Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 15.7s

Alternatives: 11

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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. div-sub N/A

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

   3. metadata-eval N/A

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

   4. pow-sub N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{k}}\right), \color{blue}{\left(\frac{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}}\right)}\right) \]

   8. inv-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(\sqrt{k}\right)}^{-1}\right), \left(\frac{\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}}\right)\right) \]

   9. pow1/2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left({k}^{\frac{1}{2}}\right)}^{-1}\right), \left(\frac{{\color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}}^{\left(\frac{k}{2}\right)}}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}}\right)\right) \]

   10. pow-pow N/A

       \[\leadsto \mathsf{/.f64}\left(\left({k}^{\left(\frac{1}{2} \cdot -1\right)}\right), \left(\frac{\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}}\right)\right) \]

   11. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, \left(\frac{1}{2} \cdot -1\right)\right), \left(\frac{\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \left(\frac{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\color{blue}{\left(\frac{k}{2}\right)}}}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}}\right)\right) \]

   13. clear-num N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \left(\frac{1}{\color{blue}{\frac{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}}}\right)\right) \]

   14. pow-sub N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \left(\frac{1}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\right)\right) \]

   15. inv-pow N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \left({\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}\right)}^{\color{blue}{-1}}\right)\right) \]

4. Applied egg-rr 99.6%

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

5. Taylor expanded in k around inf

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

   1. associate-*r/ N/A

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

   2. *-rgt-identity N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{k}}\right), \color{blue}{\left(e^{\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{2} \cdot k - \frac{1}{2}\right)}\right)}\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{k}\right)\right), \left(e^{\color{blue}{\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{2} \cdot k - \frac{1}{2}\right)}}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \left(e^{\color{blue}{\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(\frac{1}{2} \cdot k - \frac{1}{2}\right)}\right)\right) \]

   6. exp-to-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \left({\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{\color{blue}{\left(\frac{1}{2} \cdot k - \frac{1}{2}\right)}}\right)\right) \]

   7. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \mathsf{pow.f64}\left(\left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left(\frac{1}{2} \cdot k - \frac{1}{2}\right)}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \mathsf{pow.f64}\left(\left(\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right), \left(\color{blue}{\frac{1}{2} \cdot k} - \frac{1}{2}\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \mathsf{pow.f64}\left(\left(n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \left(\color{blue}{\frac{1}{2} \cdot k} - \frac{1}{2}\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \mathsf{pow.f64}\left(\left(n \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), \left(\frac{1}{2} \cdot \color{blue}{k} - \frac{1}{2}\right)\right)\right) \]

   11. rem-exp-log N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \mathsf{pow.f64}\left(\left(n \cdot e^{\log \left(2 \cdot \mathsf{PI}\left(\right)\right)}\right), \left(\frac{1}{2} \cdot \color{blue}{k} - \frac{1}{2}\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \left(e^{\log \left(2 \cdot \mathsf{PI}\left(\right)\right)}\right)\right), \left(\color{blue}{\frac{1}{2} \cdot k} - \frac{1}{2}\right)\right)\right) \]

   13. rem-exp-log N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), \left(\frac{1}{2} \cdot \color{blue}{k} - \frac{1}{2}\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \left(\frac{1}{2} \cdot \color{blue}{k} - \frac{1}{2}\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), 2\right)\right), \left(\frac{1}{2} \cdot \color{blue}{k} - \frac{1}{2}\right)\right)\right) \]

   16. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \left(\frac{1}{2} \cdot k - \frac{1}{2}\right)\right)\right) \]

   17. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \left(\frac{1}{2} \cdot k + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \left(\frac{1}{2} \cdot k + \frac{-1}{2}\right)\right)\right) \]

   19. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{2} \cdot k}\right)\right)\right) \]

   20. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{2} \cdot k\right)}\right)\right)\right) \]

   21. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(k \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   22. *-lowering-*.f64 99.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, k\right)\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(k, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

7. Simplified 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*r* N/A

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

   2. div-sub N/A

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

   3. metadata-eval N/A

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

   4. pow-sub N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{k}}\right), \color{blue}{\left(\frac{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}}\right)}\right) \]

   8. inv-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(\sqrt{k}\right)}^{-1}\right), \left(\frac{\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}}\right)\right) \]

   9. pow1/2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left({k}^{\frac{1}{2}}\right)}^{-1}\right), \left(\frac{{\color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}}^{\left(\frac{k}{2}\right)}}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}}\right)\right) \]

   10. pow-pow N/A

       \[\leadsto \mathsf{/.f64}\left(\left({k}^{\left(\frac{1}{2} \cdot -1\right)}\right), \left(\frac{\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}}\right)\right) \]

   11. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, \left(\frac{1}{2} \cdot -1\right)\right), \left(\frac{\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \left(\frac{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\color{blue}{\left(\frac{k}{2}\right)}}}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}}\right)\right) \]

   13. clear-num N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \left(\frac{1}{\color{blue}{\frac{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}}}}\right)\right) \]

   14. pow-sub N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \left(\frac{1}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}}}\right)\right) \]

   15. inv-pow N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(k, \frac{-1}{2}\right), \left({\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}\right)}^{\color{blue}{-1}}\right)\right) \]

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*r* N/A

      \[\leadsto {\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \]

   3. div-sub N/A

      \[\leadsto {\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \]

   4. metadata-eval N/A

      \[\leadsto {\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{1}{2} - \frac{k}{2}\right)}\right), \color{blue}{\left(\frac{1}{\sqrt{k}}\right)}\right) \]

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ N/A

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

   2. *-lft-identity N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right), \color{blue}{\left(\sqrt{k}\right)}\right) \]

   4. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{\color{blue}{k}}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   8. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   9. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\frac{1}{2} - \frac{k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{\_.f64}\left(\left(\frac{1}{2}\right), \left(\frac{k}{2}\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{k}{2}\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, 2\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   13. sqrt-lowering-sqrt.f64 99.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, 2\right)\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 5: 49.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[n], $MachinePrecision] / N[Sqrt[N[(k / N[(Pi / 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. Taylor expanded in k around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 35.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 35.5%

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

   1. *-commutative N/A

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

   2. sqrt-unprod N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)}{k}\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

   9. associate-/r/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{k}{2}}\right)\right) \]

   10. clear-num N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{\frac{2}{k}}}\right)\right) \]

   11. associate-/r/ N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2} \cdot k}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2} \cdot k}\right)\right) \]

   13. associate-/r* N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2}}}{k}\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2}}\right), k\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), \frac{1}{2}\right), k\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot n\right), \frac{1}{2}\right), k\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right), \frac{1}{2}\right), k\right)\right) \]

   18. PI-lowering-PI.f64 35.6%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right), \frac{1}{2}\right), k\right)\right) \]

7. Applied egg-rr 35.6%

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

   1. div-inv N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot \frac{1}{\frac{1}{2}}}{k}\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot \frac{2}{k}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{k} \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(n \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right)\right)\right) \]

   9. PI-lowering-PI.f64 35.6%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

9. Applied egg-rr 35.6%

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

    1. clear-num N/A

       \[\leadsto \sqrt{\frac{1}{\frac{k}{2}} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \]

    2. associate-/r/ N/A

       \[\leadsto \sqrt{\frac{1}{\frac{\frac{k}{2}}{n \cdot \mathsf{PI}\left(\right)}}} \]

    3. div-inv N/A

       \[\leadsto \sqrt{\frac{1}{\frac{k \cdot \frac{1}{2}}{n \cdot \mathsf{PI}\left(\right)}}} \]

    4. metadata-eval N/A

       \[\leadsto \sqrt{\frac{1}{\frac{k \cdot \frac{1}{2}}{n \cdot \mathsf{PI}\left(\right)}}} \]

    5. frac-times N/A

       \[\leadsto \sqrt{\frac{1}{\frac{k}{n} \cdot \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}}} \]

    6. associate-/r/ N/A

       \[\leadsto \sqrt{\frac{1}{\frac{k}{\frac{n}{\frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}}}}} \]

    7. clear-num N/A

       \[\leadsto \sqrt{\frac{\frac{n}{\frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}}}{k}} \]

    8. associate-/l/ N/A

       \[\leadsto \sqrt{\frac{n}{k \cdot \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}}} \]

    9. sqrt-div N/A

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

    10. pow1/2 N/A

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

    11. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\left({n}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{k \cdot \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}}\right)}\right) \]

    12. pow1/2 N/A

        \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{n}\right), \left(\sqrt{\color{blue}{k \cdot \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}}}\right)\right) \]

    13. sqrt-lowering-sqrt.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(n\right), \left(\sqrt{\color{blue}{k \cdot \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}}}\right)\right) \]

    14. sqrt-lowering-sqrt.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(k \cdot \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

    15. clear-num N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(k \cdot \frac{1}{\frac{\mathsf{PI}\left(\right)}{\frac{1}{2}}}\right)\right)\right) \]

    16. un-div-inv N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\frac{k}{\frac{\mathsf{PI}\left(\right)}{\frac{1}{2}}}\right)\right)\right) \]

    17. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(k, \left(\frac{\mathsf{PI}\left(\right)}{\frac{1}{2}}\right)\right)\right)\right) \]

    18. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{PI}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

    19. PI-lowering-PI.f64 47.3%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{2}\right)\right)\right)\right) \]

11. Applied egg-rr 47.3%

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

Alternative 6: 49.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 35.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 35.5%

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

   1. *-commutative N/A

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

   2. sqrt-unprod N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)}{k}\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

   9. associate-/r/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{k}{2}}\right)\right) \]

   10. clear-num N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{\frac{2}{k}}}\right)\right) \]

   11. associate-/r/ N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2} \cdot k}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2} \cdot k}\right)\right) \]

   13. associate-/r* N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2}}}{k}\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2}}\right), k\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), \frac{1}{2}\right), k\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot n\right), \frac{1}{2}\right), k\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right), \frac{1}{2}\right), k\right)\right) \]

   18. PI-lowering-PI.f64 35.6%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right), \frac{1}{2}\right), k\right)\right) \]

7. Applied egg-rr 35.6%

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

   1. div-inv N/A

      \[\leadsto \sqrt{\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot \frac{1}{\frac{1}{2}}}{k}} \]

   2. *-commutative N/A

      \[\leadsto \sqrt{\frac{\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{\frac{1}{2}}}{k}} \]

   3. metadata-eval N/A

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

   4. associate-*r* N/A

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

   5. associate-/l* N/A

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

   6. sqrt-prod N/A

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

   7. pow1/2 N/A

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

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({n}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{\frac{\mathsf{PI}\left(\right) \cdot 2}{k}}\right)}\right) \]

   9. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{n}\right), \left(\sqrt{\color{blue}{\frac{\mathsf{PI}\left(\right) \cdot 2}{k}}}\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \left(\sqrt{\color{blue}{\frac{\mathsf{PI}\left(\right) \cdot 2}{k}}}\right)\right) \]

   11. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot 2}{k}\right)\right)\right) \]

   12. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{2}{k}\right)\right)\right) \]

   13. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{\frac{k}{2}}\right)\right)\right) \]

   14. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{\frac{k}{2}}\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), \left(\frac{k}{2}\right)\right)\right)\right) \]

   16. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{k}{2}\right)\right)\right)\right) \]

   17. /-lowering-/.f64 47.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(k, 2\right)\right)\right)\right) \]

9. Applied egg-rr 47.2%

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

Alternative 7: 49.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 35.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 35.5%

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

   1. sqrt-unprod N/A

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

   2. associate-*l/ N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. sqrt-prod N/A

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

   9. pow1/2 N/A

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

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left({n}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{\frac{2 \cdot \mathsf{PI}\left(\right)}{k}}\right)}\right) \]

   11. pow1/2 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{n}\right), \left(\sqrt{\color{blue}{\frac{2 \cdot \mathsf{PI}\left(\right)}{k}}}\right)\right) \]

   12. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \left(\sqrt{\color{blue}{\frac{2 \cdot \mathsf{PI}\left(\right)}{k}}}\right)\right) \]

   13. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \mathsf{PI}\left(\right)}{k}\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot 2}{k}\right)\right)\right) \]

   15. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{k} \cdot 2\right)\right)\right) \]

   16. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{k}{\mathsf{PI}\left(\right)}} \cdot 2\right)\right)\right) \]

   17. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\frac{1 \cdot 2}{\frac{k}{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

   19. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{k}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

   20. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   21. PI-lowering-PI.f64 47.2%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

7. Applied egg-rr 47.2%

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

Alternative 8: 38.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 35.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 35.5%

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

   1. sqrt-unprod N/A

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

   2. associate-*l/ N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. pow1/2 N/A

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

   6. metadata-eval N/A

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

   7. pow-pow N/A

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

   8. inv-pow N/A

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

   9. clear-num N/A

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

   10. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\left(\frac{k}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}\right), \color{blue}{\frac{-1}{2}}\right) \]

   11. associate-/r* N/A

       \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\frac{k}{2}}{\mathsf{PI}\left(\right) \cdot n}\right), \frac{-1}{2}\right) \]

   12. associate-/l/ N/A

       \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\frac{\frac{k}{2}}{n}}{\mathsf{PI}\left(\right)}\right), \frac{-1}{2}\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{k}{2}}{n}\right), \mathsf{PI}\left(\right)\right), \frac{-1}{2}\right) \]

   14. associate-/l/ N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{k}{n \cdot 2}\right), \mathsf{PI}\left(\right)\right), \frac{-1}{2}\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(k, \left(n \cdot 2\right)\right), \mathsf{PI}\left(\right)\right), \frac{-1}{2}\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(k, \left(2 \cdot n\right)\right), \mathsf{PI}\left(\right)\right), \frac{-1}{2}\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(2, n\right)\right), \mathsf{PI}\left(\right)\right), \frac{-1}{2}\right) \]

   18. PI-lowering-PI.f64 36.1%

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(2, n\right)\right), \mathsf{PI.f64}\left(\right)\right), \frac{-1}{2}\right) \]

7. Applied egg-rr 36.1%

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

8. Final simplification 36.1%

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

Alternative 9: 38.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 35.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 35.5%

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

   1. *-commutative N/A

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

   2. sqrt-unprod N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)}{k}\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

   9. associate-/r/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{k}{2}}\right)\right) \]

   10. clear-num N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{\frac{2}{k}}}\right)\right) \]

   11. associate-/r/ N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2} \cdot k}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2} \cdot k}\right)\right) \]

   13. associate-/r* N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2}}}{k}\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2}}\right), k\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), \frac{1}{2}\right), k\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot n\right), \frac{1}{2}\right), k\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right), \frac{1}{2}\right), k\right)\right) \]

   18. PI-lowering-PI.f64 35.6%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right), \frac{1}{2}\right), k\right)\right) \]

7. Applied egg-rr 35.6%

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

   1. clear-num N/A

      \[\leadsto \sqrt{\frac{1}{\frac{k}{\frac{\mathsf{PI}\left(\right) \cdot n}{\frac{1}{2}}}}} \]

   2. inv-pow N/A

      \[\leadsto \sqrt{{\left(\frac{k}{\frac{\mathsf{PI}\left(\right) \cdot n}{\frac{1}{2}}}\right)}^{-1}} \]

   3. sqrt-pow1 N/A

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

   4. metadata-eval N/A

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

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{k}{\frac{\mathsf{PI}\left(\right) \cdot n}{\frac{1}{2}}}\right), \color{blue}{\frac{-1}{2}}\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(\frac{\mathsf{PI}\left(\right) \cdot n}{\frac{1}{2}}\right)\right), \frac{-1}{2}\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(\frac{1}{\frac{\frac{1}{2}}{\mathsf{PI}\left(\right) \cdot n}}\right)\right), \frac{-1}{2}\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(\frac{1}{\frac{\frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}}{n}}\right)\right), \frac{-1}{2}\right) \]

   9. clear-num N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(\frac{n}{\frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}}\right)\right), \frac{-1}{2}\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{/.f64}\left(n, \left(\frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}\right)\right)\right), \frac{-1}{2}\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{PI}\left(\right)\right)\right)\right), \frac{-1}{2}\right) \]

   12. PI-lowering-PI.f64 36.0%

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{/.f64}\left(n, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{PI.f64}\left(\right)\right)\right)\right), \frac{-1}{2}\right) \]

9. Applied egg-rr 36.0%

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

Alternative 10: 37.5% accurate, 2.0× 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 99.5%

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

3. Taylor expanded in k around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 35.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 35.5%

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

   1. *-commutative N/A

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

   2. sqrt-unprod N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)}{k}\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

   9. associate-/r/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{k}{2}}\right)\right) \]

   10. clear-num N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{\frac{2}{k}}}\right)\right) \]

   11. associate-/r/ N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2} \cdot k}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2} \cdot k}\right)\right) \]

   13. associate-/r* N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2}}}{k}\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2}}\right), k\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), \frac{1}{2}\right), k\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot n\right), \frac{1}{2}\right), k\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right), \frac{1}{2}\right), k\right)\right) \]

   18. PI-lowering-PI.f64 35.6%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right), \frac{1}{2}\right), k\right)\right) \]

7. Applied egg-rr 35.6%

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

   1. div-inv N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot \frac{1}{\frac{1}{2}}}{k}\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot \frac{2}{k}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{k} \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(n \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right)\right)\right) \]

   9. PI-lowering-PI.f64 35.6%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

9. Applied egg-rr 35.6%

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

Alternative 11: 37.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 35.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 35.5%

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

   1. *-commutative N/A

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

   2. sqrt-unprod N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)}{k}\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

   9. associate-/r/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{k}{2}}\right)\right) \]

   10. clear-num N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{\frac{2}{k}}}\right)\right) \]

   11. associate-/r/ N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2} \cdot k}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2} \cdot k}\right)\right) \]

   13. associate-/r* N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2}}}{k}\right)\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{\frac{1}{2}}\right), k\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), \frac{1}{2}\right), k\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot n\right), \frac{1}{2}\right), k\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right), \frac{1}{2}\right), k\right)\right) \]

   18. PI-lowering-PI.f64 35.6%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right), \frac{1}{2}\right), k\right)\right) \]

7. Applied egg-rr 35.6%

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

   1. div-inv N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot \frac{1}{\frac{1}{2}}}{k}\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot \frac{2}{k}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{k} \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \left(n \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right)\right)\right) \]

   9. PI-lowering-PI.f64 35.6%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

9. Applied egg-rr 35.6%

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

    1. clear-num N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{k}{2}} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    2. associate-/r/ N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{\frac{k}{2}}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

    3. div-inv N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{k \cdot \frac{1}{2}}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

    4. metadata-eval N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{k \cdot \frac{1}{2}}{n \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

    5. frac-times N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{k}{n} \cdot \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}}\right)\right) \]

    6. associate-/l/ N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{\frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}}}{\frac{k}{n}}\right)\right) \]

    7. clear-num N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{\frac{1}{2}}}{\frac{k}{n}}\right)\right) \]

    8. div-inv N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \frac{1}{\frac{1}{2}}}{\frac{k}{n}}\right)\right) \]

    9. metadata-eval N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot 2}{\frac{k}{n}}\right)\right) \]

    10. associate-/l* N/A

        \[\leadsto \mathsf{sqrt.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{2}{\frac{k}{n}}\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{2}{\frac{k}{n}}\right)\right)\right) \]

    12. PI-lowering-PI.f64 N/A

        \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{2}{\frac{k}{n}}\right)\right)\right) \]

    13. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(2, \left(\frac{k}{n}\right)\right)\right)\right) \]

    14. /-lowering-/.f64 35.5%

        \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(k, n\right)\right)\right)\right) \]

11. Applied egg-rr 35.5%

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

Reproduce

herbie shell --seed 2024164 
(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))))