}^{0.16666666666666666} \]

   13. Simplified 25.4%

       \[\leadsto \color{blue}{{\left(8 \cdot {\left(n \cdot \frac{\pi}{k}\right)}^{3}\right)}^{0.16666666666666666}} \]
3. Recombined 2 regimes into one program.
4. 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/ 99.5%

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

   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. Add Preprocessing

Alternative 5: 49.0% 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 44.5%

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

   1. associate-/l* 44.6%

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

5. Simplified 44.6%

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

   1. pow1 44.6%

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

   2. sqrt-unprod 44.7%

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

7. Applied egg-rr 44.7%

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

   1. unpow1 44.7%

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

   2. associate-*l* 44.7%

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

9. Simplified 44.7%

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

    1. sqrt-prod 61.0%

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

    2. *-commutative 61.0%

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

    3. *-commutative 61.0%

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

11. Applied egg-rr 61.0%

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

Alternative 6: 38.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-/r/ 99.4%

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

   2. associate-*r* 99.4%

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

   3. div-sub 99.4%

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

   4. metadata-eval 99.4%

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

   5. sub-neg 99.4%

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

   6. div-inv 99.4%

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

   7. metadata-eval 99.4%

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

   8. distribute-rgt-neg-in 99.4%

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

   9. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in k around 0 45.4%

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

   1. associate-*r/ 45.5%

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

   2. *-rgt-identity 45.5%

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

7. Simplified 45.5%

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

   1. inv-pow 45.5%

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

   2. sqrt-undiv 45.5%

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

   3. sqrt-pow2 45.6%

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

   4. div-inv 45.6%

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

   5. associate-/r* 45.6%

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

   6. metadata-eval 45.6%

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

   7. metadata-eval 45.6%

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

9. Applied egg-rr 45.6%

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

10. Final simplification 45.6%

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

Alternative 7: 38.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-/r/ 99.4%

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

   2. associate-*r* 99.4%

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

   3. div-sub 99.4%

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

   4. metadata-eval 99.4%

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

   5. sub-neg 99.4%

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

   6. div-inv 99.4%

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

   7. metadata-eval 99.4%

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

   8. distribute-rgt-neg-in 99.4%

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

   9. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in k around 0 45.4%

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

   1. associate-*r/ 45.5%

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

   2. *-rgt-identity 45.5%

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

7. Simplified 45.5%

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

   1. inv-pow 45.5%

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

   2. sqrt-undiv 45.5%

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

   3. sqrt-pow2 45.6%

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

   4. div-inv 45.6%

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

   5. associate-/r* 45.6%

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

   6. metadata-eval 45.6%

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

   7. metadata-eval 45.6%

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

9. Applied egg-rr 45.6%

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

    1. associate-*l/ 45.6%

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

    2. associate-/l* 45.6%

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

11. Simplified 45.6%

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

Alternative 8: 38.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-/r/ 99.4%

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

   2. associate-*r* 99.4%

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

   3. div-sub 99.4%

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

   4. metadata-eval 99.4%

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

   5. sub-neg 99.4%

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

   6. div-inv 99.4%

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

   7. metadata-eval 99.4%

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

   8. distribute-rgt-neg-in 99.4%

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

   9. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in k around 0 45.4%

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

   1. associate-*r/ 45.5%

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

   2. *-rgt-identity 45.5%

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

7. Simplified 45.5%

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

   1. inv-pow 45.5%

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

   2. sqrt-undiv 45.5%

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

   3. sqrt-pow2 45.6%

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

   4. div-inv 45.6%

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

   5. associate-/r* 45.6%

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

   6. metadata-eval 45.6%

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

   7. metadata-eval 45.6%

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

9. Applied egg-rr 45.6%

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

    1. *-commutative 45.6%

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

    2. associate-/r* 45.6%

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

11. Simplified 45.6%

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

12. Final simplification 45.6%

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

Alternative 9: 37.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(n * N[(2.0 * 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 44.5%

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

   1. associate-/l* 44.6%

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

5. Simplified 44.6%

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

   1. pow1 44.6%

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

   2. sqrt-unprod 44.7%

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

7. Applied egg-rr 44.7%

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

   1. unpow1 44.7%

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

   2. associate-*l* 44.7%

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

9. Simplified 44.7%

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

10. Final simplification 44.7%

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

Alternative 10: 37.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(n * N[(Pi * N[(2.0 / 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. Add Preprocessing

3. Taylor expanded in k around 0 44.5%

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

   1. associate-/l* 44.6%

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

5. Simplified 44.6%

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

   1. pow1 44.6%

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

   2. sqrt-unprod 44.7%

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

7. Applied egg-rr 44.7%

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

   1. unpow1 44.7%

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

   2. associate-*l* 44.7%

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

9. Simplified 44.7%

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

    1. *-commutative 44.7%

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

    2. clear-num 44.6%

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

    3. un-div-inv 44.6%

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

11. Applied egg-rr 44.6%

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

    1. associate-/r/ 44.6%

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

13. Applied egg-rr 44.6%

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

14. Final simplification 44.6%

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

Reproduce

herbie shell --seed 2024143 
(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))))