Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.1%

Time: 19.6s

Alternatives: 13

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: 99.5% 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: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.9e-59)
   (* (sqrt (* 2.0 (/ PI k))) (sqrt n))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.9e-59) {
		tmp = sqrt((2.0 * (((double) M_PI) / k))) * sqrt(n);
	} else {
		tmp = sqrt((pow((((double) M_PI) * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.9e-59) {
		tmp = Math.sqrt((2.0 * (Math.PI / k))) * Math.sqrt(n);
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.9e-59:
		tmp = math.sqrt((2.0 * (math.pi / k))) * math.sqrt(n)
	else:
		tmp = math.sqrt((math.pow((math.pi * (2.0 * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.9e-59)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi / k))) * sqrt(n));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(2.0 * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.9e-59)
		tmp = sqrt((2.0 * (pi / k))) * sqrt(n);
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.9e-59], N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.90000000000000016e-59

   1. Initial program 98.0%

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

   3. Taylor expanded in k around 0 65.4%

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

      1. *-commutative 65.4%

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

      2. associate-/l* 65.4%

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

   5. Simplified 65.4%

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

      1. sqrt-unprod 65.6%

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

   7. Applied egg-rr 65.6%

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

      1. *-commutative 65.6%

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

      2. associate-*r* 65.6%

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

      3. sqrt-prod 99.4%

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

   9. Applied egg-rr 99.4%

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

   if 2.90000000000000016e-59 < k

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 99.5%

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

      2. sqrt-unprod 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*r* 99.5%

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

      5. div-sub 99.5%

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

      6. metadata-eval 99.5%

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

      7. div-inv 99.5%

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

      8. *-commutative 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.9 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. associate-*r* 98.8%

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

   2. div-sub 98.8%

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

   3. metadata-eval 98.8%

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

   4. pow-div 98.8%

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

   5. pow1/2 98.8%

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

   6. associate-*r/ 98.8%

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

   7. inv-pow 98.8%

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

   8. sqrt-pow2 98.9%

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

   9. metadata-eval 98.9%

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

   10. div-inv 98.9%

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

   11. metadata-eval 98.9%

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

4. Applied egg-rr 98.9%

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

Alternative 3: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(2.0 * Pi), $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 98.8%

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

   1. associate-*r* 98.8%

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

   2. div-sub 98.8%

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

   3. metadata-eval 98.8%

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

   4. sub-neg 98.8%

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

   5. unpow-prod-up 98.8%

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

   6. pow1/2 98.8%

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

   7. div-inv 98.8%

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

   8. metadata-eval 98.8%

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

   9. distribute-rgt-neg-in 98.8%

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

   10. metadata-eval 98.8%

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

4. Applied egg-rr 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*l* 98.8%

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

   3. *-commutative 98.8%

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

   4. associate-*l* 98.8%

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

6. Simplified 98.8%

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

8. Applied egg-rr 98.9%

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

9. Final simplification 98.9%

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

Alternative 4: 60.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(\pi, \frac{n}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.45e+22)
   (* (sqrt (* 2.0 (/ PI k))) (sqrt n))
   (sqrt (* 2.0 (+ -1.0 (fma PI (/ n k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.45e+22) {
		tmp = sqrt((2.0 * (((double) M_PI) / k))) * sqrt(n);
	} else {
		tmp = sqrt((2.0 * (-1.0 + fma(((double) M_PI), (n / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.45e+22)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi / k))) * sqrt(n));
	else
		tmp = sqrt(Float64(2.0 * Float64(-1.0 + fma(pi, Float64(n / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.45e+22], N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(-1.0 + N[(Pi * N[(n / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.45e22

   1. Initial program 98.0%

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

   3. Taylor expanded in k around 0 64.6%

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

      1. *-commutative 64.6%

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

      2. associate-/l* 64.6%

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

   5. Simplified 64.6%

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

      1. sqrt-unprod 64.8%

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

   7. Applied egg-rr 64.8%

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

      1. *-commutative 64.8%

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

      2. associate-*r* 64.8%

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

      3. sqrt-prod 92.5%

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

   9. Applied egg-rr 92.5%

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

   if 1.45e22 < 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. *-commutative 2.6%

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

      2. associate-/l* 2.6%

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

   5. Simplified 2.6%

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

      1. sqrt-unprod 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. expm1-log1p-u 2.6%

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

      2. expm1-undefine 21.8%

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

      3. clear-num 21.8%

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

      4. div-inv 21.8%

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

   9. Applied egg-rr 21.8%

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

       1. sub-neg 21.8%

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

       2. metadata-eval 21.8%

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

       3. +-commutative 21.8%

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

       4. log1p-undefine 21.8%

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

       5. rem-exp-log 21.8%

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

       6. +-commutative 21.8%

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

       7. associate-/r/ 21.8%

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

       8. *-commutative 21.8%

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

       9. fma-define 21.8%

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

   11. Simplified 21.8%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(-1 + \mathsf{fma}\left(\pi, \frac{n}{k}, 1\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. associate-*l/ 98.8%

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

   2. *-lft-identity 98.8%

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

   3. associate-*l* 98.8%

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

   4. div-sub 98.8%

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

   5. metadata-eval 98.8%

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

3. Simplified 98.8%

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

   1. div-inv 98.8%

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

   2. div-inv 98.8%

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

   3. metadata-eval 98.8%

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

   4. inv-pow 98.8%

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

   5. sqrt-pow2 98.8%

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

   6. metadata-eval 98.8%

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

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. associate-*l/ 98.8%

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

   2. *-lft-identity 98.8%

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

   3. associate-*l* 98.8%

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

   4. div-sub 98.8%

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

   5. metadata-eval 98.8%

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

3. Simplified 98.8%

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

Alternative 7: 49.6% 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 98.8%

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

3. Taylor expanded in k around 0 40.6%

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

   1. *-commutative 40.6%

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

   2. associate-/l* 40.6%

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

5. Simplified 40.6%

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

   1. sqrt-unprod 40.7%

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

7. Applied egg-rr 40.7%

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

   1. *-commutative 40.7%

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

   2. associate-*r* 40.7%

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

   3. sqrt-prod 57.8%

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

9. Applied egg-rr 57.8%

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

Alternative 8: 49.6% 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 98.8%

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

3. Taylor expanded in k around 0 40.6%

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

   1. *-commutative 40.6%

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

   2. associate-/l* 40.6%

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

5. Simplified 40.6%

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

   1. sqrt-unprod 40.7%

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

7. Applied egg-rr 40.7%

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

   1. *-commutative 40.7%

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

   2. associate-*r* 40.7%

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

   3. *-commutative 40.7%

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

   4. sqrt-prod 57.8%

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

9. Applied egg-rr 57.8%

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

    1. associate-*r/ 57.8%

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

    2. *-commutative 57.8%

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

    3. associate-/l* 57.7%

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

11. Simplified 57.7%

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

Alternative 9: 38.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. associate-/r/ 98.7%

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

   2. associate-*r* 98.7%

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

   3. div-sub 98.7%

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

   4. metadata-eval 98.7%

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

   5. div-inv 98.7%

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

   6. metadata-eval 98.7%

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

4. Applied egg-rr 98.7%

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

5. Taylor expanded in k around 0 41.3%

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

   1. associate-*r/ 41.4%

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

   2. *-rgt-identity 41.4%

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

   3. *-commutative 41.4%

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

   4. associate-/r* 41.4%

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

7. Simplified 41.4%

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

   1. inv-pow 41.4%

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

   2. add-sqr-sqrt 41.2%

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

   3. sqrt-unprod 41.3%

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

   4. sqrt-pow2 41.3%

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

   5. sqrt-undiv 41.4%

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

   6. sqrt-undiv 41.4%

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

   7. add-sqr-sqrt 41.5%

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

   8. div-inv 41.5%

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

   9. associate-/r* 41.5%

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

   10. metadata-eval 41.5%

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

   11. metadata-eval 41.5%

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

9. Applied egg-rr 41.5%

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

    1. associate-*l/ 41.5%

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

    2. *-commutative 41.5%

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

11. Simplified 41.5%

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

12. Final simplification 41.5%

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

Alternative 10: 38.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. associate-/r/ 98.7%

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

   2. associate-*r* 98.7%

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

   3. div-sub 98.7%

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

   4. metadata-eval 98.7%

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

   5. div-inv 98.7%

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

   6. metadata-eval 98.7%

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

4. Applied egg-rr 98.7%

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

5. Taylor expanded in k around 0 41.3%

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

   1. associate-*r/ 41.4%

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

   2. *-rgt-identity 41.4%

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

   3. *-commutative 41.4%

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

   4. associate-/r* 41.4%

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

7. Simplified 41.4%

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

   1. inv-pow 41.4%

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

   2. add-sqr-sqrt 41.2%

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

   3. sqrt-unprod 41.3%

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

   4. sqrt-pow2 41.3%

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

   5. sqrt-undiv 41.4%

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

   6. sqrt-undiv 41.4%

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

   7. add-sqr-sqrt 41.5%

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

   8. div-inv 41.5%

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

   9. associate-/r* 41.5%

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

   10. metadata-eval 41.5%

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

   11. metadata-eval 41.5%

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

9. Applied egg-rr 41.5%

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

    1. associate-*l/ 41.5%

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

    2. times-frac 41.4%

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

11. Simplified 41.4%

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

Alternative 11: 38.7% 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 98.8%

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

3. Taylor expanded in k around 0 40.6%

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

   1. *-commutative 40.6%

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

   2. associate-/l* 40.6%

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

5. Simplified 40.6%

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

   1. sqrt-unprod 40.7%

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

7. Applied egg-rr 40.7%

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

   1. *-commutative 40.7%

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

   2. associate-*r* 40.7%

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

   3. *-commutative 40.7%

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

   4. *-un-lft-identity 40.7%

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

   5. *-commutative 40.7%

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

   6. associate-*r* 40.7%

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

   7. *-commutative 40.7%

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

   8. clear-num 40.7%

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

   9. div-inv 40.7%

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

9. Applied egg-rr 40.7%

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

    1. *-lft-identity 40.7%

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

    2. associate-/r/ 40.7%

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

11. Simplified 40.7%

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

    1. associate-/r/ 40.7%

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

    2. associate-*r/ 40.7%

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

    3. *-commutative 40.7%

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

    4. clear-num 40.7%

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

    5. metadata-eval 40.7%

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

    6. un-div-inv 40.7%

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

    7. metadata-eval 40.7%

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

    8. div-inv 40.7%

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

    9. clear-num 40.7%

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

    10. add-sqr-sqrt 40.6%

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

    11. frac-times 40.7%

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

    12. sqrt-unprod 41.2%

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

    13. add-sqr-sqrt 41.4%

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

    14. inv-pow 41.4%

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

    15. sqrt-pow2 41.4%

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

13. Applied egg-rr 41.4%

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

Alternative 12: 38.1% 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 98.8%

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

3. Taylor expanded in k around 0 40.6%

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

   1. *-commutative 40.6%

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

   2. associate-/l* 40.6%

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

5. Simplified 40.6%

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

   1. sqrt-unprod 40.7%

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

7. Applied egg-rr 40.7%

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

   1. clear-num 40.7%

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

   2. div-inv 40.7%

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

9. Applied egg-rr 40.7%

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

Alternative 13: 38.1% 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 98.8%

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

3. Taylor expanded in k around 0 40.6%

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

   1. *-commutative 40.6%

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

   2. associate-/l* 40.6%

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

5. Simplified 40.6%

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

   1. sqrt-unprod 40.7%

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

7. Applied egg-rr 40.7%

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

Reproduce

herbie shell --seed 2024088 
(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))))