Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 22.2s

Alternatives: 15

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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;k \leq 10^{-24}:\\ \;\;\;\;\frac{\sqrt{t_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* n (* 2.0 PI))))
   (if (<= k 1e-24) (/ (sqrt t_0) (sqrt k)) (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = n * (2.0 * ((double) M_PI));
	double tmp;
	if (k <= 1e-24) {
		tmp = sqrt(t_0) / sqrt(k);
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = n * (2.0 * Math.PI);
	double tmp;
	if (k <= 1e-24) {
		tmp = Math.sqrt(t_0) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = n * (2.0 * math.pi)
	tmp = 0
	if k <= 1e-24:
		tmp = math.sqrt(t_0) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(n * Float64(2.0 * pi))
	tmp = 0.0
	if (k <= 1e-24)
		tmp = Float64(sqrt(t_0) / sqrt(k));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = n * (2.0 * pi);
	tmp = 0.0;
	if (k <= 1e-24)
		tmp = sqrt(t_0) / sqrt(k);
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1e-24], N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 9.99999999999999924e-25

   1. Initial program 99.2%

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

      1. associate-*l/ 99.4%

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

      2. *-lft-identity 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in k around 0 99.3%

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

      1. sqrt-unprod 99.4%

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

      2. associate-*r* 99.4%

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

   6. Applied egg-rr 99.4%

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

   if 9.99999999999999924e-25 < 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. Step-by-step derivation

      1. associate-/r/ 99.7%

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

      2. associate-*l* 99.7%

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

      3. div-sub 99.7%

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

      4. metadata-eval 99.7%

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

      5. div-inv 99.7%

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

      6. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. clear-num 99.7%

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

      2. add-sqr-sqrt 99.7%

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

      3. add-sqr-sqrt 99.7%

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

      4. expm1-log1p 99.4%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left(\pi \cdot n\right)\right)\right)\right)}}^{\left(0.5 - k \cdot 0.5\right)}}{\sqrt{k}} \]

      5. metadata-eval 99.4%

         \[\leadsto \frac{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left(\pi \cdot n\right)\right)\right)\right)}^{\left(\color{blue}{\frac{1}{2}} - k \cdot 0.5\right)}}{\sqrt{k}} \]

      6. metadata-eval 99.4%

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

      7. div-inv 99.4%

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

      8. div-sub 99.4%

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

      9. add-sqr-sqrt 99.4%

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

      10. sqrt-unprod 99.4%

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

   5. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.7%

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

      6. metadata-eval 99.7%

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

      7. mul-1-neg 99.7%

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

      8. sub-neg 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-24}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-/r/ 99.6%

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

   2. associate-*l* 99.5%

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

   3. div-sub 99.5%

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

   4. metadata-eval 99.5%

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

   5. div-inv 99.5%

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

   6. metadata-eval 99.5%

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

3. Applied egg-rr 99.5%

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

   1. inv-pow 99.5%

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

   2. div-inv 99.5%

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

   3. metadata-eval 99.5%

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

   4. unpow-prod-down 99.5%

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

   5. pow-flip 99.5%

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

   6. pow1 99.5%

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

   7. pow1/2 99.5%

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

   8. pow-flip 99.5%

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

   9. metadata-eval 99.5%

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

   10. pow-flip 99.5%

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

   11. associate-*r* 99.5%

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

   12. *-commutative 99.5%

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

   13. *-commutative 99.5%

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

   14. sub-neg 99.5%

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

   15. distribute-rgt-neg-in 99.5%

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

   16. metadata-eval 99.5%

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

   17. metadata-eval 99.5%

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

5. Applied egg-rr 99.5%

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

   1. unpow-1 99.5%

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

   2. associate-*r/ 99.6%

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

   3. *-rgt-identity 99.6%

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

   4. *-commutative 99.6%

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

   5. distribute-neg-in 99.6%

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

   6. metadata-eval 99.6%

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

   7. distribute-rgt-neg-in 99.6%

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

   8. metadata-eval 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 4: 54.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.42 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.42e+131)
   (/ (sqrt (* n (* 2.0 PI))) (sqrt k))
   (pow (pow (* n (* PI (/ 2.0 k))) 3.0) 0.16666666666666666)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.42e+131) {
		tmp = sqrt((n * (2.0 * ((double) M_PI)))) / sqrt(k);
	} else {
		tmp = pow(pow((n * (((double) M_PI) * (2.0 / k))), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.42e+131) {
		tmp = Math.sqrt((n * (2.0 * Math.PI))) / Math.sqrt(k);
	} else {
		tmp = Math.pow(Math.pow((n * (Math.PI * (2.0 / k))), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.42e+131:
		tmp = math.sqrt((n * (2.0 * math.pi))) / math.sqrt(k)
	else:
		tmp = math.pow(math.pow((n * (math.pi * (2.0 / k))), 3.0), 0.16666666666666666)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.42e+131)
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * pi))) / sqrt(k));
	else
		tmp = (Float64(n * Float64(pi * Float64(2.0 / k))) ^ 3.0) ^ 0.16666666666666666;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.42e+131)
		tmp = sqrt((n * (2.0 * pi))) / sqrt(k);
	else
		tmp = ((n * (pi * (2.0 / k))) ^ 3.0) ^ 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.42e+131], N[(N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(n * N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.42e131

   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. Step-by-step derivation

      1. associate-*l/ 99.4%

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

      2. *-lft-identity 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in k around 0 68.3%

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

      1. sqrt-unprod 68.4%

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

      2. associate-*r* 68.4%

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

   6. Applied egg-rr 68.4%

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

   if 1.42e131 < 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. Step-by-step derivation

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 2.7%

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

      1. expm1-log1p-u 2.7%

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

      2. expm1-udef 26.1%

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

   6. Applied egg-rr 26.1%

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

      1. expm1-def 2.6%

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

      2. expm1-log1p 2.6%

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

      3. associate-/l* 2.6%

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

      4. *-commutative 2.6%

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

      5. associate-/r* 2.6%

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

   8. Simplified 2.6%

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

   9. Taylor expanded in n around 0 2.6%

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

       1. *-commutative 2.6%

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

       2. *-rgt-identity 2.6%

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

       3. associate-*r/ 2.6%

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

       4. associate-*r/ 2.6%

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

       5. associate-*r/ 2.6%

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

       6. *-rgt-identity 2.6%

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

   11. Simplified 2.6%

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

       1. pow1/2 2.6%

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

       2. associate-*r* 2.6%

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

       3. associate-*r/ 2.6%

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

       4. *-commutative 2.6%

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

       5. associate-*r/ 2.6%

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

       6. *-commutative 2.6%

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

       7. associate-*r/ 2.6%

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

       8. metadata-eval 2.6%

          \[\leadsto {\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       9. pow-pow 5.2%

          \[\leadsto \color{blue}{{\left({\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       10. sqr-pow 5.2%

           \[\leadsto \color{blue}{{\left({\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{1.5}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left({\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{1.5}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

       11. pow-prod-down 13.5%

           \[\leadsto \color{blue}{{\left({\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{1.5} \cdot {\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{1.5}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

   13. Applied egg-rr 13.5%

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

       1. associate-*r* 13.5%

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

       2. associate-*r/ 13.5%

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

       3. associate-*l/ 13.5%

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

       4. *-commutative 13.5%

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

       5. associate-*l/ 13.5%

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

       6. associate-*r* 13.5%

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

       7. *-commutative 13.5%

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

       8. *-commutative 13.5%

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

   15. Simplified 13.5%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.42 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \]

Alternative 5: 49.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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. Simplified 99.6%

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

4. Taylor expanded in k around 0 50.6%

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

   1. expm1-log1p-u 47.8%

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

   2. expm1-udef 42.3%

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

6. Applied egg-rr 31.5%

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

   1. expm1-def 37.0%

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

   2. expm1-log1p 38.6%

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

   3. associate-/l* 38.7%

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

   4. *-commutative 38.7%

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

   5. associate-/r* 38.7%

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

8. Simplified 38.7%

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

   1. div-inv 38.2%

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

   2. sqrt-prod 49.9%

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

   3. div-inv 49.9%

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

   4. metadata-eval 49.9%

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

   5. clear-num 49.9%

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

   6. div-inv 49.9%

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

   7. metadata-eval 49.9%

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

   8. div-inv 49.9%

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

   9. clear-num 49.9%

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

10. Applied egg-rr 49.9%

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

11. Final simplification 49.9%

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

Alternative 6: 49.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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. Simplified 99.6%

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

4. Taylor expanded in k around 0 50.6%

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

   1. expm1-log1p-u 47.8%

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

   2. expm1-udef 42.3%

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

6. Applied egg-rr 31.5%

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

   1. expm1-def 37.0%

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

   2. expm1-log1p 38.6%

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

   3. associate-/l* 38.7%

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

   4. *-commutative 38.7%

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

   5. associate-/r* 38.7%

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

8. Simplified 38.7%

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

9. Taylor expanded in n around 0 38.6%

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

    1. *-commutative 38.6%

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

    2. *-rgt-identity 38.6%

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

    3. associate-*r/ 38.6%

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

    4. associate-*r/ 38.6%

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

    5. associate-*r/ 38.6%

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

    6. *-rgt-identity 38.6%

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

11. Simplified 38.6%

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

    1. associate-*r/ 38.6%

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

    2. associate-*r/ 38.6%

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

    3. *-commutative 38.6%

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

    4. associate-*r/ 38.6%

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

    5. sqrt-prod 50.6%

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

13. Applied egg-rr 50.6%

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

14. Final simplification 50.6%

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

Alternative 7: 49.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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. Simplified 99.6%

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

4. Taylor expanded in k around 0 50.6%

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

   1. expm1-log1p-u 47.8%

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

   2. expm1-udef 42.3%

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

6. Applied egg-rr 31.5%

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

   1. expm1-def 37.0%

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

   2. expm1-log1p 38.6%

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

   3. associate-/l* 38.7%

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

   4. *-commutative 38.7%

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

   5. associate-/r* 38.7%

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

8. Simplified 38.7%

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

   1. sqrt-div 50.6%

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

   2. associate-/r* 50.6%

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

10. Applied egg-rr 50.6%

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

11. Final simplification 50.6%

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

Alternative 8: 49.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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. Simplified 99.6%

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

4. Taylor expanded in k around 0 50.6%

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

   1. associate-/l* 50.6%

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

   2. div-inv 50.5%

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

   3. *-commutative 50.5%

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

   4. sqrt-undiv 50.6%

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

   5. div-inv 50.6%

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

   6. metadata-eval 50.6%

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

6. Applied egg-rr 50.6%

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

   1. associate-*r/ 50.6%

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

   2. *-rgt-identity 50.6%

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

   3. *-commutative 50.6%

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

8. Simplified 50.6%

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

9. Final simplification 50.6%

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

Alternative 9: 49.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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. Simplified 99.6%

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

4. Taylor expanded in k around 0 50.6%

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

   1. sqrt-unprod 50.7%

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

   2. associate-*r* 50.7%

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

6. Applied egg-rr 50.7%

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

7. Final simplification 50.7%

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

Alternative 10: 38.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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. Simplified 99.6%

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

4. Taylor expanded in k around 0 50.6%

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

   1. expm1-log1p-u 47.8%

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

   2. expm1-udef 42.3%

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

6. Applied egg-rr 31.5%

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

   1. expm1-def 37.0%

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

   2. expm1-log1p 38.6%

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

   3. associate-/l* 38.7%

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

   4. *-commutative 38.7%

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

   5. associate-/r* 38.7%

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

8. Simplified 38.7%

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

9. Taylor expanded in n around 0 38.6%

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

    1. *-commutative 38.6%

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

    2. *-rgt-identity 38.6%

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

    3. associate-*r/ 38.6%

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

    4. associate-*r/ 38.6%

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

    5. associate-*r/ 38.6%

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

    6. *-rgt-identity 38.6%

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

11. Simplified 38.6%

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

    1. associate-*r* 38.6%

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

    2. associate-*r/ 38.6%

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

    3. *-commutative 38.6%

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

    4. associate-/l* 38.7%

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

    5. sqrt-undiv 50.6%

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

    6. clear-num 50.6%

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

    7. sqrt-undiv 38.9%

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

    8. associate-/r* 38.9%

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

    9. *-commutative 38.9%

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

    10. pow1/2 38.9%

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

    11. pow-flip 38.9%

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

    12. associate-*r* 38.9%

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

    13. *-commutative 38.9%

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

    14. *-commutative 38.9%

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

    15. metadata-eval 38.9%

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

13. Applied egg-rr 38.9%

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

    1. /-rgt-identity 38.9%

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

    2. *-commutative 38.9%

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

    3. associate-*r* 38.9%

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

    4. associate-/l* 38.9%

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

    5. metadata-eval 38.9%

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

    6. associate-/l* 38.9%

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

    7. associate-*r/ 38.9%

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

    8. *-commutative 38.9%

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

15. Simplified 38.9%

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

16. Final simplification 38.9%

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

Alternative 11: 38.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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. Simplified 99.6%

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

4. Taylor expanded in k around 0 50.6%

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

   1. expm1-log1p-u 47.8%

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

   2. expm1-udef 42.3%

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

6. Applied egg-rr 31.5%

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

   1. expm1-def 37.0%

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

   2. expm1-log1p 38.6%

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

   3. associate-/l* 38.7%

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

   4. *-commutative 38.7%

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

   5. associate-/r* 38.7%

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

8. Simplified 38.7%

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

9. Taylor expanded in n around 0 38.6%

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

    1. *-commutative 38.6%

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

    2. *-rgt-identity 38.6%

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

    3. associate-*r/ 38.6%

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

    4. associate-*r/ 38.6%

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

    5. associate-*r/ 38.6%

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

    6. *-rgt-identity 38.6%

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

11. Simplified 38.6%

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

    1. associate-*r* 38.6%

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

    2. associate-*r/ 38.6%

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

    3. *-commutative 38.6%

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

    4. associate-/l* 38.7%

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

    5. sqrt-undiv 50.6%

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

    6. clear-num 50.6%

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

    7. sqrt-undiv 38.9%

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

    8. associate-/r* 38.9%

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

    9. *-commutative 38.9%

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

    10. pow1/2 38.9%

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

    11. pow-flip 38.9%

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

    12. associate-*r* 38.9%

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

    13. *-commutative 38.9%

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

    14. *-commutative 38.9%

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

    15. metadata-eval 38.9%

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

13. Applied egg-rr 38.9%

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

    1. *-commutative 38.9%

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

15. Simplified 38.9%

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

16. Final simplification 38.9%

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

Alternative 12: 37.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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. Simplified 99.6%

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

4. Taylor expanded in k around 0 50.6%

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

   1. expm1-log1p-u 47.8%

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

   2. expm1-udef 42.3%

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

6. Applied egg-rr 31.5%

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

   1. expm1-def 37.0%

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

   2. expm1-log1p 38.6%

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

   3. associate-/l* 38.7%

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

   4. *-commutative 38.7%

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

   5. associate-/r* 38.7%

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

8. Simplified 38.7%

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

9. Taylor expanded in n around 0 38.6%

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

    1. *-commutative 38.6%

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

    2. *-rgt-identity 38.6%

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

    3. associate-*r/ 38.6%

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

    4. associate-*r/ 38.6%

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

    5. associate-*r/ 38.6%

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

    6. *-rgt-identity 38.6%

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

11. Simplified 38.6%

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

12. Taylor expanded in n around 0 38.6%

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

    1. *-commutative 38.6%

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

    2. associate-*l/ 38.6%

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

    3. *-commutative 38.6%

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

14. Simplified 38.6%

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

15. Final simplification 38.6%

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

Alternative 13: 37.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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. Simplified 99.6%

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

4. Taylor expanded in k around 0 50.6%

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

   1. expm1-log1p-u 47.8%

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

   2. expm1-udef 42.3%

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

6. Applied egg-rr 31.5%

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

   1. expm1-def 37.0%

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

   2. expm1-log1p 38.6%

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

   3. associate-/l* 38.7%

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

   4. *-commutative 38.7%

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

   5. associate-/r* 38.7%

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

8. Simplified 38.7%

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

9. Taylor expanded in n around 0 38.6%

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

    1. *-commutative 38.6%

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

    2. *-rgt-identity 38.6%

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

    3. associate-*r/ 38.6%

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

    4. associate-*r/ 38.6%

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

    5. associate-*r/ 38.6%

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

    6. *-rgt-identity 38.6%

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

11. Simplified 38.6%

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

    1. clear-num 38.6%

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

    2. un-div-inv 38.6%

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

13. Applied egg-rr 38.6%

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

14. Final simplification 38.6%

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

Alternative 14: 37.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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. Simplified 99.6%

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

4. Taylor expanded in k around 0 50.6%

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

   1. expm1-log1p-u 47.8%

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

   2. expm1-udef 42.3%

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

6. Applied egg-rr 31.5%

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

   1. expm1-def 37.0%

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

   2. expm1-log1p 38.6%

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

   3. associate-/l* 38.7%

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

   4. *-commutative 38.7%

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

   5. associate-/r* 38.7%

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

8. Simplified 38.7%

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

9. Taylor expanded in k around 0 38.7%

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

    1. associate-*r/ 38.7%

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

    2. *-commutative 38.7%

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

    3. associate-*r/ 38.7%

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

11. Simplified 38.7%

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

12. Final simplification 38.7%

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

Alternative 15: 37.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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. Simplified 99.6%

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

4. Taylor expanded in k around 0 50.6%

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

   1. expm1-log1p-u 47.8%

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

   2. expm1-udef 42.3%

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

6. Applied egg-rr 31.5%

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

   1. expm1-def 37.0%

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

   2. expm1-log1p 38.6%

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

   3. associate-/l* 38.7%

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

   4. *-commutative 38.7%

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

   5. associate-/r* 38.7%

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

8. Simplified 38.7%

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

9. Final simplification 38.7%

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

Reproduce

herbie shell --seed 2023336 
(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))))