Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 17.9s

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, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*r* 99.5%

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

   2. div-sub 99.5%

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

   3. metadata-eval 99.5%

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

   4. pow-sub 99.7%

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

   5. pow1/2 99.7%

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

   6. associate-*r* 99.7%

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

   7. *-commutative 99.7%

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

   8. associate-*l* 99.7%

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

   9. associate-*r* 99.7%

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

   10. *-commutative 99.7%

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

   11. associate-*l* 99.7%

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

   12. div-inv 99.7%

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

   13. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. inv-pow 99.7%

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

   3. sqrt-pow2 99.7%

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

   4. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

   1. *-lft-identity 99.7%

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

8. Simplified 99.7%

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

   1. *-commutative 99.7%

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

   2. div-inv 99.7%

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

   3. associate-*l* 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l* 99.7%

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

   7. pow-flip 99.7%

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

   8. *-commutative 99.7%

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

   9. *-commutative 99.7%

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

   10. associate-*l* 99.7%

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

   11. distribute-rgt-neg-in 99.7%

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

   12. metadata-eval 99.7%

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

10. Applied egg-rr 99.7%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. 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. sub-neg 99.5%

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

   6. distribute-frac-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. neg-mul-1 99.5%

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

   9. *-commutative 99.5%

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

   10. associate-/l* 99.5%

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

   11. fma-define 99.5%

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

   12. metadata-eval 99.5%

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

   13. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

   2. unpow-prod-up 99.6%

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

   3. associate-*r* 99.6%

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

   4. *-commutative 99.6%

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

   5. associate-*l* 99.6%

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

   6. pow1/2 99.6%

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

   7. associate-*r* 99.6%

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

   8. *-commutative 99.6%

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

   9. associate-*l* 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;k \leq 2.75 \cdot 10^{-44}:\\ \;\;\;\;\sqrt{t\_0} \cdot \sqrt{\frac{1}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t\_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* n (* 2.0 PI))))
   (if (<= k 2.75e-44)
     (* (sqrt t_0) (sqrt (/ 1.0 k)))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = n * (2.0 * ((double) M_PI));
	double tmp;
	if (k <= 2.75e-44) {
		tmp = sqrt(t_0) * sqrt((1.0 / k));
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = n * (2.0 * Math.PI);
	double tmp;
	if (k <= 2.75e-44) {
		tmp = Math.sqrt(t_0) * Math.sqrt((1.0 / k));
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = n * (2.0 * math.pi)
	tmp = 0
	if k <= 2.75e-44:
		tmp = math.sqrt(t_0) * math.sqrt((1.0 / k))
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(n * Float64(2.0 * pi))
	tmp = 0.0
	if (k <= 2.75e-44)
		tmp = Float64(sqrt(t_0) * sqrt(Float64(1.0 / k)));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = n * (2.0 * pi);
	tmp = 0.0;
	if (k <= 2.75e-44)
		tmp = sqrt(t_0) * sqrt((1.0 / k));
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2.75e-44], N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(1.0 / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.74999999999999996e-44

   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-*r* 99.4%

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

      2. div-sub 99.4%

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

      3. metadata-eval 99.4%

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

      4. pow-sub 99.4%

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

      5. pow1/2 99.4%

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

      6. associate-*r* 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-*l* 99.4%

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

      9. associate-*r* 99.4%

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

      10. *-commutative 99.4%

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

      11. associate-*l* 99.4%

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

      12. div-inv 99.4%

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

      13. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. inv-pow 99.4%

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

      3. sqrt-pow2 99.5%

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

      4. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. *-lft-identity 99.5%

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

   8. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. div-inv 99.5%

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

      3. associate-*l* 99.5%

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

      4. *-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*l* 99.5%

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

      7. pow-flip 99.5%

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

      8. *-commutative 99.5%

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

      9. *-commutative 99.5%

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

      10. associate-*l* 99.5%

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

      11. distribute-rgt-neg-in 99.5%

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

      12. metadata-eval 99.5%

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

   10. Applied egg-rr 99.5%

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

   11. Taylor expanded in k around 0 99.4%

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

   if 2.74999999999999996e-44 < 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. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. associate-*l* 99.6%

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

      4. div-sub 99.6%

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

      5. sub-neg 99.6%

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

      6. distribute-frac-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. neg-mul-1 99.6%

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

      9. *-commutative 99.6%

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

      10. associate-/l* 99.6%

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

      11. fma-define 99.6%

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

      12. metadata-eval 99.6%

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

      13. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. add-sqr-sqrt 99.6%

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

      2. sqrt-unprod 99.6%

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

      3. frac-times 99.6%

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

      4. pow-sqr 99.6%

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

      5. associate-*r* 99.6%

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

      6. *-commutative 99.6%

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

      7. associate-*l* 99.6%

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

      8. add-sqr-sqrt 99.7%

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

   6. Applied egg-rr 99.7%

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

   8. Simplified 99.7%

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

Alternative 4: 52.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \frac{\pi}{k}\\ \mathbf{if}\;k \leq 8 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(n \cdot t\_0\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* 2.0 (/ PI k))))
   (if (<= k 8e+152)
     (* (sqrt n) (sqrt t_0))
     (pow (pow (* n t_0) 3.0) 0.16666666666666666))))

C

double code(double k, double n) {
	double t_0 = 2.0 * (((double) M_PI) / k);
	double tmp;
	if (k <= 8e+152) {
		tmp = sqrt(n) * sqrt(t_0);
	} else {
		tmp = pow(pow((n * t_0), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = 2.0 * (Math.PI / k);
	double tmp;
	if (k <= 8e+152) {
		tmp = Math.sqrt(n) * Math.sqrt(t_0);
	} else {
		tmp = Math.pow(Math.pow((n * t_0), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = 2.0 * (math.pi / k)
	tmp = 0
	if k <= 8e+152:
		tmp = math.sqrt(n) * math.sqrt(t_0)
	else:
		tmp = math.pow(math.pow((n * t_0), 3.0), 0.16666666666666666)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(2.0 * Float64(pi / k))
	tmp = 0.0
	if (k <= 8e+152)
		tmp = Float64(sqrt(n) * sqrt(t_0));
	else
		tmp = (Float64(n * t_0) ^ 3.0) ^ 0.16666666666666666;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = 2.0 * (pi / k);
	tmp = 0.0;
	if (k <= 8e+152)
		tmp = sqrt(n) * sqrt(t_0);
	else
		tmp = ((n * t_0) ^ 3.0) ^ 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 8e+152], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(n * t$95$0), $MachinePrecision], 3.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 8.0000000000000004e152

   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 52.0%

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

      1. *-commutative 52.0%

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

      2. associate-/l* 51.9%

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

   5. Simplified 51.9%

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

      1. sqrt-unprod 52.1%

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

   7. Applied egg-rr 52.1%

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

      1. *-un-lft-identity 52.1%

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

      2. *-commutative 52.1%

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

      3. associate-*l* 52.1%

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

   9. Applied egg-rr 52.1%

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

       1. *-lft-identity 52.1%

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

       2. associate-*l/ 52.1%

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

       3. *-commutative 52.1%

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

       4. associate-*r* 52.1%

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

       5. *-commutative 52.1%

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

       6. associate-/l* 52.2%

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

       7. *-commutative 52.2%

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

   11. Simplified 52.2%

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

       1. pow1/2 52.2%

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

       2. associate-*l* 52.2%

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

       3. unpow-prod-down 65.6%

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

       4. pow1/2 65.6%

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

   13. Applied egg-rr 65.6%

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

       1. unpow1/2 65.6%

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

   15. Simplified 65.6%

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

   if 8.0000000000000004e152 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.9%

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

      1. *-commutative 2.9%

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

      2. associate-/l* 2.9%

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

   5. Simplified 2.9%

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

      1. sqrt-unprod 2.9%

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

   7. Applied egg-rr 2.9%

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

      1. *-un-lft-identity 2.9%

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

      2. *-commutative 2.9%

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

      3. associate-*l* 2.9%

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

   9. Applied egg-rr 2.9%

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

       1. *-lft-identity 2.9%

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

       2. associate-*l/ 2.9%

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

       3. *-commutative 2.9%

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

       4. associate-*r* 2.9%

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

       5. *-commutative 2.9%

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

       6. associate-/l* 2.9%

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

       7. *-commutative 2.9%

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

   11. Simplified 2.9%

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

       1. pow1/2 2.9%

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

       2. associate-*r/ 2.9%

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

       3. associate-*r* 2.9%

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

       4. *-commutative 2.9%

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

       5. associate-*l/ 2.9%

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

       6. metadata-eval 2.9%

          \[\leadsto {\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       7. pow-pow 11.0%

          \[\leadsto \color{blue}{{\left({\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       8. sqr-pow 11.0%

          \[\leadsto \color{blue}{{\left({\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}^{1.5}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left({\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}^{1.5}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

       9. pow-prod-down 27.6%

          \[\leadsto \color{blue}{{\left({\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}^{1.5} \cdot {\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}^{1.5}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

   13. Applied egg-rr 27.6%

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

Alternative 5: 49.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.9 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.9e+228)
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
   (cbrt (pow (* (* n PI) (/ 2.0 k)) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.9e+228) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = cbrt(pow(((n * ((double) M_PI)) * (2.0 / k)), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.9e+228) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	} else {
		tmp = Math.cbrt(Math.pow(((n * Math.PI) * (2.0 / k)), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.9e+228)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = cbrt((Float64(Float64(n * pi) * Float64(2.0 / k)) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.9e+228], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(n * Pi), $MachinePrecision] * N[(2.0 / k), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.9000000000000002e228

   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 46.5%

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

      1. *-commutative 46.5%

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

      2. associate-/l* 46.4%

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

   5. Simplified 46.4%

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

      1. sqrt-unprod 46.6%

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

   7. Applied egg-rr 46.6%

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

      1. *-un-lft-identity 46.6%

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

      2. *-commutative 46.6%

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

      3. associate-*l* 46.6%

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

   9. Applied egg-rr 46.6%

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

       1. *-lft-identity 46.6%

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

       2. associate-*l/ 46.6%

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

       3. *-commutative 46.6%

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

       4. associate-*r* 46.6%

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

       5. *-commutative 46.6%

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

       6. associate-/l* 46.6%

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

       7. *-commutative 46.6%

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

   11. Simplified 46.6%

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

       1. pow1/2 46.6%

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

       2. associate-*l* 46.6%

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

       3. unpow-prod-down 58.6%

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

       4. pow1/2 58.6%

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

   13. Applied egg-rr 58.6%

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

       1. unpow1/2 58.6%

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

   15. Simplified 58.6%

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

   if 4.9000000000000002e228 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.8%

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

      1. *-commutative 2.8%

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

      2. associate-/l* 2.8%

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

   5. Simplified 2.8%

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

      1. sqrt-unprod 2.8%

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

   7. Applied egg-rr 2.8%

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

      1. add-cbrt-cube 17.3%

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

      2. pow1/3 17.3%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\left(\pi \cdot \frac{n}{k}\right) \cdot 2} \cdot \sqrt{\left(\pi \cdot \frac{n}{k}\right) \cdot 2}\right) \cdot \sqrt{\left(\pi \cdot \frac{n}{k}\right) \cdot 2}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 17.3%

         \[\leadsto {\left(\color{blue}{\left(\left(\pi \cdot \frac{n}{k}\right) \cdot 2\right)} \cdot \sqrt{\left(\pi \cdot \frac{n}{k}\right) \cdot 2}\right)}^{0.3333333333333333} \]

      4. pow1 17.3%

         \[\leadsto {\left(\color{blue}{{\left(\left(\pi \cdot \frac{n}{k}\right) \cdot 2\right)}^{1}} \cdot \sqrt{\left(\pi \cdot \frac{n}{k}\right) \cdot 2}\right)}^{0.3333333333333333} \]

      5. pow1/2 17.3%

         \[\leadsto {\left({\left(\left(\pi \cdot \frac{n}{k}\right) \cdot 2\right)}^{1} \cdot \color{blue}{{\left(\left(\pi \cdot \frac{n}{k}\right) \cdot 2\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 17.3%

         \[\leadsto {\color{blue}{\left({\left(\left(\pi \cdot \frac{n}{k}\right) \cdot 2\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. *-commutative 17.3%

         \[\leadsto {\left({\left(\color{blue}{\left(\frac{n}{k} \cdot \pi\right)} \cdot 2\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. associate-*l* 17.3%

         \[\leadsto {\left({\color{blue}{\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 17.3%

         \[\leadsto {\left({\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 17.3%

      \[\leadsto \color{blue}{{\left({\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   10. Step-by-step derivation

       1. unpow1/3 17.3%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}^{1.5}}} \]

       2. associate-*l/ 17.3%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{n \cdot \left(\pi \cdot 2\right)}{k}\right)}}^{1.5}} \]

       3. *-commutative 17.3%

          \[\leadsto \sqrt[3]{{\left(\frac{n \cdot \color{blue}{\left(2 \cdot \pi\right)}}{k}\right)}^{1.5}} \]

       4. associate-*r* 17.3%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(n \cdot 2\right) \cdot \pi}}{k}\right)}^{1.5}} \]

       5. *-commutative 17.3%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(2 \cdot n\right)} \cdot \pi}{k}\right)}^{1.5}} \]

       6. associate-/l* 17.3%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(2 \cdot n\right) \cdot \frac{\pi}{k}\right)}}^{1.5}} \]

       7. *-commutative 17.3%

          \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(n \cdot 2\right)} \cdot \frac{\pi}{k}\right)}^{1.5}} \]

   11. Simplified 17.3%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(n \cdot 2\right) \cdot \frac{\pi}{k}\right)}^{1.5}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 17.3%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{\left(n \cdot 2\right) \cdot \pi}{k}\right)}}^{1.5}} \]

       2. *-commutative 17.3%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\pi \cdot \left(n \cdot 2\right)}}{k}\right)}^{1.5}} \]

       3. associate-*l* 17.3%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(\pi \cdot n\right) \cdot 2}}{k}\right)}^{1.5}} \]

       4. associate-/l* 17.3%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(\pi \cdot n\right) \cdot \frac{2}{k}\right)}}^{1.5}} \]

       5. *-commutative 17.3%

          \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(n \cdot \pi\right)} \cdot \frac{2}{k}\right)}^{1.5}} \]

   13. Applied egg-rr 17.3%

       \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}}^{1.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 49.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{+216}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\left(n \cdot 2\right) \cdot \frac{\pi}{k}\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.8e+216)
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
   (cbrt (pow (* (* n 2.0) (/ PI k)) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.8e+216) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = cbrt(pow(((n * 2.0) * (((double) M_PI) / k)), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.8e+216) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	} else {
		tmp = Math.cbrt(Math.pow(((n * 2.0) * (Math.PI / k)), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.8e+216)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = cbrt((Float64(Float64(n * 2.0) * Float64(pi / k)) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.8e+216], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(n * 2.0), $MachinePrecision] * N[(Pi / k), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.8000000000000001e216

   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 47.1%

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

      1. *-commutative 47.1%

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

      2. associate-/l* 47.0%

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

   5. Simplified 47.0%

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

      1. sqrt-unprod 47.2%

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

   7. Applied egg-rr 47.2%

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

      1. *-un-lft-identity 47.2%

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

      2. *-commutative 47.2%

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

      3. associate-*l* 47.2%

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

   9. Applied egg-rr 47.2%

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

       1. *-lft-identity 47.2%

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

       2. associate-*l/ 47.2%

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

       3. *-commutative 47.2%

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

       4. associate-*r* 47.2%

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

       5. *-commutative 47.2%

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

       6. associate-/l* 47.2%

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

       7. *-commutative 47.2%

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

   11. Simplified 47.2%

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

       1. pow1/2 47.2%

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

       2. associate-*l* 47.2%

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

       3. unpow-prod-down 59.4%

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

       4. pow1/2 59.4%

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

   13. Applied egg-rr 59.4%

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

       1. unpow1/2 59.4%

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

   15. Simplified 59.4%

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

   if 1.8000000000000001e216 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.9%

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

      1. *-commutative 2.9%

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

      2. associate-/l* 2.9%

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

   5. Simplified 2.9%

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

      1. sqrt-unprod 2.9%

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

   7. Applied egg-rr 2.9%

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

      1. add-cbrt-cube 16.1%

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

      2. pow1/3 16.1%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\left(\pi \cdot \frac{n}{k}\right) \cdot 2} \cdot \sqrt{\left(\pi \cdot \frac{n}{k}\right) \cdot 2}\right) \cdot \sqrt{\left(\pi \cdot \frac{n}{k}\right) \cdot 2}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 16.1%

         \[\leadsto {\left(\color{blue}{\left(\left(\pi \cdot \frac{n}{k}\right) \cdot 2\right)} \cdot \sqrt{\left(\pi \cdot \frac{n}{k}\right) \cdot 2}\right)}^{0.3333333333333333} \]

      4. pow1 16.1%

         \[\leadsto {\left(\color{blue}{{\left(\left(\pi \cdot \frac{n}{k}\right) \cdot 2\right)}^{1}} \cdot \sqrt{\left(\pi \cdot \frac{n}{k}\right) \cdot 2}\right)}^{0.3333333333333333} \]

      5. pow1/2 16.1%

         \[\leadsto {\left({\left(\left(\pi \cdot \frac{n}{k}\right) \cdot 2\right)}^{1} \cdot \color{blue}{{\left(\left(\pi \cdot \frac{n}{k}\right) \cdot 2\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 16.1%

         \[\leadsto {\color{blue}{\left({\left(\left(\pi \cdot \frac{n}{k}\right) \cdot 2\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. *-commutative 16.1%

         \[\leadsto {\left({\left(\color{blue}{\left(\frac{n}{k} \cdot \pi\right)} \cdot 2\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. associate-*l* 16.1%

         \[\leadsto {\left({\color{blue}{\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 16.1%

         \[\leadsto {\left({\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 16.1%

      \[\leadsto \color{blue}{{\left({\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   10. Step-by-step derivation

       1. unpow1/3 16.1%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}^{1.5}}} \]

       2. associate-*l/ 16.1%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{n \cdot \left(\pi \cdot 2\right)}{k}\right)}}^{1.5}} \]

       3. *-commutative 16.1%

          \[\leadsto \sqrt[3]{{\left(\frac{n \cdot \color{blue}{\left(2 \cdot \pi\right)}}{k}\right)}^{1.5}} \]

       4. associate-*r* 16.1%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(n \cdot 2\right) \cdot \pi}}{k}\right)}^{1.5}} \]

       5. *-commutative 16.1%

          \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(2 \cdot n\right)} \cdot \pi}{k}\right)}^{1.5}} \]

       6. associate-/l* 16.1%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(2 \cdot n\right) \cdot \frac{\pi}{k}\right)}}^{1.5}} \]

       7. *-commutative 16.1%

          \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(n \cdot 2\right)} \cdot \frac{\pi}{k}\right)}^{1.5}} \]

   11. Simplified 16.1%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(n \cdot 2\right) \cdot \frac{\pi}{k}\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*l* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

   6. div-inv 99.5%

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

   7. metadata-eval 99.5%

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

   8. inv-pow 99.5%

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

   9. sqrt-pow2 99.5%

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

   10. metadata-eval 99.5%

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

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

Alternative 8: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

   5. div-sub 99.5%

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

   6. metadata-eval 99.5%

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

   7. div-inv 99.5%

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

   8. metadata-eval 99.5%

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

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

Alternative 9: 47.9% 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 99.5%

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

3. Taylor expanded in k around 0 40.8%

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

   1. *-commutative 40.8%

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

   2. associate-/l* 40.8%

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

5. Simplified 40.8%

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

   1. sqrt-unprod 40.9%

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

7. Applied egg-rr 40.9%

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

   1. *-un-lft-identity 40.9%

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

   2. *-commutative 40.9%

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

   3. associate-*l* 40.9%

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

9. Applied egg-rr 40.9%

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

    1. *-lft-identity 40.9%

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

    2. associate-*l/ 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. *-commutative 40.9%

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

    6. associate-/l* 41.0%

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

    7. *-commutative 41.0%

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

11. Simplified 41.0%

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

    1. pow1/2 41.0%

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

    2. associate-*l* 41.0%

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

    3. unpow-prod-down 51.4%

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

    4. pow1/2 51.4%

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

13. Applied egg-rr 51.4%

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

    1. unpow1/2 51.4%

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

15. Simplified 51.4%

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

Alternative 10: 36.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 40.8%

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

   1. *-commutative 40.8%

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

   2. associate-/l* 40.8%

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

5. Simplified 40.8%

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

   1. sqrt-unprod 40.9%

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

7. Applied egg-rr 40.9%

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

   1. *-un-lft-identity 40.9%

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

   2. *-commutative 40.9%

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

   3. associate-*l* 40.9%

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

9. Applied egg-rr 40.9%

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

    1. *-lft-identity 40.9%

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

    2. associate-*l/ 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. *-commutative 40.9%

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

    6. associate-/l* 41.0%

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

    7. *-commutative 41.0%

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

11. Simplified 41.0%

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

Reproduce

herbie shell --seed 2024095 
(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))))