Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.1%

Time: 11.8s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. k = 1.1639580501713055e+166
  2. n = -1.4454984843878245e-9

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: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot n\right)\\ \mathbf{if}\;k \leq 1.75 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{t\_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t\_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* 2.0 (* PI n))))
   (if (<= k 1.75e-53)
     (/ (sqrt t_0) (sqrt k))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = 2.0 * (((double) M_PI) * n);
	double tmp;
	if (k <= 1.75e-53) {
		tmp = sqrt(t_0) / sqrt(k);
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = 2.0 * (Math.PI * n);
	double tmp;
	if (k <= 1.75e-53) {
		tmp = Math.sqrt(t_0) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = 2.0 * (math.pi * n)
	tmp = 0
	if k <= 1.75e-53:
		tmp = math.sqrt(t_0) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(2.0 * Float64(pi * n))
	tmp = 0.0
	if (k <= 1.75e-53)
		tmp = Float64(sqrt(t_0) / sqrt(k));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = 2.0 * (pi * n);
	tmp = 0.0;
	if (k <= 1.75e-53)
		tmp = sqrt(t_0) / sqrt(k);
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.75e-53], N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.74999999999999997e-53

   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

   4. Applied egg-rr 90.1%

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

   5. Taylor expanded in k around 0

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

      1. div-exp N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. unpow1/2 N/A

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

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

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

      6. *-commutative N/A

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

      7. exp-to-pow N/A

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

      8. unpow1/2 N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 99.5%

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

   7. Simplified 99.5%

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

   if 1.74999999999999997e-53 < 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 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{\frac{1}{k}}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right), n\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, k\right), 2\right)\right)\right) \]
   4. Step-by-step derivation

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

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

      2. /-lowering-/.f64 99.7%

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

   5. Simplified 99.7%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. pow-unpow N/A

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

      7. pow1/2 N/A

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

      8. pow-prod-down N/A

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

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

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

   7. Applied egg-rr 99.7%

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

      1. unpow1/2 N/A

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

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

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 99.7%

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 53.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\frac{k}{\pi \cdot n}}\\ \mathbf{if}\;k \leq 1.85 \cdot 10^{+193}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(t\_0 \cdot t\_0\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (/ 2.0 (/ k (* PI n)))))
   (if (<= k 1.85e+193)
     (/ (sqrt (* 2.0 (* PI n))) (sqrt k))
     (pow (* t_0 t_0) 0.25))))

C

double code(double k, double n) {
	double t_0 = 2.0 / (k / (((double) M_PI) * n));
	double tmp;
	if (k <= 1.85e+193) {
		tmp = sqrt((2.0 * (((double) M_PI) * n))) / sqrt(k);
	} else {
		tmp = pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = 2.0 / (k / (Math.PI * n));
	double tmp;
	if (k <= 1.85e+193) {
		tmp = Math.sqrt((2.0 * (Math.PI * n))) / Math.sqrt(k);
	} else {
		tmp = Math.pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = 2.0 / (k / (math.pi * n))
	tmp = 0
	if k <= 1.85e+193:
		tmp = math.sqrt((2.0 * (math.pi * n))) / math.sqrt(k)
	else:
		tmp = math.pow((t_0 * t_0), 0.25)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(2.0 / Float64(k / Float64(pi * n)))
	tmp = 0.0
	if (k <= 1.85e+193)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi * n))) / sqrt(k));
	else
		tmp = Float64(t_0 * t_0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = 2.0 / (k / (pi * n));
	tmp = 0.0;
	if (k <= 1.85e+193)
		tmp = sqrt((2.0 * (pi * n))) / sqrt(k);
	else
		tmp = (t_0 * t_0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 / N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.85e+193], N[(N[Sqrt[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 0.25], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.8500000000000001e193

   1. Initial program 99.4%

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

   4. Applied egg-rr 51.2%

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

   5. Taylor expanded in k around 0

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

      1. div-exp N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. unpow1/2 N/A

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

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

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

      6. *-commutative N/A

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

      7. exp-to-pow N/A

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

      8. unpow1/2 N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 56.6%

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

   7. Simplified 56.6%

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

   if 1.8500000000000001e193 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0

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

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

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 2.6%

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

   5. Simplified 2.6%

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

      1. sqrt-unprod N/A

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

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

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. PI-lowering-PI.f64 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. pow1/2 N/A

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

      2. metadata-eval N/A

         \[\leadsto {\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)}^{\left(\frac{1}{4} + \color{blue}{\frac{1}{4}}\right)} \]

      3. metadata-eval N/A

         \[\leadsto {\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)}^{\left(\frac{1}{2} \cdot \frac{1}{2} + \frac{1}{4}\right)} \]

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

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

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

   9. Applied egg-rr 17.0%

      \[\leadsto \color{blue}{{\left(\frac{2}{\frac{k}{\pi \cdot n}} \cdot \frac{2}{\frac{k}{\pi \cdot n}}\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{+193}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{2}{\frac{k}{\pi \cdot n}} \cdot \frac{2}{\frac{k}{\pi \cdot n}}\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\frac{k}{\pi \cdot n}}\\ \mathbf{if}\;k \leq 1.8 \cdot 10^{+192}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(t\_0 \cdot t\_0\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (/ 2.0 (/ k (* PI n)))))
   (if (<= k 1.8e+192)
     (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
     (pow (* t_0 t_0) 0.25))))

C

double code(double k, double n) {
	double t_0 = 2.0 / (k / (((double) M_PI) * n));
	double tmp;
	if (k <= 1.8e+192) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = 2.0 / (k / (Math.PI * n));
	double tmp;
	if (k <= 1.8e+192) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	} else {
		tmp = Math.pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = 2.0 / (k / (math.pi * n))
	tmp = 0
	if k <= 1.8e+192:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (math.pi / k)))
	else:
		tmp = math.pow((t_0 * t_0), 0.25)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(2.0 / Float64(k / Float64(pi * n)))
	tmp = 0.0
	if (k <= 1.8e+192)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = Float64(t_0 * t_0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = 2.0 / (k / (pi * n));
	tmp = 0.0;
	if (k <= 1.8e+192)
		tmp = sqrt(n) * sqrt((2.0 * (pi / k)));
	else
		tmp = (t_0 * t_0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 / N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.8e+192], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 0.25], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.8000000000000001e192

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0

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

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

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 39.1%

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

   5. Simplified 39.1%

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

      1. sqrt-unprod N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

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

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

      7. pow1/2 N/A

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

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

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

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

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

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

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

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

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

      12. PI-lowering-PI.f64 56.6%

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

   7. Applied egg-rr 56.6%

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

   if 1.8000000000000001e192 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0

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

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

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 2.6%

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

   5. Simplified 2.6%

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

      1. sqrt-unprod N/A

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

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

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. PI-lowering-PI.f64 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. pow1/2 N/A

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

      2. metadata-eval N/A

         \[\leadsto {\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)}^{\left(\frac{1}{4} + \color{blue}{\frac{1}{4}}\right)} \]

      3. metadata-eval N/A

         \[\leadsto {\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)}^{\left(\frac{1}{2} \cdot \frac{1}{2} + \frac{1}{4}\right)} \]

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

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

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

   9. Applied egg-rr 17.0%

      \[\leadsto \color{blue}{{\left(\frac{2}{\frac{k}{\pi \cdot n}} \cdot \frac{2}{\frac{k}{\pi \cdot n}}\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{+192}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{2}{\frac{k}{\pi \cdot n}} \cdot \frac{2}{\frac{k}{\pi \cdot n}}\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{\frac{1}{k}}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right), n\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, k\right), 2\right)\right)\right) \]
4. Step-by-step derivation

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

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

   2. /-lowering-/.f64 99.6%

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

5. Simplified 99.6%

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

   1. *-commutative N/A

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

   2. sqrt-div N/A

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

   3. metadata-eval N/A

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

   4. un-div-inv N/A

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

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

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

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

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. div-sub N/A

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

   16. metadata-eval N/A

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

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

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

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

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

   19. sqrt-lowering-sqrt.f64 99.6%

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

7. Applied egg-rr 99.6%

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

Alternative 5: 41.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\frac{k}{\pi \cdot n}}\\ \mathbf{if}\;k \leq 1.9 \cdot 10^{+179}:\\ \;\;\;\;{\left(\frac{k}{n \cdot \left(2 \cdot \pi\right)}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(t\_0 \cdot t\_0\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (/ 2.0 (/ k (* PI n)))))
   (if (<= k 1.9e+179)
     (pow (/ k (* n (* 2.0 PI))) -0.5)
     (pow (* t_0 t_0) 0.25))))

C

double code(double k, double n) {
	double t_0 = 2.0 / (k / (((double) M_PI) * n));
	double tmp;
	if (k <= 1.9e+179) {
		tmp = pow((k / (n * (2.0 * ((double) M_PI)))), -0.5);
	} else {
		tmp = pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = 2.0 / (k / (Math.PI * n));
	double tmp;
	if (k <= 1.9e+179) {
		tmp = Math.pow((k / (n * (2.0 * Math.PI))), -0.5);
	} else {
		tmp = Math.pow((t_0 * t_0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = 2.0 / (k / (math.pi * n))
	tmp = 0
	if k <= 1.9e+179:
		tmp = math.pow((k / (n * (2.0 * math.pi))), -0.5)
	else:
		tmp = math.pow((t_0 * t_0), 0.25)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(2.0 / Float64(k / Float64(pi * n)))
	tmp = 0.0
	if (k <= 1.9e+179)
		tmp = Float64(k / Float64(n * Float64(2.0 * pi))) ^ -0.5;
	else
		tmp = Float64(t_0 * t_0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = 2.0 / (k / (pi * n));
	tmp = 0.0;
	if (k <= 1.9e+179)
		tmp = (k / (n * (2.0 * pi))) ^ -0.5;
	else
		tmp = (t_0 * t_0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 / N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.9e+179], N[Power[N[(k / N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 0.25], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.9e179

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0

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

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

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 40.6%

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

   5. Simplified 40.6%

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

      1. sqrt-unprod N/A

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

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

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. PI-lowering-PI.f64 40.7%

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

   7. Applied egg-rr 40.7%

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

      1. pow1/2 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

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

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. metadata-eval 42.0%

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

   9. Applied egg-rr 42.0%

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

   if 1.9e179 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0

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

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

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

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

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

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

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

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

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

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

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

      6. sqrt-lowering-sqrt.f64 2.5%

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

   5. Simplified 2.5%

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

      1. sqrt-unprod N/A

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

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

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. PI-lowering-PI.f64 2.5%

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

   7. Applied egg-rr 2.5%

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

      1. pow1/2 N/A

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

      2. metadata-eval N/A

         \[\leadsto {\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)}^{\left(\frac{1}{4} + \color{blue}{\frac{1}{4}}\right)} \]

      3. metadata-eval N/A

         \[\leadsto {\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)}^{\left(\frac{1}{2} \cdot \frac{1}{2} + \frac{1}{4}\right)} \]

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

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

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

   9. Applied egg-rr 14.8%

      \[\leadsto \color{blue}{{\left(\frac{2}{\frac{k}{\pi \cdot n}} \cdot \frac{2}{\frac{k}{\pi \cdot n}}\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 39.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

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

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

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

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

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

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

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

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

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

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

   6. sqrt-lowering-sqrt.f64 32.6%

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

5. Simplified 32.6%

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

   1. sqrt-unprod N/A

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

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

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

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

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

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

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

   11. *-commutative N/A

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

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

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

   13. PI-lowering-PI.f64 32.7%

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

7. Applied egg-rr 32.7%

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

   1. pow1/2 N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. pow-pow N/A

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

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

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

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

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

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

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

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

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

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

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

   12. metadata-eval 33.7%

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

9. Applied egg-rr 33.7%

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

Alternative 7: 38.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

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

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

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

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

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

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

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

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

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

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

   6. sqrt-lowering-sqrt.f64 32.6%

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

5. Simplified 32.6%

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

   1. sqrt-unprod N/A

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

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

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

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

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

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

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

   11. *-commutative N/A

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

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

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

   13. PI-lowering-PI.f64 32.7%

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

7. Applied egg-rr 32.7%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

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

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

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

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

   11. associate-*l/ N/A

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

   12. metadata-eval N/A

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

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

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

   14. PI-lowering-PI.f64 32.6%

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

9. Applied egg-rr 32.6%

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

    1. *-commutative N/A

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

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

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

    3. *-commutative N/A

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

    4. clear-num N/A

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

    5. un-div-inv N/A

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

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

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

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

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

    8. /-lowering-/.f64 32.7%

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

11. Applied egg-rr 32.7%

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

12. Final simplification 32.7%

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

Alternative 8: 38.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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

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

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

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

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

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

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

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

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

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

   6. sqrt-lowering-sqrt.f64 32.6%

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

5. Simplified 32.6%

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

   1. sqrt-unprod N/A

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

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

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

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

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

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

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

   11. *-commutative N/A

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

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

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

   13. PI-lowering-PI.f64 32.7%

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

7. Applied egg-rr 32.7%

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

   1. associate-*r* N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

   6. /-lowering-/.f64 32.6%

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

9. Applied egg-rr 32.6%

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

Alternative 9: 38.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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

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

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

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

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

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

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

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

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

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

   6. sqrt-lowering-sqrt.f64 32.6%

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

5. Simplified 32.6%

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

   1. sqrt-unprod N/A

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

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

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

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

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

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

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

   11. *-commutative N/A

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

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

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

   13. PI-lowering-PI.f64 32.7%

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

7. Applied egg-rr 32.7%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

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

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

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

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

   11. associate-*l/ N/A

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

   12. metadata-eval N/A

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

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

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

   14. PI-lowering-PI.f64 32.6%

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

9. Applied egg-rr 32.6%

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

10. Final simplification 32.6%

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

Reproduce

herbie shell --seed 2024185 
(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))))