Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 6.8s

Alternatives: 10

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 3.370677914109453e+238
  2. n = -7.924430785173645e+147

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-pow.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. +-commutative N/A

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

   6. div-add N/A

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

   7. metadata-eval N/A

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

   8. pow-add N/A

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

   9. lower-unsound-*.f64 N/A

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

3. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. div-flip-rev N/A

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

5. Applied rewrites 76.6%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. sqrt-prod N/A

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

   8. lower-unsound-*.f64 N/A

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

   9. lower-unsound-sqrt.f64 N/A

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

   10. lower-unsound-sqrt.f64 N/A

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

   11. mult-flip-rev N/A

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

   12. lower-/.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-pow.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. +-commutative N/A

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

   6. div-add N/A

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

   7. metadata-eval N/A

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

   8. pow-add N/A

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

   9. lower-unsound-*.f64 N/A

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

3. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-pow.f64 N/A

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

   5. sqr-pow N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. count-2-rev N/A

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

   12. lift-*.f64 N/A

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

5. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lower-/.f64 99.5

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

3. Applied rewrites 99.5%

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

Alternative 4: 73.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.29:\\ \;\;\;\;\frac{\sqrt{\left(\pi + \pi\right) \cdot n}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{n \cdot \sqrt{\left(n + n\right) \cdot \frac{\pi}{n \cdot n}}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 0.29)
   (/ (sqrt (* (+ PI PI) n)) (sqrt k))
   (/ (* n (sqrt (* (+ n n) (/ PI (* n n))))) (sqrt k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 0.29) {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) * n)) / sqrt(k);
	} else {
		tmp = (n * sqrt(((n + n) * (((double) M_PI) / (n * n))))) / sqrt(k);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 0.29) {
		tmp = Math.sqrt(((Math.PI + Math.PI) * n)) / Math.sqrt(k);
	} else {
		tmp = (n * Math.sqrt(((n + n) * (Math.PI / (n * n))))) / Math.sqrt(k);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 0.29:
		tmp = math.sqrt(((math.pi + math.pi) * n)) / math.sqrt(k)
	else:
		tmp = (n * math.sqrt(((n + n) * (math.pi / (n * n))))) / math.sqrt(k)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 0.29)
		tmp = Float64(sqrt(Float64(Float64(pi + pi) * n)) / sqrt(k));
	else
		tmp = Float64(Float64(n * sqrt(Float64(Float64(n + n) * Float64(pi / Float64(n * n))))) / sqrt(k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 0.29)
		tmp = sqrt(((pi + pi) * n)) / sqrt(k);
	else
		tmp = (n * sqrt(((n + n) * (pi / (n * n))))) / sqrt(k);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 0.29], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[(N[(n * N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(Pi / N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.28999999999999998

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.3

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.3%

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

   if 0.28999999999999998 < k

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.3

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

   4. Applied rewrites 49.3%

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

   5. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-sqrt.f64 49.4

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

   7. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-add N/A

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

      6. *-commutative N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-PI.f64 N/A

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

      11. associate-*r* N/A

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

      12. sqr-neg-rev N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. sqr-neg-rev N/A

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

      19. lower-*.f64 50.8

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

   9. Applied rewrites 50.8%

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

Alternative 5: 63.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\left(\pi + \pi\right) \cdot n}}{\sqrt{k}}\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(n + n\right) \cdot \pi}{\sqrt{k \cdot k}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 9.2e-57)
   (/ (sqrt (* (+ PI PI) n)) (sqrt k))
   (if (<= k 2.4e+162)
     (* (sqrt (/ (+ PI PI) (* n k))) n)
     (sqrt (/ (* (+ n n) PI) (sqrt (* k k)))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 9.2e-57) {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) * n)) / sqrt(k);
	} else if (k <= 2.4e+162) {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / (n * k))) * n;
	} else {
		tmp = sqrt((((n + n) * ((double) M_PI)) / sqrt((k * k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 9.2e-57) {
		tmp = Math.sqrt(((Math.PI + Math.PI) * n)) / Math.sqrt(k);
	} else if (k <= 2.4e+162) {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (n * k))) * n;
	} else {
		tmp = Math.sqrt((((n + n) * Math.PI) / Math.sqrt((k * k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 9.2e-57:
		tmp = math.sqrt(((math.pi + math.pi) * n)) / math.sqrt(k)
	elif k <= 2.4e+162:
		tmp = math.sqrt(((math.pi + math.pi) / (n * k))) * n
	else:
		tmp = math.sqrt((((n + n) * math.pi) / math.sqrt((k * k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 9.2e-57)
		tmp = Float64(sqrt(Float64(Float64(pi + pi) * n)) / sqrt(k));
	elseif (k <= 2.4e+162)
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / Float64(n * k))) * n);
	else
		tmp = sqrt(Float64(Float64(Float64(n + n) * pi) / sqrt(Float64(k * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 9.2e-57)
		tmp = sqrt(((pi + pi) * n)) / sqrt(k);
	elseif (k <= 2.4e+162)
		tmp = sqrt(((pi + pi) / (n * k))) * n;
	else
		tmp = sqrt((((n + n) * pi) / sqrt((k * k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 9.2e-57], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.4e+162], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / N[(n * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision], N[Sqrt[N[(N[(N[(n + n), $MachinePrecision] * Pi), $MachinePrecision] / N[Sqrt[N[(k * k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 9.2000000000000001e-57

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.3

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.3%

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

   if 9.2000000000000001e-57 < k < 2.40000000000000009e162

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.3

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

   4. Applied rewrites 49.3%

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

   5. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-sqrt.f64 49.4

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

   7. Applied rewrites 49.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 49.5%

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

   if 2.40000000000000009e162 < k

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.3

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

   4. Applied rewrites 49.3%

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

      1. lift-/.f64 N/A

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

   6. Applied rewrites 37.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-+.f64 N/A

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

      5. count-2-rev N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 37.9

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

   8. Applied rewrites 37.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. distribute-lft-in N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 37.9

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

      12. rem-square-sqrt N/A

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

      13. sqrt-unprod N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 N/A

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

   10. Applied rewrites 34.9%

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

Alternative 6: 61.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{\pi \cdot \frac{n + n}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\pi + \pi}{n \cdot k}} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= n 2e-32)
   (sqrt (* PI (/ (+ n n) k)))
   (* (sqrt (/ (+ PI PI) (* n k))) n)))

C

double code(double k, double n) {
	double tmp;
	if (n <= 2e-32) {
		tmp = sqrt((((double) M_PI) * ((n + n) / k)));
	} else {
		tmp = sqrt(((((double) M_PI) + ((double) M_PI)) / (n * k))) * n;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (n <= 2e-32) {
		tmp = Math.sqrt((Math.PI * ((n + n) / k)));
	} else {
		tmp = Math.sqrt(((Math.PI + Math.PI) / (n * k))) * n;
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if n <= 2e-32:
		tmp = math.sqrt((math.pi * ((n + n) / k)))
	else:
		tmp = math.sqrt(((math.pi + math.pi) / (n * k))) * n
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (n <= 2e-32)
		tmp = sqrt(Float64(pi * Float64(Float64(n + n) / k)));
	else
		tmp = Float64(sqrt(Float64(Float64(pi + pi) / Float64(n * k))) * n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (n <= 2e-32)
		tmp = sqrt((pi * ((n + n) / k)));
	else
		tmp = sqrt(((pi + pi) / (n * k))) * n;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[n, 2e-32], N[Sqrt[N[(Pi * N[(N[(n + n), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(Pi + Pi), $MachinePrecision] / N[(n * k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.00000000000000011e-32

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.3

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

   4. Applied rewrites 49.3%

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

      1. lift-/.f64 N/A

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

   6. Applied rewrites 37.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-+.f64 N/A

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

      5. count-2-rev N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 37.9

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

   8. Applied rewrites 37.9%

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

   if 2.00000000000000011e-32 < n

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 N/A

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

      6. lower-sqrt.f64 49.3

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

   4. Applied rewrites 49.3%

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

   5. Taylor expanded in n around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-sqrt.f64 49.4

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

   7. Applied rewrites 49.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 49.5%

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

Alternative 7: 49.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 49.3

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

4. Applied rewrites 49.3%

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

6. Applied rewrites 49.3%

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

Alternative 8: 49.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 49.3

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

4. Applied rewrites 49.3%

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

   1. lift-/.f64 N/A

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

6. Applied rewrites 37.9%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lift-+.f64 N/A

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

   7. count-2-rev N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. sqrt-prod N/A

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

   12. lower-unsound-*.f64 N/A

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

   13. lower-unsound-sqrt.f64 N/A

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

   14. count-2-rev N/A

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

   15. lower-+.f64 N/A

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

   16. lower-unsound-sqrt.f64 N/A

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

   17. lower-/.f64 49.3

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

8. Applied rewrites 49.3%

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

Alternative 9: 49.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 49.3

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

4. Applied rewrites 49.3%

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

   1. lift-/.f64 N/A

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

6. Applied rewrites 37.9%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. sqrt-prod N/A

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

   8. lower-unsound-*.f64 N/A

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

   9. lower-unsound-sqrt.f64 N/A

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

   10. lower-unsound-sqrt.f64 N/A

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

   11. mult-flip-rev N/A

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

   12. lower-/.f64 49.3

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

8. Applied rewrites 49.3%

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

Alternative 10: 37.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. lower-sqrt.f64 49.3

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

4. Applied rewrites 49.3%

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

   1. lift-/.f64 N/A

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

6. Applied rewrites 37.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-+.f64 N/A

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

   5. count-2-rev N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. lower-/.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-+.f64 37.9

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

8. Applied rewrites 37.9%

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

Reproduce

herbie shell --seed 2025155 
(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))))