Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 9.1s

Alternatives: 9

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 1.2731073527045462e+135
  2. n = -12.853075608190704

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, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. div-sub N/A

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

   8. metadata-eval N/A

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

   9. pow-sub N/A

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

   10. pow1/2 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. sqrt-prod N/A

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

   14. *-commutative N/A

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

   15. lower-/.f64 N/A

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

3. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. mul-1-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. associate-/l* N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. count-2-rev N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. associate-/l* N/A

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

   12. *-commutative N/A

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

   13. mul-1-neg N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg N/A

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

   16. sqrt-pow2 N/A

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

   17. *-commutative N/A

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

   18. sqrt-unprod N/A

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

   19. lower-pow.f64 N/A

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

7. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-div N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 99.4

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

3. Applied rewrites 99.4%

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

Alternative 3: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.5%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-sub N/A

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

   4. metadata-eval N/A

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

   5. frac-2neg N/A

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

   6. mul-1-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. associate-/l* N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. fp-cancel-sub-sign-inv N/A

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

   13. metadata-eval N/A

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

   14. +-commutative N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 4: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in k around inf

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

6. Applied rewrites 99.4%

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

Alternative 5: 98.1% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.0)
   (* (sqrt (/ PI k)) (sqrt (+ n n)))
   (/ (pow (sqrt (* (+ n n) PI)) (- k)) (sqrt k))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 98.9%

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

   2. Taylor expanded in k around 0

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-PI.f64 73.6

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

   4. Applied rewrites 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. count-2-rev N/A

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

      12. lift-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-PI.f64 73.6

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

   6. Applied rewrites 73.6%

      \[\leadsto \sqrt{\frac{n}{k} \cdot \left(\pi + \pi\right)} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. lift-+.f64 N/A

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

      14. sqrt-prod N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\pi}{k}} \cdot \sqrt{n + n} \]

      19. lower-sqrt.f64 96.5

          \[\leadsto \sqrt{\frac{\pi}{k}} \cdot \sqrt{n + n} \]

   8. Applied rewrites 96.5%

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

   if 1 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in k around inf

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in k around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{{\left(\sqrt{\left(n + n\right) \cdot \pi}\right)}^{\left(\mathsf{neg}\left(k\right)\right)}}{\sqrt{k}} \]

      2. lower-neg.f64 99.7

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

   9. Applied rewrites 99.7%

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

Alternative 6: 52.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\pi \cdot \frac{\mathsf{fma}\left(n, k, n \cdot k\right)}{k \cdot k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{n + n}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))) 0.0)
   (sqrt (* PI (/ (fma n k (* n k)) (* k k))))
   (* (sqrt (/ PI k)) (sqrt (+ n n)))))

C

double code(double k, double n) {
	double tmp;
	if (((1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0))) <= 0.0) {
		tmp = sqrt((((double) M_PI) * (fma(n, k, (n * k)) / (k * k))));
	} else {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((n + n));
	}
	return tmp;
}

Julia

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

Wolfram

code[k_, n_] := If[LessEqual[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], 0.0], N[Sqrt[N[(Pi * N[(N[(n * k + N[(n * k), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 0.0

   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. Taylor expanded in k around 0

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-PI.f64 3.2

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

   4. Applied rewrites 3.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. count-2-rev N/A

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

      12. lift-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-PI.f64 3.2

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

   6. Applied rewrites 3.2%

      \[\leadsto \sqrt{\frac{n}{k} \cdot \left(\pi + \pi\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. count-2 N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. count-2-rev N/A

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

      11. distribute-rgt-out N/A

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

      12. lift-+.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

          \[\leadsto \sqrt{\pi \cdot \frac{n + n}{k}} \]

      16. lower-/.f64 3.2

          \[\leadsto \sqrt{\pi \cdot \frac{n + n}{k}} \]

   8. Applied rewrites 3.2%

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

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\pi \cdot \frac{n + n}{k}} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\pi \cdot \frac{n + n}{k}} \]

      3. div-add N/A

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

      4. frac-add N/A

         \[\leadsto \sqrt{\pi \cdot \frac{n \cdot k + k \cdot n}{k \cdot k}} \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\pi \cdot \frac{n \cdot k + k \cdot n}{k \cdot k}} \]

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 14.2

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

   10. Applied rewrites 14.2%

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

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   1. Initial program 99.3%

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

   2. Taylor expanded in k around 0

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-PI.f64 49.5

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

   4. Applied rewrites 49.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. count-2-rev N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. count-2-rev N/A

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

      12. lift-+.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lift-PI.f64 49.5

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

   6. Applied rewrites 49.5%

      \[\leadsto \sqrt{\frac{n}{k} \cdot \left(\pi + \pi\right)} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. associate-*l/ N/A

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

      10. associate-/l* N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out N/A

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

      13. lift-+.f64 N/A

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

      14. sqrt-prod N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lift-PI.f64 N/A

          \[\leadsto \sqrt{\frac{\pi}{k}} \cdot \sqrt{n + n} \]

      19. lower-sqrt.f64 64.7

          \[\leadsto \sqrt{\frac{\pi}{k}} \cdot \sqrt{n + n} \]

   8. Applied rewrites 64.7%

      \[\leadsto \sqrt{\frac{\pi}{k}} \cdot \color{blue}{\sqrt{n + n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 49.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. sqrt-unprod N/A

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

   2. lower-sqrt.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-PI.f64 38.1

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

4. Applied rewrites 38.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2-rev N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. count-2-rev N/A

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

   12. lift-+.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-PI.f64 38.1

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

6. Applied rewrites 38.1%

   \[\leadsto \sqrt{\frac{n}{k} \cdot \left(\pi + \pi\right)} \]
7. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. distribute-lft-in N/A

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

   8. associate-*l/ N/A

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

   9. associate-*l/ N/A

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

   10. associate-/l* N/A

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

   11. associate-/l* N/A

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

   12. distribute-rgt-out N/A

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

   13. lift-+.f64 N/A

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

   14. sqrt-prod N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sqrt.f64 N/A

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

   17. lift-/.f64 N/A

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

   18. lift-PI.f64 N/A

       \[\leadsto \sqrt{\frac{\pi}{k}} \cdot \sqrt{n + n} \]

   19. lower-sqrt.f64 49.6

       \[\leadsto \sqrt{\frac{\pi}{k}} \cdot \sqrt{n + n} \]

8. Applied rewrites 49.6%

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

Alternative 8: 38.1% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. sqrt-unprod N/A

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

   2. lower-sqrt.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-PI.f64 38.1

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

4. Applied rewrites 38.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2-rev N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. associate-*r/ N/A

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

   10. count-2-rev N/A

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

   11. associate-/l* N/A

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

   12. associate-/l* N/A

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

   13. distribute-rgt-out N/A

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

   14. count-2-rev N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lift-PI.f64 N/A

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

   18. count-2-rev N/A

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

   19. lower-+.f64 38.0

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

6. Applied rewrites 38.0%

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

Alternative 9: 38.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. sqrt-unprod N/A

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

   2. lower-sqrt.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-PI.f64 38.1

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

4. Applied rewrites 38.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2-rev N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. count-2-rev N/A

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

   12. lift-+.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-PI.f64 38.1

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

6. Applied rewrites 38.1%

   \[\leadsto \sqrt{\frac{n}{k} \cdot \left(\pi + \pi\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. distribute-lft-in N/A

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

   7. count-2 N/A

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

   8. associate-*l/ N/A

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

   9. associate-*r/ N/A

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

   10. count-2-rev N/A

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

   11. distribute-rgt-out N/A

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

   12. lift-+.f64 N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. lift-PI.f64 N/A

       \[\leadsto \sqrt{\pi \cdot \frac{n + n}{k}} \]

   16. lower-/.f64 38.1

       \[\leadsto \sqrt{\pi \cdot \frac{n + n}{k}} \]

8. Applied rewrites 38.1%

   \[\leadsto \sqrt{\pi \cdot \frac{n + n}{k}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(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))))