Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 16.4s

Alternatives: 17

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 * N[(n * Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Power[t$95$0, N[(k * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]]

Derivation

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. 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. associate-*r* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

   6. pow-div 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. metadata-eval 99.7%

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

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. *-lft-identity 99.7%

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

   2. times-frac 99.6%

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

   3. associate-*l/ 99.7%

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

   4. *-lft-identity 99.7%

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

   5. *-commutative 99.7%

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

   6. *-commutative 99.7%

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

6. Simplified 99.7%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. 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. associate-*r* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

   6. pow-div 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. metadata-eval 99.7%

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

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. *-lft-identity 99.7%

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

   2. times-frac 99.6%

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

   3. associate-*l/ 99.7%

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

   4. *-lft-identity 99.7%

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

   5. *-commutative 99.7%

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

   6. *-commutative 99.7%

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

6. Simplified 99.7%

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

   1. associate-/l/ 99.7%

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

   2. div-inv 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-unpow 99.6%

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

   5. pow-prod-down 99.6%

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

8. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

   3. *-commutative 99.7%

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

   4. unpow1/2 99.7%

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

   5. *-commutative 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 99.2%

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

   2. pow2 99.2%

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

4. Applied egg-rr 99.3%

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

6. Applied egg-rr 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-/r* 99.4%

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

   3. unpow1/2 99.4%

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

   4. exp-to-pow 96.1%

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

   5. rec-exp 96.1%

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

   6. distribute-lft-neg-out 96.1%

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

   7. exp-prod 96.1%

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

   8. exp-neg 96.1%

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

   9. rem-exp-log 99.5%

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

   10. unpow1/2 99.5%

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

   11. unpow-1 99.5%

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

   12. metadata-eval 99.5%

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

   13. pow-sqr 99.5%

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

   14. rem-sqrt-square 99.5%

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

   15. metadata-eval 99.5%

       \[\leadsto \frac{\left|{k}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right|}{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(k \cdot 0.5 - 0.5\right)}} \]

   16. pow-sqr 99.3%

       \[\leadsto \frac{\left|\color{blue}{{k}^{-0.25} \cdot {k}^{-0.25}}\right|}{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(k \cdot 0.5 - 0.5\right)}} \]

   17. fabs-sqr 99.3%

       \[\leadsto \frac{\color{blue}{{k}^{-0.25} \cdot {k}^{-0.25}}}{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(k \cdot 0.5 - 0.5\right)}} \]

   18. pow-sqr 99.5%

       \[\leadsto \frac{\color{blue}{{k}^{\left(2 \cdot -0.25\right)}}}{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(k \cdot 0.5 - 0.5\right)}} \]

   19. metadata-eval 99.5%

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

   20. *-commutative 99.5%

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

   21. associate-*l* 99.5%

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

   22. fma-neg 99.5%

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

   23. metadata-eval 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 4: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 99.5%

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

4. Taylor expanded in k around inf 96.0%

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

   1. unpow-1 96.0%

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

   2. metadata-eval 96.0%

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

   3. pow-sqr 96.0%

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

   4. rem-sqrt-square 96.0%

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

   5. metadata-eval 96.0%

      \[\leadsto \left|{k}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \cdot e^{0.5 \cdot \left(\log \left(2 \cdot \left(n \cdot \pi\right)\right) \cdot \left(1 - k\right)\right)} \]

   6. pow-sqr 96.0%

      \[\leadsto \left|\color{blue}{{k}^{-0.25} \cdot {k}^{-0.25}}\right| \cdot e^{0.5 \cdot \left(\log \left(2 \cdot \left(n \cdot \pi\right)\right) \cdot \left(1 - k\right)\right)} \]

   7. fabs-sqr 96.0%

      \[\leadsto \color{blue}{\left({k}^{-0.25} \cdot {k}^{-0.25}\right)} \cdot e^{0.5 \cdot \left(\log \left(2 \cdot \left(n \cdot \pi\right)\right) \cdot \left(1 - k\right)\right)} \]

   8. pow-sqr 96.0%

      \[\leadsto \color{blue}{{k}^{\left(2 \cdot -0.25\right)}} \cdot e^{0.5 \cdot \left(\log \left(2 \cdot \left(n \cdot \pi\right)\right) \cdot \left(1 - k\right)\right)} \]

   9. metadata-eval 96.0%

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

   10. associate-*r* 96.0%

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

   11. exp-prod 96.0%

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

   12. *-commutative 96.0%

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

   13. exp-to-pow 99.5%

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

   14. unpow1/2 99.5%

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

   15. *-commutative 99.5%

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

   16. associate-*l* 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 5: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.7e-28)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (sqrt (/ (pow (* 2.0 (* n PI)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.7e-28) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = sqrt((pow((2.0 * (n * ((double) M_PI))), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.7e-28) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((2.0 * n));
	} else {
		tmp = Math.sqrt((Math.pow((2.0 * (n * Math.PI)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.7e-28:
		tmp = math.sqrt((math.pi / k)) * math.sqrt((2.0 * n))
	else:
		tmp = math.sqrt((math.pow((2.0 * (n * math.pi)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.7e-28)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = sqrt(Float64((Float64(2.0 * Float64(n * pi)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.7e-28)
		tmp = sqrt((pi / k)) * sqrt((2.0 * n));
	else
		tmp = sqrt((((2.0 * (n * pi)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.7e-28], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(2.0 * N[(n * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.7000000000000002e-28

   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. Step-by-step derivation

      1. add-sqr-sqrt 98.9%

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

      2. pow2 98.9%

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in k around 0 75.6%

      \[\leadsto {\color{blue}{\left({\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{0.25}\right)}}^{2} \]
   6. Step-by-step derivation

      1. pow-pow 76.0%

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

      2. metadata-eval 76.0%

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

      3. pow1/2 76.0%

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

      4. *-commutative 76.0%

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

      5. associate-*r/ 76.0%

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

      6. associate-*l* 76.0%

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

   7. Applied egg-rr 76.0%

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

      1. associate-*r/ 76.0%

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

      2. associate-*r/ 76.0%

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

      3. sqrt-div 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-*r* 99.4%

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

      6. *-commutative 99.4%

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

      7. sqrt-div 76.0%

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

      8. associate-*l/ 76.0%

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

      9. sqrt-prod 99.5%

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

   9. Applied egg-rr 99.5%

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

   if 3.7000000000000002e-28 < 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

   4. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

      2. distribute-lft-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 99.6%

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

      6. metadata-eval 99.6%

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

      7. neg-mul-1 99.6%

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

      8. sub-neg 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.95 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(-1 + \mathsf{fma}\left(\pi, \frac{2}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.95e+22)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (sqrt (* n (+ -1.0 (fma PI (/ 2.0 k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.95e+22) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = sqrt((n * (-1.0 + fma(((double) M_PI), (2.0 / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.95e+22)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = sqrt(Float64(n * Float64(-1.0 + fma(pi, Float64(2.0 / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.95e+22], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(n * N[(-1.0 + N[(Pi * N[(2.0 / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.9500000000000001e22

   1. Initial program 99.0%

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

      1. add-sqr-sqrt 98.6%

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

      2. pow2 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in k around 0 70.8%

      \[\leadsto {\color{blue}{\left({\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{0.25}\right)}}^{2} \]
   6. Step-by-step derivation

      1. pow-pow 71.2%

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

      2. metadata-eval 71.2%

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

      3. pow1/2 71.2%

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

      4. *-commutative 71.2%

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

      5. associate-*r/ 71.1%

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

      6. associate-*l* 71.1%

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

   7. Applied egg-rr 71.1%

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

      1. associate-*r/ 71.1%

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

      2. associate-*r/ 71.2%

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

      3. sqrt-div 91.2%

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

      4. *-commutative 91.2%

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

      5. associate-*r* 91.2%

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

      6. *-commutative 91.2%

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

      7. sqrt-div 71.2%

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

      8. associate-*l/ 71.1%

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

      9. sqrt-prod 91.3%

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

   9. Applied egg-rr 91.3%

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

   if 2.9500000000000001e22 < 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. add-sqr-sqrt 100.0%

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

      2. pow2 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in k around 0 2.6%

      \[\leadsto {\color{blue}{\left({\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{0.25}\right)}}^{2} \]
   6. Step-by-step derivation

      1. pow-pow 2.6%

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

      2. metadata-eval 2.6%

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

      3. pow1/2 2.6%

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

      4. *-commutative 2.6%

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

      5. associate-*r/ 2.6%

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

      6. associate-*l* 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. expm1-log1p-u 2.6%

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

      2. expm1-undefine 50.3%

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

      3. clear-num 50.3%

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

      4. un-div-inv 50.3%

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

      5. div-inv 50.3%

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

      6. metadata-eval 50.3%

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

   9. Applied egg-rr 50.3%

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

       1. sub-neg 50.3%

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

       2. metadata-eval 50.3%

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

       3. +-commutative 50.3%

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

       4. log1p-undefine 50.3%

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

       5. rem-exp-log 50.3%

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

       6. +-commutative 50.3%

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

       7. *-rgt-identity 50.3%

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

       8. associate-/l* 50.3%

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

       9. *-commutative 50.3%

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

       10. associate-/r* 50.3%

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

       11. metadata-eval 50.3%

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

       12. fma-define 50.3%

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

   11. Simplified 50.3%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.95 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(-1 + \mathsf{fma}\left(\pi, \frac{2}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. 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. associate-*l* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. metadata-eval 99.5%

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

   2. div-sub 99.5%

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

   3. associate-*r* 99.5%

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

   4. div-inv 99.4%

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

   5. associate-*r* 99.4%

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

   6. div-sub 99.4%

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

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. div-inv 99.4%

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

   10. metadata-eval 99.4%

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

   11. distribute-rgt-neg-in 99.4%

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

   12. metadata-eval 99.4%

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

   13. pow1/2 99.4%

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

   14. pow-flip 99.5%

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

   15. metadata-eval 99.5%

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

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%

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

Alternative 8: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. 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. associate-*l* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 9: 49.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 99.2%

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

   2. pow2 99.2%

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in k around 0 40.7%

   \[\leadsto {\color{blue}{\left({\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{0.25}\right)}}^{2} \]
6. Step-by-step derivation

   1. pow-pow 40.9%

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

   2. metadata-eval 40.9%

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

   3. pow1/2 40.9%

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

   4. *-commutative 40.9%

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

   5. associate-*r/ 40.9%

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

   6. associate-*l* 40.9%

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

7. Applied egg-rr 40.9%

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

   1. associate-*r/ 40.9%

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

   2. associate-*r/ 40.9%

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

   3. sqrt-div 52.2%

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

   4. *-commutative 52.2%

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

   5. associate-*r* 52.2%

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

   6. *-commutative 52.2%

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

   7. sqrt-div 40.9%

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

   8. associate-*l/ 40.9%

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

   9. sqrt-prod 52.2%

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

9. Applied egg-rr 52.2%

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

10. Final simplification 52.2%

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

Alternative 10: 49.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 40.8%

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

   1. *-commutative 40.8%

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

   2. associate-/l* 40.8%

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

5. Simplified 40.8%

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

   1. pow1 40.8%

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

   2. *-commutative 40.8%

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

   3. sqrt-unprod 40.9%

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

7. Applied egg-rr 40.9%

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

   1. unpow1 40.9%

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

   2. associate-*r/ 40.9%

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

   3. associate-*l/ 40.9%

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

   4. associate-/l* 40.9%

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

9. Simplified 40.9%

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

    1. pow1/2 40.9%

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

    2. associate-*l* 40.9%

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

    3. metadata-eval 40.9%

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

    4. unpow-prod-down 52.2%

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

    5. metadata-eval 52.2%

       \[\leadsto {n}^{\color{blue}{0.5}} \cdot {\left(\pi \cdot \frac{2}{k}\right)}^{\left(0.25 \cdot 2\right)} \]

    6. pow1/2 52.2%

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

    7. metadata-eval 52.2%

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

11. Applied egg-rr 52.2%

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

    1. unpow1/2 52.2%

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

    2. associate-*r/ 52.2%

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

    3. *-commutative 52.2%

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

    4. associate-*r/ 52.2%

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

13. Simplified 52.2%

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

Alternative 11: 38.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 40.8%

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

   1. *-commutative 40.8%

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

   2. associate-/l* 40.8%

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

5. Simplified 40.8%

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

   1. pow1 40.8%

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

   2. *-commutative 40.8%

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

   3. sqrt-unprod 40.9%

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

7. Applied egg-rr 40.9%

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

   1. unpow1 40.9%

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

   2. associate-*r/ 40.9%

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

   3. associate-*l/ 40.9%

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

   4. associate-/l* 40.9%

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

9. Simplified 40.9%

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

    1. associate-*r/ 40.9%

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

    2. *-commutative 40.9%

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

    3. clear-num 40.9%

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

    4. sqrt-div 42.0%

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

    5. metadata-eval 42.0%

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

    6. *-commutative 42.0%

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

    7. associate-*r* 42.0%

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

11. Applied egg-rr 42.0%

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

    1. associate-/r* 42.0%

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

13. Simplified 42.0%

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

14. Final simplification 42.0%

    \[\leadsto \frac{1}{\sqrt{\frac{\frac{k}{n}}{2 \cdot \pi}}} \]
15. Add Preprocessing

Alternative 12: 38.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 40.8%

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

   1. *-commutative 40.8%

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

   2. associate-/l* 40.8%

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

5. Simplified 40.8%

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

   1. pow1 40.8%

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

   2. *-commutative 40.8%

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

   3. sqrt-unprod 40.9%

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

7. Applied egg-rr 40.9%

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

   1. unpow1 40.9%

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

   2. associate-*r/ 40.9%

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

   3. associate-*l/ 40.9%

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

   4. associate-/l* 40.9%

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

9. Simplified 40.9%

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

    1. associate-*r/ 40.9%

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

    2. *-commutative 40.9%

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

    3. clear-num 40.9%

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

    4. sqrt-div 42.0%

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

    5. metadata-eval 42.0%

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

    6. *-commutative 42.0%

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

    7. associate-*r* 42.0%

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

11. Applied egg-rr 42.0%

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

    1. associate-*r* 42.0%

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

    2. *-commutative 42.0%

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

    3. associate-*r* 42.0%

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

    4. *-commutative 42.0%

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

13. Simplified 42.0%

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

14. Final simplification 42.0%

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

Alternative 13: 37.5% accurate, 2.0× speedup

Math

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

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

Derivation

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 40.8%

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

   1. *-commutative 40.8%

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

   2. associate-/l* 40.8%

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

5. Simplified 40.8%

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

   1. pow1 40.8%

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

   2. *-commutative 40.8%

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

   3. sqrt-unprod 40.9%

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

7. Applied egg-rr 40.9%

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

   1. unpow1 40.9%

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

   2. associate-*r/ 40.9%

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

   3. associate-*l/ 40.9%

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

   4. associate-/l* 40.9%

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

9. Simplified 40.9%

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

    1. associate-*r/ 40.9%

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

    2. associate-*r* 40.9%

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

11. Applied egg-rr 40.9%

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

12. Final simplification 40.9%

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

Alternative 14: 37.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 99.2%

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

   2. pow2 99.2%

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in k around 0 40.7%

   \[\leadsto {\color{blue}{\left({\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{0.25}\right)}}^{2} \]
6. Step-by-step derivation

   1. pow-pow 40.9%

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

   2. metadata-eval 40.9%

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

   3. pow1/2 40.9%

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

   4. *-commutative 40.9%

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

   5. associate-*r/ 40.9%

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

   6. associate-*l* 40.9%

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

7. Applied egg-rr 40.9%

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

   1. associate-*r/ 40.9%

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

   2. associate-*r/ 40.9%

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

   3. *-commutative 40.9%

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

   4. associate-*r* 40.9%

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

   5. associate-*r/ 40.9%

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

   6. *-commutative 40.9%

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

   7. associate-*r* 40.9%

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

   8. clear-num 40.9%

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

   9. div-inv 40.9%

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

   10. clear-num 40.9%

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

   11. un-div-inv 40.9%

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

9. Applied egg-rr 40.9%

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

Alternative 15: 37.5% 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 99.4%

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

3. Taylor expanded in k around 0 40.8%

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

   1. *-commutative 40.8%

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

   2. associate-/l* 40.8%

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

5. Simplified 40.8%

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

   1. *-commutative 40.8%

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

   2. sqrt-unprod 40.9%

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

7. Applied egg-rr 40.9%

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

8. Final simplification 40.9%

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

Alternative 16: 37.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 99.2%

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

   2. pow2 99.2%

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in k around 0 40.7%

   \[\leadsto {\color{blue}{\left({\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{0.25}\right)}}^{2} \]
6. Step-by-step derivation

   1. pow-pow 40.9%

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

   2. metadata-eval 40.9%

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

   3. pow1/2 40.9%

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

   4. *-commutative 40.9%

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

   5. associate-*r/ 40.9%

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

   6. associate-*l* 40.9%

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

7. Applied egg-rr 40.9%

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

8. Taylor expanded in n around 0 40.9%

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

   1. metadata-eval 40.9%

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

   2. times-frac 40.9%

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

   3. *-commutative 40.9%

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

   4. *-rgt-identity 40.9%

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

   5. associate-/l/ 40.9%

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

   6. associate-*l/ 40.9%

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

   7. *-commutative 40.9%

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

   8. associate-/l* 40.9%

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

10. Simplified 40.9%

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

Alternative 17: 37.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. add-sqr-sqrt 99.2%

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

   2. pow2 99.2%

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in k around 0 40.7%

   \[\leadsto {\color{blue}{\left({\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{0.25}\right)}}^{2} \]
6. Step-by-step derivation

   1. pow-pow 40.9%

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

   2. metadata-eval 40.9%

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

   3. pow1/2 40.9%

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

   4. *-commutative 40.9%

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

   5. associate-*r/ 40.9%

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

   6. associate-*l* 40.9%

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

7. Applied egg-rr 40.9%

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

Reproduce

herbie shell --seed 2024141 
(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))))