Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 16.9s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. unpow-prod-down 74.6%

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

   2. unpow-prod-down 99.3%

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

   3. div-sub 99.3%

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

   4. metadata-eval 99.3%

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

   5. pow-sub 99.5%

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

   6. pow1/2 99.5%

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

   7. associate-*r/ 99.5%

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

   8. inv-pow 99.5%

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

   9. sqrt-pow2 99.7%

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

   10. metadata-eval 99.7%

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

   11. associate-*l* 99.7%

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

   12. associate-*l* 99.7%

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

   13. div-inv 99.7%

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

   14. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

   1. associate-/l* 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. *-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. associate-*r* 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.6× speedup

Math

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

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(Float64(2.0 * n) * pi)
	return Float64(sqrt(t_0) / Float64((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[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Sqrt[t$95$0], $MachinePrecision] / N[(N[Power[t$95$0, N[(k * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.3%

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

   1. unpow-prod-down 74.6%

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

   2. unpow-prod-down 99.3%

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

   3. div-sub 99.3%

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

   4. metadata-eval 99.3%

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

   5. pow-sub 99.5%

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

   6. pow1/2 99.5%

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

   7. frac-times 99.6%

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

   8. *-un-lft-identity 99.6%

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

   9. associate-*l* 99.6%

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

   10. associate-*l* 99.6%

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

   11. div-inv 99.6%

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

   12. metadata-eval 99.6%

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

3. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*r* 99.6%

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

   5. *-commutative 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. expm1-log1p-u 96.0%

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

   2. expm1-udef 72.1%

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

   3. inv-pow 72.1%

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

   4. sqrt-pow2 72.1%

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

   5. metadata-eval 72.1%

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

3. Applied egg-rr 72.1%

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

   1. expm1-def 96.0%

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

   2. expm1-log1p 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(k, n)
	tmp = (k ^ -0.5) / ((2.0 * (n * pi)) ^ ((k * 0.5) - 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[(N[(k * 0.5), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. unpow-prod-down 74.6%

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

   2. unpow-prod-down 99.3%

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

   3. div-sub 99.3%

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

   4. metadata-eval 99.3%

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

   5. pow-sub 99.5%

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

   6. pow1/2 99.5%

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

   7. associate-*r/ 99.5%

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

   8. inv-pow 99.5%

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

   9. sqrt-pow2 99.7%

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

   10. metadata-eval 99.7%

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

   11. associate-*l* 99.7%

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

   12. associate-*l* 99.7%

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

   13. div-inv 99.7%

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

   14. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

   1. associate-/l* 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. *-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. associate-*r* 99.6%

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

5. Simplified 99.6%

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

   1. *-commutative 99.6%

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

   2. metadata-eval 99.6%

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

   3. div-inv 99.6%

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

   4. associate-*l* 99.6%

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

   5. *-commutative 99.6%

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

   6. pow1/2 99.6%

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

   7. associate-*l* 99.6%

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

   8. *-commutative 99.6%

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

   9. pow-div 99.5%

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

   10. div-inv 99.5%

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

   11. metadata-eval 99.5%

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

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-14}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{2 \cdot \left(n \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6.2e-14)
   (* (pow k -0.5) (sqrt (* 2.0 (* n PI))))
   (sqrt (/ (pow (* (* 2.0 n) PI) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6.2e-14) {
		tmp = pow(k, -0.5) * sqrt((2.0 * (n * ((double) M_PI))));
	} 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 <= 6.2e-14) {
		tmp = Math.pow(k, -0.5) * Math.sqrt((2.0 * (n * Math.PI)));
	} 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 <= 6.2e-14:
		tmp = math.pow(k, -0.5) * math.sqrt((2.0 * (n * math.pi)))
	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 <= 6.2e-14)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(2.0 * Float64(n * pi))));
	else
		tmp = sqrt(Float64((Float64(Float64(2.0 * n) * pi) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 6.2e-14)
		tmp = (k ^ -0.5) * sqrt((2.0 * (n * pi)));
	else
		tmp = sqrt(((((2.0 * n) * pi) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6.2e-14], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(n * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.20000000000000009e-14

   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. Step-by-step derivation

      1. associate-*l/ 99.1%

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

      2. *-lft-identity 99.1%

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

      3. sqr-pow 98.8%

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

      4. pow-sqr 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-*l/ 99.1%

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

      7. associate-/l* 99.1%

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

      8. metadata-eval 99.1%

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

      9. /-rgt-identity 99.1%

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

      10. div-sub 99.1%

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

      11. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. add-cube-cbrt 98.4%

         \[\leadsto \frac{{\color{blue}{\left(\left(\sqrt[3]{\left(2 \cdot \pi\right) \cdot n} \cdot \sqrt[3]{\left(2 \cdot \pi\right) \cdot n}\right) \cdot \sqrt[3]{\left(2 \cdot \pi\right) \cdot n}\right)}}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

      2. pow3 98.4%

         \[\leadsto \frac{{\color{blue}{\left({\left(\sqrt[3]{\left(2 \cdot \pi\right) \cdot n}\right)}^{3}\right)}}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

      3. associate-*l* 98.4%

         \[\leadsto \frac{{\left({\left(\sqrt[3]{\color{blue}{2 \cdot \left(\pi \cdot n\right)}}\right)}^{3}\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

   5. Applied egg-rr 98.4%

      \[\leadsto \frac{{\color{blue}{\left({\left(\sqrt[3]{2 \cdot \left(\pi \cdot n\right)}\right)}^{3}\right)}}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

   6. Taylor expanded in k around 0 98.7%

      \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left({1}^{0.16666666666666666} \cdot \sqrt{n \cdot \pi}\right)}}{\sqrt{k}} \]
   7. Step-by-step derivation

      1. pow-base-1 98.7%

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

      2. *-lft-identity 98.7%

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

   8. Simplified 98.7%

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

      1. div-inv 98.6%

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

      2. sqrt-unprod 98.7%

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

      3. pow1/2 98.7%

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

      4. pow-flip 98.9%

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

      5. metadata-eval 98.9%

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

   10. Applied egg-rr 98.9%

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

       1. *-commutative 98.9%

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

   12. Simplified 98.9%

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

   if 6.20000000000000009e-14 < 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. Step-by-step derivation

      1. *-commutative 99.6%

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

      2. div-sub 99.6%

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

      3. metadata-eval 99.6%

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

      4. div-inv 99.6%

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

      5. expm1-log1p-u 99.6%

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

      6. expm1-udef 96.9%

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

   3. Applied egg-rr 96.9%

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

      1. expm1-def 99.6%

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

      2. expm1-log1p 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r* 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-14}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{2 \cdot \left(n \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. sqr-pow 99.2%

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

   4. pow-sqr 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*l/ 99.4%

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

   7. associate-/l* 99.4%

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

   8. metadata-eval 99.4%

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

   9. /-rgt-identity 99.4%

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

   10. div-sub 99.4%

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

   11. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 7: 50.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. sqr-pow 99.2%

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

   4. pow-sqr 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*l/ 99.4%

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

   7. associate-/l* 99.4%

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

   8. metadata-eval 99.4%

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

   9. /-rgt-identity 99.4%

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

   10. div-sub 99.4%

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

   11. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. add-cube-cbrt 99.0%

      \[\leadsto \frac{{\color{blue}{\left(\left(\sqrt[3]{\left(2 \cdot \pi\right) \cdot n} \cdot \sqrt[3]{\left(2 \cdot \pi\right) \cdot n}\right) \cdot \sqrt[3]{\left(2 \cdot \pi\right) \cdot n}\right)}}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

   2. pow3 99.0%

      \[\leadsto \frac{{\color{blue}{\left({\left(\sqrt[3]{\left(2 \cdot \pi\right) \cdot n}\right)}^{3}\right)}}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

   3. associate-*l* 99.0%

      \[\leadsto \frac{{\left({\left(\sqrt[3]{\color{blue}{2 \cdot \left(\pi \cdot n\right)}}\right)}^{3}\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

5. Applied egg-rr 99.0%

   \[\leadsto \frac{{\color{blue}{\left({\left(\sqrt[3]{2 \cdot \left(\pi \cdot n\right)}\right)}^{3}\right)}}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

6. Taylor expanded in k around 0 49.1%

   \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left({1}^{0.16666666666666666} \cdot \sqrt{n \cdot \pi}\right)}}{\sqrt{k}} \]
7. Step-by-step derivation

   1. pow-base-1 49.1%

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

   2. *-lft-identity 49.1%

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

8. Simplified 49.1%

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

   1. div-inv 49.0%

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

   2. sqrt-unprod 49.1%

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

   3. pow1/2 49.1%

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

   4. pow-flip 49.2%

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

   5. metadata-eval 49.2%

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

10. Applied egg-rr 49.2%

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

    1. *-commutative 49.2%

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

12. Simplified 49.2%

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

13. Final simplification 49.2%

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

Alternative 8: 50.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. sqr-pow 99.2%

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

   4. pow-sqr 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*l/ 99.4%

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

   7. associate-/l* 99.4%

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

   8. metadata-eval 99.4%

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

   9. /-rgt-identity 99.4%

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

   10. div-sub 99.4%

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

   11. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. add-cube-cbrt 99.0%

      \[\leadsto \frac{{\color{blue}{\left(\left(\sqrt[3]{\left(2 \cdot \pi\right) \cdot n} \cdot \sqrt[3]{\left(2 \cdot \pi\right) \cdot n}\right) \cdot \sqrt[3]{\left(2 \cdot \pi\right) \cdot n}\right)}}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

   2. pow3 99.0%

      \[\leadsto \frac{{\color{blue}{\left({\left(\sqrt[3]{\left(2 \cdot \pi\right) \cdot n}\right)}^{3}\right)}}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

   3. associate-*l* 99.0%

      \[\leadsto \frac{{\left({\left(\sqrt[3]{\color{blue}{2 \cdot \left(\pi \cdot n\right)}}\right)}^{3}\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

5. Applied egg-rr 99.0%

   \[\leadsto \frac{{\color{blue}{\left({\left(\sqrt[3]{2 \cdot \left(\pi \cdot n\right)}\right)}^{3}\right)}}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

6. Taylor expanded in k around 0 49.1%

   \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left({1}^{0.16666666666666666} \cdot \sqrt{n \cdot \pi}\right)}}{\sqrt{k}} \]
7. Step-by-step derivation

   1. pow-base-1 49.1%

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

   2. *-lft-identity 49.1%

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

8. Simplified 49.1%

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

   1. expm1-log1p-u 46.4%

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

   2. expm1-udef 45.9%

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

   3. sqrt-unprod 45.9%

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

10. Applied egg-rr 45.9%

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

    1. expm1-def 46.4%

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

    2. expm1-log1p 49.1%

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

    3. *-commutative 49.1%

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

12. Simplified 49.1%

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

13. Final simplification 49.1%

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

Alternative 9: 38.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. sqr-pow 99.2%

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

   4. pow-sqr 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*l/ 99.4%

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

   7. associate-/l* 99.4%

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

   8. metadata-eval 99.4%

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

   9. /-rgt-identity 99.4%

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

   10. div-sub 99.4%

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

   11. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. add-cube-cbrt 99.0%

      \[\leadsto \frac{{\color{blue}{\left(\left(\sqrt[3]{\left(2 \cdot \pi\right) \cdot n} \cdot \sqrt[3]{\left(2 \cdot \pi\right) \cdot n}\right) \cdot \sqrt[3]{\left(2 \cdot \pi\right) \cdot n}\right)}}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

   2. pow3 99.0%

      \[\leadsto \frac{{\color{blue}{\left({\left(\sqrt[3]{\left(2 \cdot \pi\right) \cdot n}\right)}^{3}\right)}}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

   3. associate-*l* 99.0%

      \[\leadsto \frac{{\left({\left(\sqrt[3]{\color{blue}{2 \cdot \left(\pi \cdot n\right)}}\right)}^{3}\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

5. Applied egg-rr 99.0%

   \[\leadsto \frac{{\color{blue}{\left({\left(\sqrt[3]{2 \cdot \left(\pi \cdot n\right)}\right)}^{3}\right)}}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]

6. Taylor expanded in k around 0 49.1%

   \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left({1}^{0.16666666666666666} \cdot \sqrt{n \cdot \pi}\right)}}{\sqrt{k}} \]
7. Step-by-step derivation

   1. pow-base-1 49.1%

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

   2. *-lft-identity 49.1%

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

8. Simplified 49.1%

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

   1. expm1-log1p-u 46.4%

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

   2. expm1-udef 45.9%

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

   3. sqrt-unprod 45.9%

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

10. Applied egg-rr 45.9%

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

    1. expm1-def 46.4%

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

    2. expm1-log1p 49.1%

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

    3. *-commutative 49.1%

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

12. Simplified 49.1%

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

    1. expm1-log1p-u 46.4%

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

    2. expm1-udef 45.9%

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

    3. sqrt-undiv 35.5%

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

14. Applied egg-rr 35.5%

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

    1. expm1-def 36.0%

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

    2. expm1-log1p 37.7%

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

    3. associate-/l* 37.6%

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

    4. *-commutative 37.6%

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

    5. associate-/r/ 37.6%

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

16. Simplified 37.6%

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

17. Final simplification 37.6%

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

Alternative 10: 38.3% accurate, 1.5× 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 99.3%

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

   1. *-commutative 99.3%

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

   2. div-sub 99.3%

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

   3. metadata-eval 99.3%

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

   4. div-inv 99.4%

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

   5. expm1-log1p-u 96.6%

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

   6. expm1-udef 86.5%

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

3. Applied egg-rr 76.1%

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

   1. expm1-def 86.2%

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

   2. expm1-log1p 87.9%

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

   3. *-commutative 87.9%

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

   4. associate-*r* 87.9%

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

5. Simplified 87.9%

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

   1. associate-*l* 87.9%

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

   2. *-commutative 87.9%

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

   3. sub-neg 87.9%

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

   4. unpow-prod-up 87.7%

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

   5. pow1 87.7%

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

7. Applied egg-rr 87.7%

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

8. Taylor expanded in k around 0 37.7%

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

   1. *-commutative 37.7%

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

   2. associate-*r* 37.7%

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

10. Simplified 37.7%

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

11. Final simplification 37.7%

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

Reproduce

herbie shell --seed 2023273 
(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))))