Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.2%

Time: 12.9s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. k = 1.5656706559262654e+253
  2. n = -4.923311931548317e-13

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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.7e-45)
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
   (sqrt (/ (pow (* 2.0 (* PI n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.7e-45) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = sqrt((pow((2.0 * (((double) M_PI) * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.7e-45) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	} else {
		tmp = Math.sqrt((Math.pow((2.0 * (Math.PI * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.7e-45:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (math.pi / k)))
	else:
		tmp = math.sqrt((math.pow((2.0 * (math.pi * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.7e-45)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = sqrt(Float64((Float64(2.0 * Float64(pi * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.7e-45)
		tmp = sqrt(n) * sqrt((2.0 * (pi / k)));
	else
		tmp = sqrt((((2.0 * (pi * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.7e-45], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.7e-45

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 67.9%

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

      1. *-commutative 67.9%

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

      2. associate-/l* 67.9%

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

   5. Simplified 67.9%

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

      1. pow1 67.9%

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

      2. sqrt-unprod 68.0%

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

   7. Applied egg-rr 68.0%

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

      1. unpow1 68.0%

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

      2. associate-*r* 68.0%

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

      3. *-commutative 68.0%

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

      4. associate-*l* 68.0%

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

   9. Simplified 68.0%

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

       1. sqrt-prod 99.5%

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

       2. clear-num 99.5%

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

       3. un-div-inv 99.5%

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

   11. Applied egg-rr 99.5%

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

       1. associate-/r/ 99.4%

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

       2. associate-*l/ 99.5%

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

       3. associate-/l* 99.5%

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

   13. Simplified 99.5%

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

   if 3.7e-45 < k

   1. Initial program 99.7%

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

   4. Applied egg-rr 99.8%

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

      1. distribute-rgt-in 99.8%

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

      2. metadata-eval 99.8%

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

      3. associate-*l* 99.8%

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

      4. metadata-eval 99.8%

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

      5. *-commutative 99.8%

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

      6. neg-mul-1 99.8%

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

      7. sub-neg 99.8%

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

      8. *-commutative 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{+25}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \mathsf{fma}\left(\pi, n \cdot \frac{2}{k}, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.35e+25)
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
   (sqrt (+ 1.0 (fma PI (* n (/ 2.0 k)) -1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.35e+25) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = sqrt((1.0 + fma(((double) M_PI), (n * (2.0 / k)), -1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.35e+25)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = sqrt(Float64(1.0 + fma(pi, Float64(n * Float64(2.0 / k)), -1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.35e+25], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(1.0 + N[(Pi * N[(n * N[(2.0 / k), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.35e25

   1. Initial program 99.0%

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

   3. Taylor expanded in k around 0 64.3%

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

      1. *-commutative 64.3%

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

      2. associate-/l* 64.2%

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

   5. Simplified 64.2%

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

      1. pow1 64.2%

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

      2. sqrt-unprod 64.4%

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

   7. Applied egg-rr 64.4%

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

      1. unpow1 64.4%

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

      2. associate-*r* 64.4%

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

      3. *-commutative 64.4%

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

      4. associate-*l* 64.4%

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

   9. Simplified 64.4%

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

       1. sqrt-prod 88.7%

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

       2. clear-num 88.7%

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

       3. un-div-inv 88.7%

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

   11. Applied egg-rr 88.7%

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

       1. associate-/r/ 88.6%

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

       2. associate-*l/ 88.7%

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

       3. associate-/l* 88.7%

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

   13. Simplified 88.7%

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

   if 1.35e25 < 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 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

   5. Simplified 1.9%

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

      1. pow1 1.9%

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

      2. sqrt-unprod 1.9%

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

   7. Applied egg-rr 1.9%

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

      1. unpow1 1.9%

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

      2. associate-*r* 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*l* 1.9%

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

   9. Simplified 1.9%

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

   10. Taylor expanded in n around 0 1.9%

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

       1. *-commutative 1.9%

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

       2. associate-*r/ 1.9%

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

   12. Simplified 1.9%

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

       1. associate-*r* 1.9%

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

       2. clear-num 1.9%

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

       3. associate-/r/ 1.9%

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

       4. associate-*l* 1.9%

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

       5. associate-*r* 1.9%

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

       6. expm1-log1p-u 1.9%

          \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \left(\pi \cdot \frac{1}{k}\right)\right) \cdot n\right)\right)}} \]

       7. *-commutative 1.9%

          \[\leadsto \sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{n \cdot \left(2 \cdot \left(\pi \cdot \frac{1}{k}\right)\right)}\right)\right)} \]

       8. expm1-undefine 41.0%

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

       9. *-commutative 41.0%

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

       10. associate-*r* 41.0%

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

       11. *-commutative 41.0%

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

       12. associate-*l* 41.0%

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

       13. div-inv 41.0%

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

       14. associate-*l* 41.0%

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

   14. Applied egg-rr 41.0%

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

       1. log1p-undefine 41.0%

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

       2. rem-exp-log 41.0%

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

       3. associate-+r- 41.0%

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

       4. *-commutative 41.0%

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

       5. associate-*l* 41.0%

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

       6. associate-/l* 41.0%

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

       7. fmm-def 41.0%

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

       8. *-commutative 41.0%

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

       9. associate-/l* 41.0%

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

       10. metadata-eval 41.0%

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

   16. Simplified 41.0%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. associate-*l* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. metadata-eval 99.6%

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

   2. div-sub 99.6%

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

   3. associate-*r* 99.6%

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

   4. div-inv 99.6%

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

   5. associate-*r* 99.6%

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

   6. div-sub 99.6%

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

   7. metadata-eval 99.6%

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

   8. sub-neg 99.6%

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

   9. div-inv 99.6%

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

   10. metadata-eval 99.6%

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

   11. distribute-rgt-neg-in 99.6%

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

   12. metadata-eval 99.6%

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

   13. pow1/2 99.6%

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

   14. pow-flip 99.6%

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

   15. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. associate-*l* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

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

Alternative 5: 40.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 28.9%

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

   1. *-commutative 28.9%

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

   2. associate-/l* 28.9%

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

5. Simplified 28.9%

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

   1. pow1 28.9%

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

   2. sqrt-unprod 29.0%

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

7. Applied egg-rr 29.0%

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

   1. unpow1 29.0%

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

   2. associate-*r* 29.0%

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

   3. *-commutative 29.0%

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

   4. associate-*l* 29.0%

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

9. Simplified 29.0%

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

    1. sqrt-prod 39.5%

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

    2. clear-num 39.5%

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

    3. un-div-inv 39.5%

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

11. Applied egg-rr 39.5%

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

    1. associate-/r/ 39.5%

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

    2. associate-*l/ 39.5%

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

    3. associate-/l* 39.5%

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

13. Simplified 39.5%

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

Alternative 6: 30.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 28.9%

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

   1. *-commutative 28.9%

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

   2. associate-/l* 28.9%

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

5. Simplified 28.9%

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

   1. pow1 28.9%

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

   2. sqrt-unprod 29.0%

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

7. Applied egg-rr 29.0%

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

   1. unpow1 29.0%

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

   2. associate-*r* 29.0%

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

   3. *-commutative 29.0%

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

   4. associate-*l* 29.0%

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

9. Simplified 29.0%

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

10. Taylor expanded in n around 0 29.0%

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

    1. *-commutative 29.0%

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

    2. associate-*r/ 29.0%

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

12. Simplified 29.0%

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

    1. pow1 29.0%

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

    2. metadata-eval 29.0%

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

    3. pow-sqr 29.0%

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

    4. pow-prod-down 23.6%

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

    5. *-commutative 23.6%

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

    6. *-commutative 23.6%

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

    7. swap-sqr 23.6%

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

    8. pow2 23.6%

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

    9. metadata-eval 23.6%

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

14. Applied egg-rr 23.6%

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

    1. unpow1/2 23.6%

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

    2. unpow2 23.6%

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

    3. metadata-eval 23.6%

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

    4. swap-sqr 23.6%

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

    5. *-commutative 23.6%

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

    6. associate-*r* 23.6%

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

    7. *-commutative 23.6%

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

    8. associate-*r* 23.6%

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

    9. rem-sqrt-square 29.6%

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

    10. associate-*l/ 29.5%

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

    11. associate-/l* 29.6%

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

    12. associate-/l* 29.6%

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

16. Simplified 29.6%

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

Alternative 7: 30.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

4. Applied egg-rr 89.1%

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

   1. distribute-rgt-in 89.1%

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

   2. metadata-eval 89.1%

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

   3. associate-*l* 89.1%

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

   4. metadata-eval 89.1%

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

   5. *-commutative 89.1%

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

   6. neg-mul-1 89.1%

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

   7. sub-neg 89.1%

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

   8. *-commutative 89.1%

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

6. Simplified 89.1%

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

   1. clear-num 89.1%

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

   2. sqrt-div 89.7%

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

   3. metadata-eval 89.7%

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

   4. associate-*r* 89.7%

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

   5. *-commutative 89.7%

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

8. Applied egg-rr 89.7%

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

9. Taylor expanded in k around 0 29.6%

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

    1. associate-*r* 29.6%

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

    2. *-commutative 29.6%

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

    3. *-commutative 29.6%

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

11. Simplified 29.6%

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

12. Final simplification 29.6%

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

Alternative 8: 30.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

4. Applied egg-rr 89.1%

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

   1. distribute-rgt-in 89.1%

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

   2. metadata-eval 89.1%

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

   3. associate-*l* 89.1%

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

   4. metadata-eval 89.1%

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

   5. *-commutative 89.1%

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

   6. neg-mul-1 89.1%

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

   7. sub-neg 89.1%

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

   8. *-commutative 89.1%

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

6. Simplified 89.1%

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

   1. clear-num 89.1%

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

   2. sqrt-div 89.7%

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

   3. metadata-eval 89.7%

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

   4. associate-*r* 89.7%

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

   5. *-commutative 89.7%

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

8. Applied egg-rr 89.7%

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

9. Taylor expanded in k around 0 29.6%

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

    1. associate-*r/ 29.6%

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

    2. *-commutative 29.6%

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

    3. associate-*r/ 29.5%

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

    4. associate-/r* 29.6%

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

11. Simplified 29.6%

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

Alternative 9: 30.1% accurate, 2.0× speedup

Math

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

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 28.9%

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

   1. *-commutative 28.9%

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

   2. associate-/l* 28.9%

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

5. Simplified 28.9%

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

   1. pow1 28.9%

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

   2. sqrt-unprod 29.0%

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

7. Applied egg-rr 29.0%

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

   1. unpow1 29.0%

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

   2. associate-*r* 29.0%

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

   3. *-commutative 29.0%

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

   4. associate-*l* 29.0%

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

9. Simplified 29.0%

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

    1. associate-*r* 29.0%

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

    2. *-commutative 29.0%

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

    3. clear-num 29.0%

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

    4. un-div-inv 29.0%

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

11. Applied egg-rr 29.0%

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

Alternative 10: 30.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 28.9%

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

   1. *-commutative 28.9%

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

   2. associate-/l* 28.9%

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

5. Simplified 28.9%

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

   1. pow1 28.9%

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

   2. sqrt-unprod 29.0%

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

7. Applied egg-rr 29.0%

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

   1. unpow1 29.0%

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

   2. associate-*r* 29.0%

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

   3. *-commutative 29.0%

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

   4. associate-*l* 29.0%

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

9. Simplified 29.0%

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

10. Taylor expanded in n around 0 29.0%

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

    1. *-commutative 29.0%

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

    2. associate-*r/ 29.0%

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

12. Simplified 29.0%

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

13. Taylor expanded in n around 0 29.0%

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

    1. associate-*r/ 29.0%

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

    2. *-commutative 29.0%

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

    3. associate-*r* 29.0%

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

    4. associate-/l* 29.0%

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

    5. associate-/l* 29.0%

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

15. Simplified 29.0%

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

Alternative 11: 30.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 28.9%

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

   1. *-commutative 28.9%

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

   2. associate-/l* 28.9%

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

5. Simplified 28.9%

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

   1. pow1 28.9%

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

   2. sqrt-unprod 29.0%

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

7. Applied egg-rr 29.0%

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

   1. unpow1 29.0%

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

   2. associate-*r* 29.0%

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

   3. *-commutative 29.0%

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

   4. associate-*l* 29.0%

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

9. Simplified 29.0%

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

10. Taylor expanded in n around 0 29.0%

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

    1. *-commutative 29.0%

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

    2. associate-*r/ 29.0%

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

12. Simplified 29.0%

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

Reproduce

herbie shell --seed 2024172 
(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))))