Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.6%

Time: 11.6s

Alternatives: 10

Speedup: N/A×

• could not determine a ground truth

  1. k = 5.0878570325285684e+266
  2. n = -4.5409513115274597e-20

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: 99.6% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. 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. lower-/.f64 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. *-lft-identity N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. sqr-pow N/A

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

   13. lower-*.f64 N/A

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

6. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-/.f64 N/A

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. associate-/l/ N/A

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

10. Applied rewrites 99.7%

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

Alternative 2: 99.4% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[Power[k, -1.0], $MachinePrecision], 0.5], $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 99.5%

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

   1. 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. pow1/2 N/A

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

   6. lower-pow.f64 N/A

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

   7. inv-pow N/A

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

   8. lower-pow.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 3: 99.4% accurate, N/A× 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]

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. Add Preprocessing

Alternative 4: 99.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(\pi \cdot 2\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\\ \frac{1}{\sqrt{k}} \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (pow (* (* PI 2.0) n) (/ (/ (- 1.0 k) 2.0) 2.0))))
   (* (/ 1.0 (sqrt k)) (* t_0 t_0))))

C

double code(double k, double n) {
	double t_0 = pow(((((double) M_PI) * 2.0) * n), (((1.0 - k) / 2.0) / 2.0));
	return (1.0 / sqrt(k)) * (t_0 * t_0);
}

Java

public static double code(double k, double n) {
	double t_0 = Math.pow(((Math.PI * 2.0) * n), (((1.0 - k) / 2.0) / 2.0));
	return (1.0 / Math.sqrt(k)) * (t_0 * t_0);
}

Python

def code(k, n):
	t_0 = math.pow(((math.pi * 2.0) * n), (((1.0 - k) / 2.0) / 2.0))
	return (1.0 / math.sqrt(k)) * (t_0 * t_0)

Julia

function code(k, n)
	t_0 = Float64(Float64(pi * 2.0) * n) ^ Float64(Float64(Float64(1.0 - k) / 2.0) / 2.0)
	return Float64(Float64(1.0 / sqrt(k)) * Float64(t_0 * t_0))
end

MATLAB

function tmp = code(k, n)
	t_0 = ((pi * 2.0) * n) ^ (((1.0 - k) / 2.0) / 2.0);
	tmp = (1.0 / sqrt(k)) * (t_0 * t_0);
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[Power[N[(N[(Pi * 2.0), $MachinePrecision] * n), $MachinePrecision], N[(N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $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. 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. sqr-pow N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift--.f64 N/A

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

   17. lower-pow.f64 N/A

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

4. Applied rewrites 99.3%

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

Alternative 5: 99.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot 2\right) \cdot n\\ t_1 := \log t\_0\\ t_2 := \sqrt{\pi \cdot n}\\ t_3 := {\left(e^{0.5}\right)}^{\left(\frac{\mathsf{fma}\left(-1, k, 1\right) \cdot \log \left(n \cdot \left(\pi \cdot 2\right)\right)}{2}\right)}\\ \mathbf{if}\;k \leq 3.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\sqrt{k}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(k \cdot {t\_1}^{3}\right) \cdot \sqrt{2}\right) \cdot t\_2, -0.020833333333333332, \left(0.125 \cdot t\_2\right) \cdot \left({t\_1}^{2} \cdot \sqrt{2}\right)\right), k, \left(-0.5 \cdot t\_2\right) \cdot \left(t\_1 \cdot \sqrt{2}\right)\right), k, \sqrt{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left({k}^{-1}\right) \cdot 0.5} \cdot \left(t\_3 \cdot t\_3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* PI 2.0) n))
        (t_1 (log t_0))
        (t_2 (sqrt (* PI n)))
        (t_3
         (pow (exp 0.5) (/ (* (fma -1.0 k 1.0) (log (* n (* PI 2.0)))) 2.0))))
   (if (<= k 3.7e-6)
     (*
      (/ 1.0 (sqrt k))
      (fma
       (fma
        (fma
         (* (* (* k (pow t_1 3.0)) (sqrt 2.0)) t_2)
         -0.020833333333333332
         (* (* 0.125 t_2) (* (pow t_1 2.0) (sqrt 2.0))))
        k
        (* (* -0.5 t_2) (* t_1 (sqrt 2.0))))
       k
       (sqrt t_0)))
     (* (exp (* (log (pow k -1.0)) 0.5)) (* t_3 t_3)))))

C

double code(double k, double n) {
	double t_0 = (((double) M_PI) * 2.0) * n;
	double t_1 = log(t_0);
	double t_2 = sqrt((((double) M_PI) * n));
	double t_3 = pow(exp(0.5), ((fma(-1.0, k, 1.0) * log((n * (((double) M_PI) * 2.0)))) / 2.0));
	double tmp;
	if (k <= 3.7e-6) {
		tmp = (1.0 / sqrt(k)) * fma(fma(fma((((k * pow(t_1, 3.0)) * sqrt(2.0)) * t_2), -0.020833333333333332, ((0.125 * t_2) * (pow(t_1, 2.0) * sqrt(2.0)))), k, ((-0.5 * t_2) * (t_1 * sqrt(2.0)))), k, sqrt(t_0));
	} else {
		tmp = exp((log(pow(k, -1.0)) * 0.5)) * (t_3 * t_3);
	}
	return tmp;
}

Julia

function code(k, n)
	t_0 = Float64(Float64(pi * 2.0) * n)
	t_1 = log(t_0)
	t_2 = sqrt(Float64(pi * n))
	t_3 = exp(0.5) ^ Float64(Float64(fma(-1.0, k, 1.0) * log(Float64(n * Float64(pi * 2.0)))) / 2.0)
	tmp = 0.0
	if (k <= 3.7e-6)
		tmp = Float64(Float64(1.0 / sqrt(k)) * fma(fma(fma(Float64(Float64(Float64(k * (t_1 ^ 3.0)) * sqrt(2.0)) * t_2), -0.020833333333333332, Float64(Float64(0.125 * t_2) * Float64((t_1 ^ 2.0) * sqrt(2.0)))), k, Float64(Float64(-0.5 * t_2) * Float64(t_1 * sqrt(2.0)))), k, sqrt(t_0)));
	else
		tmp = Float64(exp(Float64(log((k ^ -1.0)) * 0.5)) * Float64(t_3 * t_3));
	end
	return tmp
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(Pi * 2.0), $MachinePrecision] * n), $MachinePrecision]}, Block[{t$95$1 = N[Log[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(Pi * n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Exp[0.5], $MachinePrecision], N[(N[(N[(-1.0 * k + 1.0), $MachinePrecision] * N[Log[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[k, 3.7e-6], N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(k * N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * -0.020833333333333332 + N[(N[(0.125 * t$95$2), $MachinePrecision] * N[(N[Power[t$95$1, 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + N[(N[(-0.5 * t$95$2), $MachinePrecision] * N[(t$95$1 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[N[Power[k, -1.0], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.7000000000000002e-6

   1. Initial program 99.2%

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

   3. Taylor expanded in k around 0

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\left(k \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{n \cdot \mathsf{PI}\left(\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)\right) + k \cdot \left(\frac{-1}{48} \cdot \left(\left(k \cdot \left({\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{3} \cdot \sqrt{2}\right)\right) \cdot \sqrt{n \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{8} \cdot \left(\sqrt{n \cdot \mathsf{PI}\left(\right)} \cdot \left({\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \sqrt{2}\right)\right)\right)\right) + \sqrt{n \cdot \mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)} \]

   5. Applied rewrites 99.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(k \cdot {\log \left(\left(\pi \cdot 2\right) \cdot n\right)}^{3}\right) \cdot \sqrt{2}\right) \cdot \sqrt{\pi \cdot n}, -0.020833333333333332, \left(0.125 \cdot \sqrt{\pi \cdot n}\right) \cdot \left({\log \left(\left(\pi \cdot 2\right) \cdot n\right)}^{2} \cdot \sqrt{2}\right)\right), k, \left(-0.5 \cdot \sqrt{\pi \cdot n}\right) \cdot \left(\log \left(\left(\pi \cdot 2\right) \cdot n\right) \cdot \sqrt{2}\right)\right), k, \sqrt{\left(\pi \cdot 2\right) \cdot n}\right)} \]

   if 3.7000000000000002e-6 < k

   1. Initial program 99.7%

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

   3. Taylor expanded in k around -inf

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

      1. exp-prod N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. sqr-pow N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-div N/A

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

      5. pow1/2 N/A

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

      6. pow-to-exp N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 99.7

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

   9. Applied rewrites 99.7%

      \[\leadsto \color{blue}{e^{\log \left({k}^{-1}\right) \cdot 0.5}} \cdot \left({\left(e^{0.5}\right)}^{\left(\frac{\mathsf{fma}\left(-1, k, 1\right) \cdot \log \left(n \cdot \left(\pi \cdot 2\right)\right)}{2}\right)} \cdot {\left(e^{0.5}\right)}^{\left(\frac{\mathsf{fma}\left(-1, k, 1\right) \cdot \log \left(n \cdot \left(\pi \cdot 2\right)\right)}{2}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 95.8% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. exp-prod N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-log.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-PI.f64 96.0

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

5. Applied rewrites 96.0%

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

Alternative 7: 95.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{0.5}\right)}^{\left(\frac{\mathsf{fma}\left(-1, k, 1\right) \cdot \log \left(n \cdot \left(\pi \cdot 2\right)\right)}{2}\right)}\\ \frac{1}{\sqrt{k}} \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0
         (pow (exp 0.5) (/ (* (fma -1.0 k 1.0) (log (* n (* PI 2.0)))) 2.0))))
   (* (/ 1.0 (sqrt k)) (* t_0 t_0))))

C

double code(double k, double n) {
	double t_0 = pow(exp(0.5), ((fma(-1.0, k, 1.0) * log((n * (((double) M_PI) * 2.0)))) / 2.0));
	return (1.0 / sqrt(k)) * (t_0 * t_0);
}

Julia

function code(k, n)
	t_0 = exp(0.5) ^ Float64(Float64(fma(-1.0, k, 1.0) * log(Float64(n * Float64(pi * 2.0)))) / 2.0)
	return Float64(Float64(1.0 / sqrt(k)) * Float64(t_0 * t_0))
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[Power[N[Exp[0.5], $MachinePrecision], N[(N[(N[(-1.0 * k + 1.0), $MachinePrecision] * N[Log[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $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 -inf

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

   1. exp-prod N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-log.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-PI.f64 96.0

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

5. Applied rewrites 96.0%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. sqr-pow N/A

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

   9. lower-*.f64 N/A

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

7. Applied rewrites 96.0%

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

Alternative 8: 95.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{0.5}\right)}^{\left(\frac{\mathsf{fma}\left(-1, k, 1\right) \cdot \log \left(n \cdot \left(\pi \cdot 2\right)\right)}{2}\right)}\\ e^{\log \left({k}^{-1}\right) \cdot 0.5} \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0
         (pow (exp 0.5) (/ (* (fma -1.0 k 1.0) (log (* n (* PI 2.0)))) 2.0))))
   (* (exp (* (log (pow k -1.0)) 0.5)) (* t_0 t_0))))

C

double code(double k, double n) {
	double t_0 = pow(exp(0.5), ((fma(-1.0, k, 1.0) * log((n * (((double) M_PI) * 2.0)))) / 2.0));
	return exp((log(pow(k, -1.0)) * 0.5)) * (t_0 * t_0);
}

Julia

function code(k, n)
	t_0 = exp(0.5) ^ Float64(Float64(fma(-1.0, k, 1.0) * log(Float64(n * Float64(pi * 2.0)))) / 2.0)
	return Float64(exp(Float64(log((k ^ -1.0)) * 0.5)) * Float64(t_0 * t_0))
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[Power[N[Exp[0.5], $MachinePrecision], N[(N[(N[(-1.0 * k + 1.0), $MachinePrecision] * N[Log[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[Exp[N[(N[Log[N[Power[k, -1.0], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * t$95$0), $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 -inf

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

   1. exp-prod N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-log.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-PI.f64 96.0

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

5. Applied rewrites 96.0%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. sqr-pow N/A

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

   9. lower-*.f64 N/A

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

7. Applied rewrites 96.0%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-div N/A

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

   5. pow1/2 N/A

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

   6. pow-to-exp N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. inv-pow N/A

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

   11. lower-pow.f64 95.5

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

9. Applied rewrites 95.5%

   \[\leadsto \color{blue}{e^{\log \left({k}^{-1}\right) \cdot 0.5}} \cdot \left({\left(e^{0.5}\right)}^{\left(\frac{\mathsf{fma}\left(-1, k, 1\right) \cdot \log \left(n \cdot \left(\pi \cdot 2\right)\right)}{2}\right)} \cdot {\left(e^{0.5}\right)}^{\left(\frac{\mathsf{fma}\left(-1, k, 1\right) \cdot \log \left(n \cdot \left(\pi \cdot 2\right)\right)}{2}\right)}\right) \]
10. Add Preprocessing

Alternative 9: 65.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\left(\pi \cdot 2\right) \cdot n\right)\\ t_1 := \left(\pi \cdot n\right) \cdot k\\ t_2 := \sqrt{t\_1}\\ \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 4 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\frac{\pi \cdot n}{k} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.020833333333333332 \cdot \sqrt{\left({k}^{3} \cdot n\right) \cdot \pi}, {t\_0}^{3} \cdot \sqrt{2}, \left(0.125 \cdot t\_2\right) \cdot \left({t\_0}^{2} \cdot \sqrt{2}\right)\right), k, \left(-0.5 \cdot t\_2\right) \cdot \left(t\_0 \cdot \sqrt{2}\right)\right), k, \sqrt{t\_1 \cdot 2}\right)}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (log (* (* PI 2.0) n))) (t_1 (* (* PI n) k)) (t_2 (sqrt t_1)))
   (if (<=
        (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0)))
        4e+149)
     (sqrt (* (/ (* PI n) k) 2.0))
     (/
      (fma
       (fma
        (fma
         (* -0.020833333333333332 (sqrt (* (* (pow k 3.0) n) PI)))
         (* (pow t_0 3.0) (sqrt 2.0))
         (* (* 0.125 t_2) (* (pow t_0 2.0) (sqrt 2.0))))
        k
        (* (* -0.5 t_2) (* t_0 (sqrt 2.0))))
       k
       (sqrt (* t_1 2.0)))
      k))))

C

double code(double k, double n) {
	double t_0 = log(((((double) M_PI) * 2.0) * n));
	double t_1 = (((double) M_PI) * n) * k;
	double t_2 = sqrt(t_1);
	double tmp;
	if (((1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0))) <= 4e+149) {
		tmp = sqrt((((((double) M_PI) * n) / k) * 2.0));
	} else {
		tmp = fma(fma(fma((-0.020833333333333332 * sqrt(((pow(k, 3.0) * n) * ((double) M_PI)))), (pow(t_0, 3.0) * sqrt(2.0)), ((0.125 * t_2) * (pow(t_0, 2.0) * sqrt(2.0)))), k, ((-0.5 * t_2) * (t_0 * sqrt(2.0)))), k, sqrt((t_1 * 2.0))) / k;
	}
	return tmp;
}

Julia

function code(k, n)
	t_0 = log(Float64(Float64(pi * 2.0) * n))
	t_1 = Float64(Float64(pi * n) * k)
	t_2 = sqrt(t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0))) <= 4e+149)
		tmp = sqrt(Float64(Float64(Float64(pi * n) / k) * 2.0));
	else
		tmp = Float64(fma(fma(fma(Float64(-0.020833333333333332 * sqrt(Float64(Float64((k ^ 3.0) * n) * pi))), Float64((t_0 ^ 3.0) * sqrt(2.0)), Float64(Float64(0.125 * t_2) * Float64((t_0 ^ 2.0) * sqrt(2.0)))), k, Float64(Float64(-0.5 * t_2) * Float64(t_0 * sqrt(2.0)))), k, sqrt(Float64(t_1 * 2.0))) / k);
	end
	return tmp
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[Log[N[(N[(Pi * 2.0), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * n), $MachinePrecision] * k), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$1], $MachinePrecision]}, 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], 4e+149], N[Sqrt[N[(N[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[(N[(-0.020833333333333332 * N[Sqrt[N[(N[(N[Power[k, 3.0], $MachinePrecision] * n), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$0, 3.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.125 * t$95$2), $MachinePrecision] * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + N[(N[(-0.5 * t$95$2), $MachinePrecision] * N[(t$95$0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * k + N[Sqrt[N[(t$95$1 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k), $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)))) < 4.0000000000000002e149

   1. Initial program 99.3%

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

   3. 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. 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. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 55.4

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

   5. Applied rewrites 55.4%

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

   if 4.0000000000000002e149 < (*.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 99.8%

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

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{k \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)\right) + k \cdot \left(\frac{-1}{48} \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left({\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{3} \cdot \sqrt{2}\right)\right) + \frac{1}{8} \cdot \left(\sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left({\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \sqrt{2}\right)\right)\right)\right) + \sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{k}} \]

   5. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.020833333333333332 \cdot \sqrt{\left({k}^{3} \cdot n\right) \cdot \pi}, {\log \left(\left(\pi \cdot 2\right) \cdot n\right)}^{3} \cdot \sqrt{2}, \left(0.125 \cdot \sqrt{\left(\pi \cdot n\right) \cdot k}\right) \cdot \left({\log \left(\left(\pi \cdot 2\right) \cdot n\right)}^{2} \cdot \sqrt{2}\right)\right), k, \left(-0.5 \cdot \sqrt{\left(\pi \cdot n\right) \cdot k}\right) \cdot \left(\log \left(\left(\pi \cdot 2\right) \cdot n\right) \cdot \sqrt{2}\right)\right), k, \sqrt{\left(\left(\pi \cdot n\right) \cdot k\right) \cdot 2}\right)}{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 37.2% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision] * 2.0), $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. 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. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 37.8

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

5. Applied rewrites 37.8%

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

Reproduce

herbie shell --seed 2025057 
(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))))