Migdal et al, Equation (51)

Percentage Accurate: 98.4% → 99.5%

Time: 15.2s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. k = 4.94066e-324
  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: 98.4% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ {k}^{-0.5} \cdot {\left(\left|\pi \cdot \left(n \cdot 2\right)\right|\right)}^{\left(\frac{1 - k}{2}\right)} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (pow k -0.5) (pow (fabs (* PI (* n 2.0))) (/ (- 1.0 k) 2.0))))

C

double code(double k, double n) {
	return pow(k, -0.5) * pow(fabs((((double) M_PI) * (n * 2.0))), ((1.0 - k) / 2.0));
}

Java

public static double code(double k, double n) {
	return Math.pow(k, -0.5) * Math.pow(Math.abs((Math.PI * (n * 2.0))), ((1.0 - k) / 2.0));
}

Python

def code(k, n):
	return math.pow(k, -0.5) * math.pow(math.fabs((math.pi * (n * 2.0))), ((1.0 - k) / 2.0))

Julia

function code(k, n)
	return Float64((k ^ -0.5) * (abs(Float64(pi * Float64(n * 2.0))) ^ Float64(Float64(1.0 - k) / 2.0)))
end

MATLAB

function tmp = code(k, n)
	tmp = (k ^ -0.5) * (abs((pi * (n * 2.0))) ^ ((1.0 - k) / 2.0));
end

Wolfram

code[k_, n_] := N[(N[Power[k, -0.5], $MachinePrecision] * N[Power[N[Abs[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

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

   1. associate-*r* 98.3%

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

   2. add-sqr-sqrt 83.8%

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

   3. sqrt-unprod 81.1%

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

   4. *-commutative 81.1%

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

   5. *-commutative 81.1%

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

   6. swap-sqr 81.1%

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

   7. pow2 81.1%

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

   8. metadata-eval 81.1%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{{\left(\pi \cdot n\right)}^{2} \cdot \color{blue}{4}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

4. Applied egg-rr 81.1%

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

   1. *-commutative 81.1%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{\color{blue}{4 \cdot {\left(\pi \cdot n\right)}^{2}}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. metadata-eval 81.1%

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

   3. unpow2 81.1%

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

   4. swap-sqr 81.1%

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

   5. rem-sqrt-square 99.1%

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

   6. *-commutative 99.1%

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

   7. associate-*l* 99.1%

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

6. Simplified 99.1%

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

   1. *-un-lft-identity 99.1%

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

   2. inv-pow 99.1%

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

   3. sqrt-pow2 99.1%

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

   4. metadata-eval 99.1%

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

8. Applied egg-rr 99.1%

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

   1. *-lft-identity 99.1%

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

10. Simplified 99.1%

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

11. Final simplification 99.1%

    \[\leadsto {k}^{-0.5} \cdot {\left(\left|\pi \cdot \left(n \cdot 2\right)\right|\right)}^{\left(\frac{1 - k}{2}\right)} \]
12. Add Preprocessing

Alternative 2: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;k \leq 2.4 \cdot 10^{-68}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t\_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* n 2.0))))
   (if (<= k 2.4e-68)
     (* (pow k -0.5) (sqrt t_0))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (n * 2.0);
	double tmp;
	if (k <= 2.4e-68) {
		tmp = pow(k, -0.5) * sqrt(t_0);
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (n * 2.0);
	double tmp;
	if (k <= 2.4e-68) {
		tmp = Math.pow(k, -0.5) * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.pi * (n * 2.0)
	tmp = 0
	if k <= 2.4e-68:
		tmp = math.pow(k, -0.5) * math.sqrt(t_0)
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(n * 2.0))
	tmp = 0.0
	if (k <= 2.4e-68)
		tmp = Float64((k ^ -0.5) * sqrt(t_0));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = pi * (n * 2.0);
	tmp = 0.0;
	if (k <= 2.4e-68)
		tmp = (k ^ -0.5) * sqrt(t_0);
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2.4e-68], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[t$95$0], $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 < 2.39999999999999991e-68

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0 64.9%

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

      1. *-commutative 64.9%

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

      2. associate-/l* 65.0%

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

   5. Simplified 65.0%

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

      1. pow1 65.0%

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

      2. sqrt-unprod 64.9%

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

      3. associate-*r* 64.9%

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

      4. *-commutative 64.9%

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

   7. Applied egg-rr 64.9%

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

      1. unpow1 64.9%

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

      2. associate-*r/ 64.8%

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

      3. *-commutative 64.8%

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

      4. associate-/l* 64.9%

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

      5. *-commutative 64.9%

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

   9. Simplified 64.9%

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

       1. associate-*r/ 64.8%

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

       2. *-commutative 64.8%

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

       3. div-inv 64.9%

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

       4. sqrt-unprod 99.5%

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

       5. inv-pow 99.5%

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

       6. sqrt-pow1 99.5%

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

       7. metadata-eval 99.5%

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

       8. *-commutative 99.5%

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

   11. Applied egg-rr 99.5%

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

   if 2.39999999999999991e-68 < k

   1. Initial program 97.9%

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

      1. add-sqr-sqrt 97.3%

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

      2. sqrt-unprod 97.4%

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

      3. *-commutative 97.4%

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

      4. associate-*r* 97.4%

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

      5. div-sub 97.4%

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

      6. metadata-eval 97.4%

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

      7. div-inv 97.4%

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

      8. *-commutative 97.4%

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

   4. Applied egg-rr 97.4%

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

   6. Simplified 97.4%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-68}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.6 \cdot 10^{+94}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \mathsf{fma}\left(n, \pi \cdot \frac{2}{k}, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 9.6e+94)
   (* (pow k -0.5) (sqrt (* PI (* n 2.0))))
   (sqrt (+ 1.0 (fma n (* PI (/ 2.0 k)) -1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 9.6e+94) {
		tmp = pow(k, -0.5) * sqrt((((double) M_PI) * (n * 2.0)));
	} else {
		tmp = sqrt((1.0 + fma(n, (((double) M_PI) * (2.0 / k)), -1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 9.6e+94)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(pi * Float64(n * 2.0))));
	else
		tmp = sqrt(Float64(1.0 + fma(n, Float64(pi * Float64(2.0 / k)), -1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 9.6e+94], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(1.0 + N[(n * N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.5999999999999993e94

   1. Initial program 96.7%

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

   3. Taylor expanded in k around 0 49.4%

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

      1. *-commutative 49.4%

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

      2. associate-/l* 49.4%

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

   5. Simplified 49.4%

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

      1. pow1 49.4%

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

      2. sqrt-unprod 49.4%

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

      3. associate-*r* 49.4%

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

      4. *-commutative 49.4%

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

   7. Applied egg-rr 49.4%

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

      1. unpow1 49.4%

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

      2. associate-*r/ 49.4%

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

      3. *-commutative 49.4%

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

      4. associate-/l* 49.4%

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

      5. *-commutative 49.4%

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

   9. Simplified 49.4%

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

       1. associate-*r/ 49.4%

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

       2. *-commutative 49.4%

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

       3. div-inv 49.4%

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

       4. sqrt-unprod 66.6%

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

       5. inv-pow 66.6%

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

       6. sqrt-pow1 66.6%

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

       7. metadata-eval 66.6%

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

       8. *-commutative 66.6%

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

   11. Applied egg-rr 66.6%

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

   if 9.5999999999999993e94 < 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. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. add-sqr-sqrt 78.4%

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

      3. sqrt-unprod 100.0%

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

      4. *-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. swap-sqr 100.0%

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

      7. pow2 100.0%

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

      8. metadata-eval 100.0%

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{{\left(\pi \cdot n\right)}^{2} \cdot \color{blue}{4}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\sqrt{\color{blue}{4 \cdot {\left(\pi \cdot n\right)}^{2}}}\right)}^{\left(\frac{1 - k}{2}\right)} \]

      2. metadata-eval 100.0%

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

      3. unpow2 100.0%

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

      4. swap-sqr 100.0%

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

      5. rem-sqrt-square 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. inv-pow 100.0%

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

      3. sqrt-pow2 100.0%

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

      4. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in k around 0 2.8%

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

       1. fabs-mul 2.8%

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

       2. metadata-eval 2.8%

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

   13. Simplified 2.8%

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

       1. expm1-log1p-u 2.8%

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

       2. expm1-undefine 33.4%

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

       3. add-sqr-sqrt 33.4%

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

       4. add-sqr-sqrt 33.4%

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

       5. associate-/l* 33.4%

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

       6. add-sqr-sqrt 18.8%

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

       7. fabs-sqr 18.8%

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

       8. add-sqr-sqrt 33.2%

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

       9. associate-/l* 33.2%

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

   15. Applied egg-rr 33.2%

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

       1. log1p-undefine 33.2%

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

       2. rem-exp-log 33.2%

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

       3. associate-+r- 33.2%

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

       4. *-commutative 33.2%

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

       5. associate-*l* 33.2%

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

       6. *-commutative 33.2%

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

       7. associate-*r/ 33.2%

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

       8. *-commutative 33.2%

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

       9. associate-*r/ 33.2%

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

       10. fma-neg 33.2%

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

       11. metadata-eval 33.2%

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

   17. Simplified 33.2%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.6 \cdot 10^{+94}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(n \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \mathsf{fma}\left(n, \pi \cdot \frac{2}{k}, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

   1. associate-*l/ 98.3%

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

   2. *-lft-identity 98.3%

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

   3. associate-*l* 98.3%

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

   4. div-sub 98.3%

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

   5. metadata-eval 98.3%

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

3. Simplified 98.3%

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

   1. div-inv 98.3%

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

   2. div-inv 98.3%

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

   3. metadata-eval 98.3%

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

   4. inv-pow 98.3%

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

   5. sqrt-pow2 98.3%

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

   6. metadata-eval 98.3%

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

6. Applied egg-rr 98.3%

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

7. Final simplification 98.3%

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

Alternative 5: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

   1. associate-*l/ 98.3%

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

   2. *-lft-identity 98.3%

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

   3. associate-*l* 98.3%

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

   4. div-sub 98.3%

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

   5. metadata-eval 98.3%

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

3. Simplified 98.3%

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

5. Final simplification 98.3%

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

Alternative 6: 39.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in k around 0 26.3%

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

   1. *-commutative 26.3%

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

   2. associate-/l* 26.3%

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

5. Simplified 26.3%

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

   1. pow1 26.3%

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

   2. sqrt-unprod 26.3%

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

   3. associate-*r* 26.3%

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

   4. *-commutative 26.3%

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

7. Applied egg-rr 26.3%

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

   1. unpow1 26.3%

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

   2. associate-*r/ 26.3%

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

   3. *-commutative 26.3%

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

   4. associate-/l* 26.3%

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

   5. *-commutative 26.3%

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

9. Simplified 26.3%

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

    1. associate-*r/ 26.3%

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

    2. *-commutative 26.3%

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

    3. div-inv 26.3%

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

    4. sqrt-unprod 35.1%

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

    5. inv-pow 35.1%

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

    6. sqrt-pow1 35.1%

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

    7. metadata-eval 35.1%

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

    8. *-commutative 35.1%

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

11. Applied egg-rr 35.1%

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

12. Final simplification 35.1%

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

Alternative 7: 39.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in k around 0 26.3%

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

   1. *-commutative 26.3%

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

   2. associate-/l* 26.3%

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

5. Simplified 26.3%

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

   1. pow1 26.3%

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

   2. sqrt-unprod 26.3%

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

   3. associate-*r* 26.3%

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

   4. *-commutative 26.3%

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

7. Applied egg-rr 26.3%

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

   1. unpow1 26.3%

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

   2. associate-*r/ 26.3%

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

   3. *-commutative 26.3%

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

   4. associate-/l* 26.3%

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

   5. *-commutative 26.3%

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

9. Simplified 26.3%

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

10. Taylor expanded in n around 0 26.3%

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

    1. div-inv 26.3%

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

    2. associate-*r* 26.3%

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

    3. *-commutative 26.3%

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

    4. associate-*r* 26.3%

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

    5. div-inv 26.3%

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

    6. sqrt-prod 35.1%

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

    7. *-commutative 35.1%

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

12. Applied egg-rr 35.1%

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

13. Final simplification 35.1%

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

Alternative 8: 39.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in k around 0 35.0%

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

   1. associate-*l/ 35.0%

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

   2. *-un-lft-identity 35.0%

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

   3. sqrt-unprod 35.1%

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

   4. *-commutative 35.1%

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

   5. associate-*r* 35.1%

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

5. Applied egg-rr 35.1%

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

6. Final simplification 35.1%

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

Alternative 9: 30.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in k around 0 26.3%

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

   1. *-commutative 26.3%

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

   2. associate-/l* 26.3%

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

5. Simplified 26.3%

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

   1. pow1 26.3%

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

   2. sqrt-unprod 26.3%

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

   3. associate-*r* 26.3%

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

   4. *-commutative 26.3%

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

7. Applied egg-rr 26.3%

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

   1. unpow1 26.3%

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

   2. associate-*r/ 26.3%

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

   3. *-commutative 26.3%

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

   4. associate-/l* 26.3%

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

   5. *-commutative 26.3%

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

9. Simplified 26.3%

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

10. Taylor expanded in n around 0 26.3%

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

    1. sqrt-prod 26.3%

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

    2. add-sqr-sqrt 26.2%

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

    3. fabs-sqr 26.2%

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

    4. add-sqr-sqrt 26.8%

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

    5. sqrt-prod 26.8%

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

    6. associate-/l* 26.8%

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

    7. clear-num 26.7%

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

    8. sqrt-div 27.3%

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

    9. metadata-eval 27.3%

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

    10. *-un-lft-identity 27.3%

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

    11. add-sqr-sqrt 26.8%

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

    12. fabs-sqr 26.8%

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

    13. add-sqr-sqrt 26.8%

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

    14. times-frac 26.8%

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

    15. metadata-eval 26.8%

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

    16. *-commutative 26.8%

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

12. Applied egg-rr 26.8%

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

    1. associate-*r/ 26.8%

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

    2. *-commutative 26.8%

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

    3. associate-/l* 26.8%

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

    4. *-commutative 26.8%

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

14. Simplified 26.8%

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

15. Final simplification 26.8%

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

Alternative 10: 30.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in k around 0 26.3%

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

   1. *-commutative 26.3%

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

   2. associate-/l* 26.3%

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

5. Simplified 26.3%

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

   1. pow1 26.3%

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

   2. sqrt-unprod 26.3%

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

   3. associate-*r* 26.3%

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

   4. *-commutative 26.3%

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

7. Applied egg-rr 26.3%

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

   1. unpow1 26.3%

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

   2. associate-*r/ 26.3%

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

   3. *-commutative 26.3%

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

   4. associate-/l* 26.3%

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

   5. *-commutative 26.3%

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

9. Simplified 26.3%

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

    1. associate-*r/ 26.3%

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

    2. *-commutative 26.3%

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

    3. sqrt-undiv 35.1%

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

    4. clear-num 35.1%

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

    5. sqrt-undiv 26.8%

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

11. Applied egg-rr 26.8%

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

12. Final simplification 26.8%

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

Alternative 11: 30.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in k around 0 26.3%

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

   1. *-commutative 26.3%

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

   2. associate-/l* 26.3%

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

5. Simplified 26.3%

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

   1. pow1 26.3%

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

   2. sqrt-unprod 26.3%

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

   3. associate-*r* 26.3%

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

   4. *-commutative 26.3%

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

7. Applied egg-rr 26.3%

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

   1. unpow1 26.3%

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

   2. associate-*r/ 26.3%

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

   3. *-commutative 26.3%

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

   4. associate-/l* 26.3%

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

   5. *-commutative 26.3%

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

9. Simplified 26.3%

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

10. Taylor expanded in n around 0 26.3%

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

11. Final simplification 26.3%

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

Alternative 12: 30.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in k around 0 26.3%

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

   1. *-commutative 26.3%

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

   2. associate-/l* 26.3%

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

5. Simplified 26.3%

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

   1. pow1 26.3%

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

   2. sqrt-unprod 26.3%

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

   3. associate-*r* 26.3%

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

   4. *-commutative 26.3%

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

7. Applied egg-rr 26.3%

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

   1. unpow1 26.3%

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

   2. associate-*r/ 26.3%

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

   3. *-commutative 26.3%

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

   4. associate-/l* 26.3%

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

   5. *-commutative 26.3%

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

9. Simplified 26.3%

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

10. Final simplification 26.3%

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

Alternative 13: 30.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in k around 0 26.3%

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

   1. *-commutative 26.3%

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

   2. associate-/l* 26.3%

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

5. Simplified 26.3%

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

   1. sqrt-unprod 26.3%

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

   2. associate-*r* 26.3%

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

   3. *-commutative 26.3%

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

7. Applied egg-rr 26.3%

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

8. Final simplification 26.3%

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

Reproduce

herbie shell --seed 2024060 
(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))))