Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 15.8s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[k, -0.5], $MachinePrecision] / N[Power[N[(Pi * N[(n * 2.0), $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. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r/ 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*r* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

   6. associate-*r* 99.5%

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

   7. *-commutative 99.5%

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

   8. associate-*l* 99.5%

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

   9. sub-neg 99.5%

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

   10. div-inv 99.5%

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

   11. metadata-eval 99.5%

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

   12. distribute-rgt-neg-in 99.5%

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

   13. metadata-eval 99.5%

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

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{k}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 + k \cdot -0.5\right)}}}} \]
5. 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 99.4%

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

   3. unpow-prod-down 99.4%

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

   4. sqrt-pow2 99.5%

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

   5. metadata-eval 99.5%

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

   6. associate-*r* 99.5%

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

   7. *-commutative 99.5%

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

   8. *-commutative 99.5%

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

   9. pow-flip 99.5%

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

   10. *-commutative 99.5%

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

   11. +-commutative 99.5%

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

   12. fma-def 99.5%

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

6. Applied egg-rr 99.5%

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

   1. unpow-1 99.5%

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

   2. associate-*r/ 99.5%

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

   3. *-rgt-identity 99.5%

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

   4. *-commutative 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-*l* 99.5%

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

   7. *-commutative 99.5%

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

   8. neg-sub0 99.5%

      \[\leadsto \frac{{k}^{-0.5}}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\color{blue}{\left(0 - \mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}} \]

   9. metadata-eval 99.5%

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

   10. fma-udef 99.5%

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

   11. *-commutative 99.5%

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

   12. +-commutative 99.5%

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

   13. *-commutative 99.5%

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

   14. associate--r+ 99.5%

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

   15. metadata-eval 99.5%

       \[\leadsto \frac{{k}^{-0.5}}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(\left(\color{blue}{0} - 0.5\right) - k \cdot -0.5\right)}} \]

   16. metadata-eval 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 53.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{+114}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(n \cdot \frac{\pi}{k}\right)}^{3} \cdot 8\right)}^{0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.3e+114)
   (* (pow k -0.5) (sqrt (* n (* PI 2.0))))
   (pow (* (pow (* n (/ PI k)) 3.0) 8.0) 0.16666666666666666)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.3e+114) {
		tmp = pow(k, -0.5) * sqrt((n * (((double) M_PI) * 2.0)));
	} else {
		tmp = pow((pow((n * (((double) M_PI) / k)), 3.0) * 8.0), 0.16666666666666666);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.3e+114) {
		tmp = Math.pow(k, -0.5) * Math.sqrt((n * (Math.PI * 2.0)));
	} else {
		tmp = Math.pow((Math.pow((n * (Math.PI / k)), 3.0) * 8.0), 0.16666666666666666);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.3e+114:
		tmp = math.pow(k, -0.5) * math.sqrt((n * (math.pi * 2.0)))
	else:
		tmp = math.pow((math.pow((n * (math.pi / k)), 3.0) * 8.0), 0.16666666666666666)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.3e+114)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(n * Float64(pi * 2.0))));
	else
		tmp = Float64((Float64(n * Float64(pi / k)) ^ 3.0) * 8.0) ^ 0.16666666666666666;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.3e+114)
		tmp = (k ^ -0.5) * sqrt((n * (pi * 2.0)));
	else
		tmp = (((n * (pi / k)) ^ 3.0) * 8.0) ^ 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.3e+114], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * 8.0), $MachinePrecision], 0.16666666666666666], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3e114

   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 73.1%

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

      1. expm1-log1p-u 70.3%

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

      2. expm1-udef 42.5%

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

      3. sqrt-unprod 42.5%

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

      4. *-commutative 42.5%

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

      5. *-commutative 42.5%

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

      6. associate-*r* 42.5%

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

      7. *-commutative 42.5%

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

      8. *-commutative 42.5%

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

   5. Applied egg-rr 42.5%

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

      1. expm1-def 70.4%

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

      2. expm1-log1p 73.2%

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

   7. Simplified 73.2%

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

      1. expm1-log1p-u 67.7%

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

      2. expm1-udef 77.3%

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

      3. inv-pow 77.3%

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

      4. sqrt-pow2 77.3%

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

      5. metadata-eval 77.3%

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

   9. Applied egg-rr 77.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-0.5}\right)} - 1\right)} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)} \]
   10. Step-by-step derivation

       1. expm1-def 67.7%

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

       2. expm1-log1p 73.2%

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

   11. Simplified 73.2%

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

   if 1.3e114 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.7%

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

      1. expm1-log1p-u 2.7%

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

      2. expm1-udef 33.3%

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

      3. associate-*l/ 33.3%

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

      4. *-un-lft-identity 33.3%

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

      5. sqrt-unprod 33.3%

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

      6. *-commutative 33.3%

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

      7. *-commutative 33.3%

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

      8. sqrt-undiv 33.3%

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

      9. associate-*r* 33.3%

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

      10. *-commutative 33.3%

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

      11. *-commutative 33.3%

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

   5. Applied egg-rr 33.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{n \cdot \left(\pi \cdot 2\right)}{k}}\right)} - 1} \]
   6. 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-*r* 2.6%

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

      4. *-commutative 2.6%

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

      5. *-commutative 2.6%

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

      6. *-commutative 2.6%

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

      7. associate-*r/ 2.6%

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

      8. associate-/l* 2.6%

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

   7. Simplified 2.6%

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

   8. Taylor expanded in n around 0 2.6%

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

      1. associate-*r/ 2.6%

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

   10. Simplified 2.6%

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

   12. Applied egg-rr 20.4%

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

       1. associate-*r* 20.4%

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

       2. cube-prod 20.4%

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

       3. associate-*r/ 20.4%

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

       4. associate-/l* 20.4%

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

       5. associate-/r/ 20.4%

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

       6. *-commutative 20.4%

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

       7. metadata-eval 20.4%

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

   14. Simplified 20.4%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{+114}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(n \cdot \frac{\pi}{k}\right)}^{3} \cdot 8\right)}^{0.16666666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[k, -0.5], $MachinePrecision] * N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(N[(k * -0.5), $MachinePrecision] + 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/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. sqr-pow 99.3%

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

   4. pow-sqr 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-*l* 99.5%

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

   7. associate-*r/ 99.5%

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

   8. *-commutative 99.5%

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

   9. associate-/l* 99.5%

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

   10. metadata-eval 99.5%

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

   11. /-rgt-identity 99.5%

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

   12. div-sub 99.5%

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

   13. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.5%

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

   2. associate-*r* 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

   5. sub-neg 99.5%

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

   6. div-inv 99.5%

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

   7. metadata-eval 99.5%

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

   8. distribute-rgt-neg-in 99.5%

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

   9. metadata-eval 99.5%

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

   10. inv-pow 99.5%

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

   11. sqrt-pow2 99.5%

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

   12. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 - N[(k / 2.0), $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/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. sqr-pow 99.3%

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

   4. pow-sqr 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-*l* 99.5%

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

   7. associate-*r/ 99.5%

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

   8. *-commutative 99.5%

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

   9. associate-/l* 99.5%

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

   10. metadata-eval 99.5%

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

   11. /-rgt-identity 99.5%

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

   12. div-sub 99.5%

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

   13. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 5: 48.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in k around 0 52.5%

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

   1. expm1-log1p-u 50.5%

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

   2. expm1-udef 30.8%

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

   3. sqrt-unprod 30.8%

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

   4. *-commutative 30.8%

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

   5. *-commutative 30.8%

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

   6. associate-*r* 30.8%

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

   7. *-commutative 30.8%

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

   8. *-commutative 30.8%

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

5. Applied egg-rr 30.8%

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

   1. expm1-def 50.5%

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

   2. expm1-log1p 52.5%

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

7. Simplified 52.5%

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

   1. expm1-log1p-u 48.7%

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

   2. expm1-udef 69.0%

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

   3. inv-pow 69.0%

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

   4. sqrt-pow2 69.0%

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

   5. metadata-eval 69.0%

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

9. Applied egg-rr 69.0%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-0.5}\right)} - 1\right)} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)} \]
10. Step-by-step derivation

    1. expm1-def 48.7%

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

    2. expm1-log1p 52.6%

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

11. Simplified 52.6%

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

12. Final simplification 52.6%

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

Alternative 6: 37.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in k around 0 52.5%

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

   1. expm1-log1p-u 49.1%

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

   2. expm1-udef 51.5%

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

   3. associate-*l/ 51.5%

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

   4. *-un-lft-identity 51.5%

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

   5. sqrt-unprod 51.5%

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

   6. *-commutative 51.5%

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

   7. *-commutative 51.5%

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

   8. sqrt-undiv 38.7%

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

   9. associate-*r* 38.7%

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

   10. *-commutative 38.7%

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

   11. *-commutative 38.7%

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

5. Applied egg-rr 38.7%

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

   1. expm1-def 36.4%

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

   2. expm1-log1p 38.4%

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

   3. associate-*r* 38.4%

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

   4. *-commutative 38.4%

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

   5. *-commutative 38.4%

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

   6. *-commutative 38.4%

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

   7. associate-*r/ 38.4%

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

   8. associate-/l* 38.4%

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

7. Simplified 38.4%

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

   1. associate-/r/ 38.4%

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

   2. add-sqr-sqrt 38.3%

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

   3. pow1/2 38.3%

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

   4. pow1/2 38.3%

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

   5. pow-prod-down 22.2%

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

   6. pow2 22.2%

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

   7. *-commutative 22.2%

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

9. Applied egg-rr 22.2%

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

    1. unpow1/2 22.2%

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

    2. unpow2 22.2%

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

    3. rem-sqrt-square 38.4%

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

    4. associate-*r/ 38.4%

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

    5. *-commutative 38.4%

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

    6. associate-*r/ 38.4%

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

11. Simplified 38.4%

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

12. Final simplification 38.4%

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

Alternative 7: 48.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in k around 0 52.5%

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

   1. expm1-log1p-u 49.1%

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

   2. expm1-udef 51.5%

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

   3. associate-*l/ 51.5%

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

   4. *-un-lft-identity 51.5%

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

   5. sqrt-unprod 51.5%

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

   6. *-commutative 51.5%

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

   7. *-commutative 51.5%

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

   8. sqrt-undiv 38.7%

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

   9. associate-*r* 38.7%

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

   10. *-commutative 38.7%

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

   11. *-commutative 38.7%

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

5. Applied egg-rr 38.7%

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

   1. expm1-def 36.4%

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

   2. expm1-log1p 38.4%

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

   3. associate-*r* 38.4%

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

   4. *-commutative 38.4%

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

   5. *-commutative 38.4%

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

   6. *-commutative 38.4%

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

   7. associate-*r/ 38.4%

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

   8. associate-/l* 38.4%

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

7. Simplified 38.4%

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

8. Taylor expanded in n around 0 38.4%

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

   1. associate-*r/ 38.4%

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

10. Simplified 38.4%

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

    1. associate-*r* 38.4%

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

    2. *-commutative 38.4%

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

    3. sqrt-prod 52.5%

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

12. Applied egg-rr 52.5%

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

13. Final simplification 52.5%

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

Alternative 8: 48.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in k around 0 52.5%

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

   1. associate-*l/ 52.4%

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

   2. *-un-lft-identity 52.4%

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

   3. sqrt-unprod 52.5%

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

   4. *-commutative 52.5%

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

   5. *-commutative 52.5%

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

   6. associate-*r* 52.5%

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

   7. *-commutative 52.5%

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

   8. *-commutative 52.5%

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

5. Applied egg-rr 52.5%

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

6. Final simplification 52.5%

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

Alternative 9: 37.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 52.5%

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

   1. expm1-log1p-u 49.1%

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

   2. expm1-udef 51.5%

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

   3. associate-*l/ 51.5%

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

   4. *-un-lft-identity 51.5%

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

   5. sqrt-unprod 51.5%

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

   6. *-commutative 51.5%

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

   7. *-commutative 51.5%

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

   8. sqrt-undiv 38.7%

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

   9. associate-*r* 38.7%

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

   10. *-commutative 38.7%

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

   11. *-commutative 38.7%

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

5. Applied egg-rr 38.7%

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

   1. expm1-def 36.4%

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

   2. expm1-log1p 38.4%

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

   3. associate-*r* 38.4%

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

   4. *-commutative 38.4%

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

   5. *-commutative 38.4%

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

   6. *-commutative 38.4%

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

   7. associate-*r/ 38.4%

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

   8. associate-/l* 38.4%

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

7. Simplified 38.4%

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

8. Taylor expanded in n around 0 38.4%

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

   1. associate-*r/ 38.4%

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

10. Simplified 38.4%

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

11. Final simplification 38.4%

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

Alternative 10: 37.7% accurate, 2.0× speedup

Math

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

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(Float64(pi * 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[(N[(Pi * 2.0), $MachinePrecision] / 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. Add Preprocessing

3. Taylor expanded in k around 0 52.5%

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

   1. expm1-log1p-u 49.1%

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

   2. expm1-udef 51.5%

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

   3. associate-*l/ 51.5%

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

   4. *-un-lft-identity 51.5%

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

   5. sqrt-unprod 51.5%

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

   6. *-commutative 51.5%

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

   7. *-commutative 51.5%

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

   8. sqrt-undiv 38.7%

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

   9. associate-*r* 38.7%

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

   10. *-commutative 38.7%

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

   11. *-commutative 38.7%

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

5. Applied egg-rr 38.7%

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

   1. expm1-def 36.4%

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

   2. expm1-log1p 38.4%

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

   3. *-rgt-identity 38.4%

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

   4. associate-*r/ 38.4%

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

   5. associate-*l* 38.4%

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

   6. associate-*r/ 38.4%

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

   7. *-commutative 38.4%

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

   8. associate-*r* 38.4%

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

   9. *-rgt-identity 38.4%

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

   10. *-commutative 38.4%

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

7. Simplified 38.4%

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

8. Final simplification 38.4%

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

Reproduce

herbie shell --seed 2024027 
(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))))