Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 15.6s

Alternatives: 8

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot n\right)\\ \frac{\sqrt{t\_0}}{\sqrt{k} \cdot {t\_0}^{\left(k \cdot 0.5\right)}} \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* 2.0 n))))
   (/ (sqrt t_0) (* (sqrt k) (pow t_0 (* k 0.5))))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (2.0 * n);
	return sqrt(t_0) / (sqrt(k) * pow(t_0, (k * 0.5)));
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (2.0 * n);
	return Math.sqrt(t_0) / (Math.sqrt(k) * Math.pow(t_0, (k * 0.5)));
}

Python

def code(k, n):
	t_0 = math.pi * (2.0 * n)
	return math.sqrt(t_0) / (math.sqrt(k) * math.pow(t_0, (k * 0.5)))

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(2.0 * n))
	return Float64(sqrt(t_0) / Float64(sqrt(k) * (t_0 ^ Float64(k * 0.5))))
end

MATLAB

function tmp = code(k, n)
	t_0 = pi * (2.0 * n);
	tmp = sqrt(t_0) / (sqrt(k) * (t_0 ^ (k * 0.5)));
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, N[(N[Sqrt[t$95$0], $MachinePrecision] / N[(N[Sqrt[k], $MachinePrecision] * N[Power[t$95$0, 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-*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. *-un-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. *-commutative 99.5%

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

   4. associate-*r* 99.5%

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

   5. div-sub 99.5%

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

   6. metadata-eval 99.5%

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

   7. pow-sub 99.7%

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

   8. pow1/2 99.7%

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

   9. associate-/l/ 99.7%

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

   10. associate-*r* 99.7%

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

   11. *-commutative 99.7%

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

   12. associate-*l* 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*r* 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-*l* 99.7%

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

   4. associate-*r* 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l* 99.7%

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

   7. *-commutative 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] * N[Power[k, -0.5], $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-*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. *-un-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. *-commutative 99.5%

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

   4. associate-*r* 99.5%

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

   5. div-sub 99.5%

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

   6. metadata-eval 99.5%

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

   7. pow-sub 99.7%

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

   8. pow1/2 99.7%

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

   9. associate-/l/ 99.7%

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

   10. associate-*r* 99.7%

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

   11. *-commutative 99.7%

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

   12. associate-*l* 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*r* 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-*l* 99.7%

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

   4. associate-*r* 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l* 99.7%

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

   7. *-commutative 99.7%

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

6. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. frac-times 99.7%

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

   3. metadata-eval 99.7%

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

   4. sqrt-div 99.7%

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

   5. pow1 99.7%

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

   6. pow-unpow 99.7%

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

   7. pow1/2 99.7%

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

   8. pow-sub 99.5%

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

   9. *-commutative 99.5%

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

   10. sqrt-div 99.5%

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

   11. metadata-eval 99.5%

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

   12. inv-pow 99.5%

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

   13. sqrt-pow2 99.6%

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

   14. metadata-eval 99.6%

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

   15. metadata-eval 99.6%

       \[\leadsto {\left(\sqrt{\pi \cdot \left(2 \cdot n\right)}\right)}^{\left(1 - k\right)} \cdot {k}^{\color{blue}{\left(0.25 \cdot -2\right)}} \]

   16. pow-pow 99.3%

       \[\leadsto {\left(\sqrt{\pi \cdot \left(2 \cdot n\right)}\right)}^{\left(1 - k\right)} \cdot \color{blue}{{\left({k}^{0.25}\right)}^{-2}} \]

8. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. *-commutative 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2e-49)
   (* (sqrt (* 2.0 (/ PI k))) (sqrt n))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2e-49) {
		tmp = sqrt((2.0 * (((double) M_PI) / k))) * sqrt(n);
	} else {
		tmp = sqrt((pow((((double) M_PI) * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2e-49) {
		tmp = Math.sqrt((2.0 * (Math.PI / k))) * Math.sqrt(n);
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2e-49:
		tmp = math.sqrt((2.0 * (math.pi / k))) * math.sqrt(n)
	else:
		tmp = math.sqrt((math.pow((math.pi * (2.0 * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2e-49)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi / k))) * sqrt(n));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(2.0 * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2e-49)
		tmp = sqrt((2.0 * (pi / k))) * sqrt(n);
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2e-49], N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999987e-49

   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 99.1%

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

      1. expm1-log1p-u 93.7%

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

      2. expm1-udef 72.7%

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

      3. associate-*l/ 72.7%

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

      4. *-un-lft-identity 72.7%

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

      5. sqrt-unprod 72.7%

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

      6. *-commutative 72.7%

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

      7. *-commutative 72.7%

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

      8. sqrt-undiv 54.2%

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

      9. associate-*r* 54.2%

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

      10. *-commutative 54.2%

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

      11. *-commutative 54.2%

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

   5. Applied egg-rr 54.2%

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

      1. expm1-def 75.3%

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

      2. expm1-log1p 79.1%

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

      3. *-rgt-identity 79.1%

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

      4. associate-*r/ 79.0%

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

      5. associate-*r* 79.0%

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

      6. *-commutative 79.0%

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

      7. *-commutative 79.0%

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

      8. associate-*l* 79.0%

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

      9. *-commutative 79.0%

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

      10. associate-*r* 79.0%

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

      11. associate-*l/ 79.2%

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

      12. *-lft-identity 79.2%

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

   7. Simplified 79.2%

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

      1. associate-*r* 79.2%

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

      2. sqrt-prod 99.6%

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

   9. Applied egg-rr 99.6%

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

   if 1.99999999999999987e-49 < k

   1. Initial program 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*r* 99.6%

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

      5. div-sub 99.6%

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

      6. metadata-eval 99.6%

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

      7. pow-sub 99.9%

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

      8. pow1/2 99.9%

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

      9. associate-/l/ 99.9%

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

      10. associate-*r* 99.9%

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

      11. *-commutative 99.9%

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

      12. associate-*l* 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*l* 99.9%

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

      7. *-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. frac-times 99.9%

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

      3. metadata-eval 99.9%

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

      4. sqrt-div 99.9%

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

      5. pow1 99.9%

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

      6. pow-unpow 99.9%

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

      7. pow1/2 99.9%

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

      8. pow-sub 99.6%

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

      9. *-commutative 99.6%

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

      10. sqrt-div 99.6%

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

      11. metadata-eval 99.6%

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

      12. inv-pow 99.6%

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

      13. sqrt-pow2 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

          \[\leadsto {\left(\sqrt{\pi \cdot \left(2 \cdot n\right)}\right)}^{\left(1 - k\right)} \cdot {k}^{\color{blue}{\left(0.25 \cdot -2\right)}} \]

      16. pow-pow 99.6%

          \[\leadsto {\left(\sqrt{\pi \cdot \left(2 \cdot n\right)}\right)}^{\left(1 - k\right)} \cdot \color{blue}{{\left({k}^{0.25}\right)}^{-2}} \]

   8. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

   10. Simplified 99.6%

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

       1. add-sqr-sqrt 99.6%

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

       2. sqrt-unprod 99.6%

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

       3. swap-sqr 99.6%

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

       4. pow-prod-down 99.6%

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

       5. add-sqr-sqrt 99.6%

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

       6. associate-*l* 99.6%

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

       7. pow-prod-up 99.6%

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

       8. metadata-eval 99.6%

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

       9. inv-pow 99.6%

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

   12. Applied egg-rr 99.6%

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

       1. associate-*r/ 99.6%

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

       2. *-rgt-identity 99.6%

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

       3. associate-*r* 99.6%

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

       4. *-commutative 99.6%

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

   14. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% 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 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 99.4%

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

   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%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(Pi * N[(2.0 * n), $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 99.4%

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

   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%

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

Alternative 6: 48.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $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 48.8%

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

   1. expm1-log1p-u 46.4%

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

   2. expm1-udef 42.2%

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

   3. associate-*l/ 42.2%

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

   4. *-un-lft-identity 42.2%

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

   5. sqrt-unprod 42.2%

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

   6. *-commutative 42.2%

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

   7. *-commutative 42.2%

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

   8. sqrt-undiv 34.2%

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

   9. associate-*r* 34.2%

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

   10. *-commutative 34.2%

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

   11. *-commutative 34.2%

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

5. Applied egg-rr 34.2%

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

   1. expm1-def 38.5%

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

   2. expm1-log1p 40.2%

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

   3. *-rgt-identity 40.2%

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

   4. associate-*r/ 40.1%

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

   5. associate-*r* 40.1%

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

   6. *-commutative 40.1%

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

   7. *-commutative 40.1%

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

   8. associate-*l* 40.1%

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

   9. *-commutative 40.1%

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

   10. associate-*r* 40.1%

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

   11. associate-*l/ 40.2%

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

   12. *-lft-identity 40.2%

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

7. Simplified 40.2%

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

   1. associate-*r* 40.2%

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

   2. sqrt-prod 49.0%

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

9. Applied egg-rr 49.0%

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

10. Final simplification 49.0%

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

Alternative 7: 37.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 48.8%

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

   1. expm1-log1p-u 46.4%

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

   2. expm1-udef 42.2%

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

   3. associate-*l/ 42.2%

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

   4. *-un-lft-identity 42.2%

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

   5. sqrt-unprod 42.2%

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

   6. *-commutative 42.2%

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

   7. *-commutative 42.2%

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

   8. sqrt-undiv 34.2%

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

   9. associate-*r* 34.2%

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

   10. *-commutative 34.2%

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

   11. *-commutative 34.2%

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

5. Applied egg-rr 34.2%

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

   1. expm1-def 38.5%

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

   2. expm1-log1p 40.2%

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

   3. *-rgt-identity 40.2%

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

   4. associate-*r/ 40.1%

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

   5. associate-*r* 40.1%

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

   6. *-commutative 40.1%

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

   7. *-commutative 40.1%

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

   8. associate-*l* 40.1%

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

   9. *-commutative 40.1%

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

   10. associate-*r* 40.1%

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

   11. associate-*l/ 40.2%

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

   12. *-lft-identity 40.2%

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

7. Simplified 40.2%

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

8. Taylor expanded in k around 0 40.2%

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

   1. associate-*l/ 40.2%

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

10. Simplified 40.2%

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

11. Final simplification 40.2%

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

Alternative 8: 37.3% 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 48.8%

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

   1. expm1-log1p-u 46.4%

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

   2. expm1-udef 42.2%

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

   3. associate-*l/ 42.2%

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

   4. *-un-lft-identity 42.2%

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

   5. sqrt-unprod 42.2%

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

   6. *-commutative 42.2%

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

   7. *-commutative 42.2%

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

   8. sqrt-undiv 34.2%

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

   9. associate-*r* 34.2%

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

   10. *-commutative 34.2%

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

   11. *-commutative 34.2%

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

5. Applied egg-rr 34.2%

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

   1. expm1-def 38.5%

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

   2. expm1-log1p 40.2%

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

   3. *-rgt-identity 40.2%

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

   4. associate-*r/ 40.1%

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

   5. associate-*r* 40.1%

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

   6. *-commutative 40.1%

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

   7. *-commutative 40.1%

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

   8. associate-*l* 40.1%

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

   9. *-commutative 40.1%

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

   10. associate-*r* 40.1%

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

   11. associate-*l/ 40.2%

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

   12. *-lft-identity 40.2%

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

7. Simplified 40.2%

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

8. Final simplification 40.2%

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

Reproduce

herbie shell --seed 2024026 
(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))))