Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.6%

Time: 12.1s

Alternatives: 8

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 2.2717806892373905e+255
  2. n = -183.16729146234204

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, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-sqrt.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. associate-*l/ N/A

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

   11. neg-mul-1 N/A

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

   12. frac-2neg N/A

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

4. Applied egg-rr 99.7%

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

Alternative 2: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-sqrt.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. associate-*l/ N/A

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

   11. neg-mul-1 N/A

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

   12. frac-2neg N/A

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

4. Applied egg-rr 99.7%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqrt-prod N/A

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

   4. pow1/2 N/A

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

   5. pow1/2 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. times-frac N/A

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

   13. lower-*.f64 N/A

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

6. Applied egg-rr 99.6%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-sqrt.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. associate-*l/ N/A

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

   11. neg-mul-1 N/A

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

   12. frac-2neg N/A

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

4. Applied egg-rr 99.7%

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

6. Applied egg-rr 99.5%

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

Alternative 4: 49.5% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

5. Simplified 47.1%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. unpow-prod-down N/A

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

   12. associate-*l* N/A

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

   13. lift-/.f64 N/A

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

   14. div-inv N/A

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

   15. lower-*.f64 N/A

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

   16. unpow1/2 N/A

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

   17. lower-sqrt.f64 N/A

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

7. Applied egg-rr 47.2%

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

Alternative 5: 37.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

5. Simplified 47.1%

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

   1. lift-sqrt.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-PI.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. pow1/2 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. associate-*l/ N/A

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

   11. neg-mul-1 N/A

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

   12. frac-2neg N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. sqrt-undiv N/A

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

   16. lower-sqrt.f64 N/A

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

   17. lower-/.f64 38.0

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

7. Applied egg-rr 38.0%

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

8. Taylor expanded in n around 0

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

   1. lower-*.f64 N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-PI.f64 38.0

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

10. Simplified 38.0%

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

    1. lift-PI.f64 N/A

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

    2. clear-num N/A

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

    3. un-div-inv N/A

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

    4. associate-*r/ N/A

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

    5. lift-*.f64 N/A

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

    6. lower-/.f64 N/A

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

    7. lower-/.f64 38.1

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

12. Applied egg-rr 38.1%

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

Alternative 6: 37.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

5. Simplified 47.1%

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

   1. lift-sqrt.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-PI.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. pow1/2 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. associate-*l/ N/A

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

   11. neg-mul-1 N/A

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

   12. frac-2neg N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. sqrt-undiv N/A

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

   16. lower-sqrt.f64 N/A

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

   17. lower-/.f64 38.0

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

7. Applied egg-rr 38.0%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 38.1

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 38.1

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

9. Applied egg-rr 38.1%

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

Alternative 7: 37.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

5. Simplified 47.1%

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

   1. lift-sqrt.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-PI.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. pow1/2 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. associate-*l/ N/A

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

   11. neg-mul-1 N/A

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

   12. frac-2neg N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. sqrt-undiv N/A

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

   16. lower-sqrt.f64 N/A

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

   17. lower-/.f64 38.0

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

7. Applied egg-rr 38.0%

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

8. Taylor expanded in n around 0

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

   1. lower-*.f64 N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-PI.f64 38.0

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

10. Simplified 38.0%

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

Alternative 8: 9.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(k, n)
    real(8), intent (in) :: k
    real(8), intent (in) :: n
    code = sqrt((2.0d0 * (n * (1.0d0 / k))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

5. Simplified 47.1%

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

   1. lift-sqrt.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-PI.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. pow1/2 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. associate-*l/ N/A

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

   11. neg-mul-1 N/A

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

   12. frac-2neg N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. sqrt-undiv N/A

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

   16. lower-sqrt.f64 N/A

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

   17. lower-/.f64 38.0

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

7. Applied egg-rr 38.0%

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

8. Taylor expanded in n around 0

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

   1. lower-*.f64 N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-PI.f64 38.0

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

10. Simplified 38.0%

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

11. Taylor expanded in k around 0

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

    1. lower-/.f64 9.1

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

13. Simplified 9.1%

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

Reproduce

herbie shell --seed 2024214 
(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))))