Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 19.2s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. associate-*l/ 98.7%

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

   2. *-lft-identity 98.7%

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

   3. associate-*l* 98.7%

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

   4. div-sub 98.7%

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

   5. metadata-eval 98.7%

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

3. Simplified 98.7%

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

   1. div-inv 98.6%

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

   2. div-inv 98.6%

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

   3. metadata-eval 98.6%

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

   4. inv-pow 98.6%

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

   5. sqrt-pow2 98.7%

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

   6. metadata-eval 98.7%

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

6. Applied egg-rr 98.7%

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

7. Final simplification 98.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.15e-79)
   (* (sqrt (* 2.0 n)) (sqrt (/ PI k)))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.15e-79) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	} 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 <= 1.15e-79) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((Math.PI / k));
	} 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 <= 1.15e-79:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((math.pi / k))
	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 <= 1.15e-79)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	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 <= 1.15e-79)
		tmp = sqrt((2.0 * n)) * sqrt((pi / k));
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.15e-79], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $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 < 1.15000000000000006e-79

   1. Initial program 98.3%

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

   3. Taylor expanded in k around 0 67.6%

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

      1. *-commutative 67.6%

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

      2. associate-/l* 67.6%

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

   5. Simplified 67.6%

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

      1. pow1 67.6%

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

      2. sqrt-unprod 67.7%

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

      3. associate-*r/ 67.7%

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

      4. *-commutative 67.7%

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

      5. associate-/l* 67.8%

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

   7. Applied egg-rr 67.8%

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

      1. unpow1 67.8%

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

      2. *-commutative 67.8%

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

      3. associate-*l* 67.8%

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

   9. Simplified 67.8%

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

   10. Taylor expanded in n around 0 67.7%

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

       1. associate-/l* 67.7%

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

   12. Simplified 67.7%

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

       1. associate-*r* 67.7%

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

       2. *-commutative 67.7%

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

       3. sqrt-prod 98.4%

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

   14. Applied egg-rr 98.4%

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

   if 1.15000000000000006e-79 < k

   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. add-sqr-sqrt 98.7%

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

      2. sqrt-unprod 98.2%

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

      3. *-commutative 98.2%

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

      4. associate-*r* 98.2%

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

      5. div-sub 98.2%

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

      6. metadata-eval 98.2%

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

      7. div-inv 98.2%

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

      8. *-commutative 98.2%

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

   4. Applied egg-rr 98.3%

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

   6. Simplified 98.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \mathsf{fma}\left(n, \frac{2 \cdot \pi}{k}, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.8e+112)
   (* (sqrt (* 2.0 n)) (sqrt (/ PI k)))
   (sqrt (+ 1.0 (fma n (/ (* 2.0 PI) k) -1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.8e+112) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	} else {
		tmp = sqrt((1.0 + fma(n, ((2.0 * ((double) M_PI)) / k), -1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.8e+112)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	else
		tmp = sqrt(Float64(1.0 + fma(n, Float64(Float64(2.0 * pi) / k), -1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.8e+112], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(1.0 + N[(n * N[(N[(2.0 * Pi), $MachinePrecision] / k), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.80000000000000008e112

   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 56.6%

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

      1. *-commutative 56.6%

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

      2. associate-/l* 56.6%

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

   5. Simplified 56.6%

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

      1. pow1 56.6%

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

      2. sqrt-unprod 56.7%

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

      3. associate-*r/ 56.7%

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

      4. *-commutative 56.7%

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

      5. associate-/l* 56.7%

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

   7. Applied egg-rr 56.7%

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

      1. unpow1 56.7%

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

      2. *-commutative 56.7%

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

      3. associate-*l* 56.7%

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

   9. Simplified 56.7%

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

   10. Taylor expanded in n around 0 56.7%

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

       1. associate-/l* 56.7%

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

   12. Simplified 56.7%

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

       1. associate-*r* 56.7%

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

       2. *-commutative 56.7%

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

       3. sqrt-prod 72.3%

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

   14. Applied egg-rr 72.3%

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

   if 3.80000000000000008e112 < 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. pow1 2.6%

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

      2. sqrt-unprod 2.6%

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

      3. associate-*r/ 2.6%

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

      4. *-commutative 2.6%

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

      5. associate-/l* 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. unpow1 2.6%

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

      2. *-commutative 2.6%

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

      3. associate-*l* 2.6%

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

   9. Simplified 2.6%

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

   10. Taylor expanded in n around 0 2.6%

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

       1. associate-/l* 2.6%

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

   12. Simplified 2.6%

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

       1. associate-*r* 2.6%

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

       2. *-commutative 2.6%

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

       3. div-inv 2.6%

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

       4. associate-*r* 2.6%

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

       5. associate-*r* 2.6%

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

       6. *-commutative 2.6%

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

       7. associate-*r* 2.6%

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

       8. *-commutative 2.6%

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

       9. div-inv 2.6%

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

       10. associate-*r* 2.6%

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

       11. expm1-log1p-u 2.6%

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

       12. expm1-undefine 29.6%

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

   14. Applied egg-rr 29.6%

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

       1. log1p-undefine 29.6%

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

       2. rem-exp-log 29.6%

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

       3. associate-+r- 29.6%

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

       4. associate-*l* 29.6%

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

       5. fma-neg 29.6%

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

       6. associate-*r/ 29.6%

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

       7. metadata-eval 29.6%

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

   16. Simplified 29.6%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \mathsf{fma}\left(n, \frac{2 \cdot \pi}{k}, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. associate-*l/ 98.7%

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

   2. *-lft-identity 98.7%

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

   3. associate-*l* 98.7%

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

   4. div-sub 98.7%

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

   5. metadata-eval 98.7%

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

3. Simplified 98.7%

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

5. Final simplification 98.7%

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

Alternative 5: 49.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in k around 0 40.2%

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

   1. *-commutative 40.2%

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

   2. associate-/l* 40.1%

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

5. Simplified 40.1%

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

   1. pow1 40.1%

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

   2. sqrt-unprod 40.2%

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

   3. associate-*r/ 40.2%

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

   4. *-commutative 40.2%

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

   5. associate-/l* 40.3%

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

7. Applied egg-rr 40.3%

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

   1. unpow1 40.3%

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

   2. *-commutative 40.3%

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

   3. associate-*l* 40.3%

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

9. Simplified 40.3%

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

10. Taylor expanded in n around 0 40.2%

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

    1. associate-/l* 40.2%

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

12. Simplified 40.2%

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

    1. associate-*r* 40.2%

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

    2. *-commutative 40.2%

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

    3. sqrt-prod 51.1%

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

14. Applied egg-rr 51.1%

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

15. Final simplification 51.1%

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

Alternative 6: 38.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. associate-/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 40.7%

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

   1. associate-*r/ 40.7%

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

   2. *-rgt-identity 40.7%

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

   3. *-commutative 40.7%

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

   4. associate-/r* 40.9%

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

7. Simplified 40.9%

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

   1. inv-pow 40.9%

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

   2. sqrt-undiv 40.9%

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

   3. sqrt-pow2 40.9%

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

   4. associate-/l/ 40.8%

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

   5. metadata-eval 40.8%

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

9. Applied egg-rr 40.8%

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

    1. associate-/r* 40.8%

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

    2. *-commutative 40.8%

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

    3. associate-*l* 40.8%

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

11. Simplified 40.8%

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

12. Final simplification 40.8%

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

Alternative 7: 38.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

   1. associate-/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 40.7%

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

   1. associate-*r/ 40.7%

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

   2. *-rgt-identity 40.7%

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

   3. *-commutative 40.7%

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

   4. associate-/r* 40.9%

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

7. Simplified 40.9%

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

   1. inv-pow 40.9%

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

   2. sqrt-undiv 40.9%

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

   3. sqrt-pow2 40.9%

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

   4. associate-/l/ 40.8%

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

   5. metadata-eval 40.8%

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

9. Applied egg-rr 40.8%

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

    1. associate-/r* 40.9%

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

11. Simplified 40.9%

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

12. Final simplification 40.9%

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

Alternative 8: 37.5% 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 98.6%

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

3. Taylor expanded in k around 0 40.2%

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

   1. *-commutative 40.2%

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

   2. associate-/l* 40.1%

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

5. Simplified 40.1%

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

   1. pow1 40.1%

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

   2. sqrt-unprod 40.2%

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

   3. associate-*r/ 40.2%

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

   4. *-commutative 40.2%

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

   5. associate-/l* 40.3%

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

7. Applied egg-rr 40.3%

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

   1. unpow1 40.3%

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

   2. *-commutative 40.3%

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

   3. associate-*l* 40.3%

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

9. Simplified 40.3%

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

10. Taylor expanded in n around 0 40.2%

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

    1. associate-/l* 40.2%

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

12. Simplified 40.2%

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

13. Final simplification 40.2%

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

Alternative 9: 37.5% 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 98.6%

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

3. Taylor expanded in k around 0 40.2%

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

   1. *-commutative 40.2%

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

   2. associate-/l* 40.1%

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

5. Simplified 40.1%

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

   1. pow1 40.1%

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

   2. sqrt-unprod 40.2%

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

   3. associate-*r/ 40.2%

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

   4. *-commutative 40.2%

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

   5. associate-/l* 40.3%

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

7. Applied egg-rr 40.3%

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

   1. unpow1 40.3%

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

   2. *-commutative 40.3%

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

   3. associate-*l* 40.3%

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

9. Simplified 40.3%

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

10. Taylor expanded in n around 0 40.2%

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

    1. associate-/l* 40.2%

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

12. Simplified 40.2%

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

    1. clear-num 40.2%

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

    2. un-div-inv 40.2%

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

14. Applied egg-rr 40.2%

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

15. Final simplification 40.2%

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

Alternative 10: 37.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in k around 0 40.2%

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

   1. *-commutative 40.2%

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

   2. associate-/l* 40.1%

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

5. Simplified 40.1%

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

   1. pow1 40.1%

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

   2. sqrt-unprod 40.2%

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

   3. associate-*r/ 40.2%

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

   4. *-commutative 40.2%

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

   5. associate-/l* 40.3%

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

7. Applied egg-rr 40.3%

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

   1. unpow1 40.3%

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

   2. *-commutative 40.3%

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

   3. associate-*l* 40.3%

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

9. Simplified 40.3%

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

10. Taylor expanded in n around 0 40.2%

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

    1. associate-*r/ 40.2%

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

    2. *-commutative 40.2%

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

    3. associate-/l* 40.2%

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

    4. *-commutative 40.2%

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

    5. associate-*r* 40.3%

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

12. Simplified 40.3%

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

13. Final simplification 40.3%

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

Reproduce

herbie shell --seed 2024071 
(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))))