Migdal et al, Equation (51)

Percentage Accurate: 99.6% → 99.5%

Time: 10.6s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. k = 8.535938569456363e+132
  2. n = -8.633098564341318e-19

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: 99.6% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-36}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.2e-36)
   (* (sqrt (* 2.0 n)) (sqrt (/ PI k)))
   (sqrt (/ (pow (* n (* 2.0 PI)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.2e-36) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	} else {
		tmp = sqrt((pow((n * (2.0 * ((double) M_PI))), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.2e-36) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((Math.PI / k));
	} else {
		tmp = Math.sqrt((Math.pow((n * (2.0 * Math.PI)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 4.2e-36:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((math.pi / k))
	else:
		tmp = math.sqrt((math.pow((n * (2.0 * math.pi)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.2e-36)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	else
		tmp = sqrt(Float64((Float64(n * Float64(2.0 * pi)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 4.2e-36)
		tmp = sqrt((2.0 * n)) * sqrt((pi / k));
	else
		tmp = sqrt((((n * (2.0 * pi)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.2e-36], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.19999999999999982e-36

   1. Initial program 99.2%

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

      1. associate-*l/ 99.4%

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

      2. *-lft-identity 99.4%

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

      3. associate-*l* 99.4%

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

      4. div-sub 99.4%

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

      5. sub-neg 99.4%

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

      6. distribute-frac-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. neg-mul-1 99.4%

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

      9. *-commutative 99.4%

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

      10. associate-/l* 99.4%

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

      11. fma-define 99.4%

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

      12. metadata-eval 99.4%

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

      13. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in k around 0 70.5%

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

      1. associate-/l* 70.5%

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

   7. Simplified 70.5%

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

      1. *-commutative 70.5%

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

      2. sqrt-unprod 70.7%

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

   9. Applied egg-rr 70.7%

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

       1. pow1/2 70.7%

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

       2. associate-*r* 70.7%

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

       3. unpow-prod-down 99.5%

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

       4. pow1/2 99.5%

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

   11. Applied egg-rr 99.5%

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

       1. unpow1/2 99.5%

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

   13. Simplified 99.5%

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

   if 4.19999999999999982e-36 < k

   1. Initial program 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

      3. associate-*l* 99.8%

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

      4. div-sub 99.8%

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

      5. sub-neg 99.8%

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

      6. distribute-frac-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. neg-mul-1 99.8%

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

      9. *-commutative 99.8%

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

      10. associate-/l* 99.8%

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

      11. fma-define 99.8%

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

      12. metadata-eval 99.8%

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

      13. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. frac-times 99.8%

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

      4. pow-sqr 99.8%

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

      5. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

   8. Simplified 99.8%

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

Alternative 2: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2200000:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2200000.0) (* (sqrt (* 2.0 n)) (sqrt (/ PI k))) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2200000.0) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	} else {
		tmp = sqrt(0.0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2200000.0) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((Math.PI / k));
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2200000.0:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((math.pi / k))
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2200000.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	else
		tmp = sqrt(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2200000.0)
		tmp = sqrt((2.0 * n)) * sqrt((pi / k));
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2200000.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.2e6

   1. Initial program 99.0%

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

      1. associate-*l/ 99.1%

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

      2. *-lft-identity 99.1%

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

      3. associate-*l* 99.1%

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

      4. div-sub 99.1%

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

      5. sub-neg 99.1%

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

      6. distribute-frac-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. neg-mul-1 99.1%

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

      9. *-commutative 99.1%

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

      10. associate-/l* 99.1%

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

      11. fma-define 99.1%

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

      12. metadata-eval 99.1%

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

      13. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in k around 0 69.6%

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

      1. associate-/l* 69.6%

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

   7. Simplified 69.6%

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

      1. *-commutative 69.6%

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

      2. sqrt-unprod 69.8%

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

   9. Applied egg-rr 69.8%

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

       1. pow1/2 69.8%

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

       2. associate-*r* 69.8%

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

       3. unpow-prod-down 96.1%

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

       4. pow1/2 96.1%

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

   11. Applied egg-rr 96.1%

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

       1. unpow1/2 96.1%

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

   13. Simplified 96.1%

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

   if 2.2e6 < k

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. div-sub 100.0%

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

      5. sub-neg 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-/l* 100.0%

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

      11. fma-define 100.0%

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

      12. metadata-eval 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 2.0%

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

      1. associate-/l* 2.0%

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

   7. Simplified 2.0%

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

      1. *-commutative 2.0%

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

      2. sqrt-unprod 2.0%

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

   9. Applied egg-rr 2.0%

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

       1. expm1-log1p-u 2.0%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)\right)}} \]

       2. expm1-undefine 41.5%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} - 1\right)}} \]

   11. Applied egg-rr 41.5%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} - 1\right)}} \]
   12. Step-by-step derivation

       1. sub-neg 41.5%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} + \left(-1\right)\right)}} \]

       2. metadata-eval 41.5%

          \[\leadsto \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} + \color{blue}{-1}\right)} \]

       3. +-commutative 41.5%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)}\right)}} \]

       4. log1p-undefine 41.5%

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

       5. rem-exp-log 41.5%

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

       6. +-commutative 41.5%

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

       7. associate-*r/ 41.5%

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

       8. *-commutative 41.5%

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

       9. associate-/l* 41.5%

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

       10. fma-define 41.5%

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

   13. Simplified 41.5%

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

   14. Taylor expanded in n around 0 69.8%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2200000:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2200000:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\frac{2 \cdot \pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2200000.0) (* (sqrt n) (sqrt (/ (* 2.0 PI) k))) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2200000.0) {
		tmp = sqrt(n) * sqrt(((2.0 * ((double) M_PI)) / k));
	} else {
		tmp = sqrt(0.0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2200000.0) {
		tmp = Math.sqrt(n) * Math.sqrt(((2.0 * Math.PI) / k));
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2200000.0:
		tmp = math.sqrt(n) * math.sqrt(((2.0 * math.pi) / k))
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2200000.0)
		tmp = Float64(sqrt(n) * sqrt(Float64(Float64(2.0 * pi) / k)));
	else
		tmp = sqrt(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2200000.0)
		tmp = sqrt(n) * sqrt(((2.0 * pi) / k));
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2200000.0], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.2e6

   1. Initial program 99.0%

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

      1. associate-*l/ 99.1%

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

      2. *-lft-identity 99.1%

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

      3. associate-*l* 99.1%

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

      4. div-sub 99.1%

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

      5. sub-neg 99.1%

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

      6. distribute-frac-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. neg-mul-1 99.1%

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

      9. *-commutative 99.1%

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

      10. associate-/l* 99.1%

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

      11. fma-define 99.1%

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

      12. metadata-eval 99.1%

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

      13. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in k around 0 69.6%

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

      1. associate-/l* 69.6%

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

   7. Simplified 69.6%

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

      1. *-commutative 69.6%

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

      2. sqrt-unprod 69.8%

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

   9. Applied egg-rr 69.8%

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

       1. associate-*r* 69.8%

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

       2. associate-*r/ 69.8%

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

       3. *-commutative 69.8%

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

       4. associate-*r* 69.8%

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

       5. pow1/2 69.7%

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

       6. associate-/l* 69.8%

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

       7. unpow-prod-down 96.0%

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

       8. pow1/2 96.0%

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

       9. associate-/l* 96.0%

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

   11. Applied egg-rr 96.0%

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

       1. unpow1/2 96.0%

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

       2. associate-*r/ 96.0%

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

   13. Simplified 96.0%

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

   if 2.2e6 < k

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. div-sub 100.0%

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

      5. sub-neg 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-/l* 100.0%

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

      11. fma-define 100.0%

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

      12. metadata-eval 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 2.0%

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

      1. associate-/l* 2.0%

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

   7. Simplified 2.0%

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

      1. *-commutative 2.0%

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

      2. sqrt-unprod 2.0%

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

   9. Applied egg-rr 2.0%

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

       1. expm1-log1p-u 2.0%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)\right)}} \]

       2. expm1-undefine 41.5%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} - 1\right)}} \]

   11. Applied egg-rr 41.5%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} - 1\right)}} \]
   12. Step-by-step derivation

       1. sub-neg 41.5%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} + \left(-1\right)\right)}} \]

       2. metadata-eval 41.5%

          \[\leadsto \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} + \color{blue}{-1}\right)} \]

       3. +-commutative 41.5%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)}\right)}} \]

       4. log1p-undefine 41.5%

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

       5. rem-exp-log 41.5%

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

       6. +-commutative 41.5%

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

       7. associate-*r/ 41.5%

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

       8. *-commutative 41.5%

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

       9. associate-/l* 41.5%

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

       10. fma-define 41.5%

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

   13. Simplified 41.5%

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

   14. Taylor expanded in n around 0 69.8%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2200000:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\frac{2 \cdot \pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.7%

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

   2. *-un-lft-identity 99.7%

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

   3. associate-*r* 99.7%

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

   4. div-sub 99.7%

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

   5. metadata-eval 99.7%

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

   6. div-inv 99.7%

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

   7. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 5: 69.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2200000:\\ \;\;\;\;{\left(\frac{k}{2 \cdot \left(\pi \cdot n\right)}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2200000.0) (pow (/ k (* 2.0 (* PI n))) -0.5) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2200000.0) {
		tmp = pow((k / (2.0 * (((double) M_PI) * n))), -0.5);
	} else {
		tmp = sqrt(0.0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2200000.0) {
		tmp = Math.pow((k / (2.0 * (Math.PI * n))), -0.5);
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2200000.0:
		tmp = math.pow((k / (2.0 * (math.pi * n))), -0.5)
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2200000.0)
		tmp = Float64(k / Float64(2.0 * Float64(pi * n))) ^ -0.5;
	else
		tmp = sqrt(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2200000.0)
		tmp = (k / (2.0 * (pi * n))) ^ -0.5;
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2200000.0], N[Power[N[(k / N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.2e6

   1. Initial program 99.0%

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

      1. associate-*l/ 99.1%

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

      2. *-lft-identity 99.1%

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

      3. associate-*l* 99.1%

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

      4. div-sub 99.1%

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

      5. sub-neg 99.1%

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

      6. distribute-frac-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. neg-mul-1 99.1%

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

      9. *-commutative 99.1%

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

      10. associate-/l* 99.1%

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

      11. fma-define 99.1%

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

      12. metadata-eval 99.1%

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

      13. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in k around 0 69.6%

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

      1. associate-/l* 69.6%

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

   7. Simplified 69.6%

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

      1. pow1 69.6%

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

      2. *-commutative 69.6%

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

      3. sqrt-unprod 69.8%

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

   9. Applied egg-rr 69.8%

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

       1. unpow1 69.8%

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

       2. associate-*r/ 69.8%

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

       3. *-commutative 69.8%

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

       4. associate-/l* 69.9%

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

   11. Simplified 69.9%

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

       1. associate-*r* 69.9%

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

       2. associate-*r/ 69.8%

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

       3. *-commutative 69.8%

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

       4. clear-num 69.8%

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

       5. metadata-eval 69.8%

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

       6. add-sqr-sqrt 69.5%

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

       7. frac-times 69.6%

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

       8. sqrt-unprod 70.2%

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

       9. add-sqr-sqrt 70.5%

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

       10. pow1/2 70.5%

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

       11. pow-flip 70.7%

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

       12. *-commutative 70.7%

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

       13. associate-*l* 70.7%

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

       14. *-commutative 70.7%

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

       15. metadata-eval 70.7%

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

   13. Applied egg-rr 70.7%

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

   if 2.2e6 < k

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. div-sub 100.0%

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

      5. sub-neg 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-/l* 100.0%

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

      11. fma-define 100.0%

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

      12. metadata-eval 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 2.0%

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

      1. associate-/l* 2.0%

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

   7. Simplified 2.0%

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

      1. *-commutative 2.0%

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

      2. sqrt-unprod 2.0%

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

   9. Applied egg-rr 2.0%

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

       1. expm1-log1p-u 2.0%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)\right)}} \]

       2. expm1-undefine 41.5%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} - 1\right)}} \]

   11. Applied egg-rr 41.5%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} - 1\right)}} \]
   12. Step-by-step derivation

       1. sub-neg 41.5%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} + \left(-1\right)\right)}} \]

       2. metadata-eval 41.5%

          \[\leadsto \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} + \color{blue}{-1}\right)} \]

       3. +-commutative 41.5%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)}\right)}} \]

       4. log1p-undefine 41.5%

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

       5. rem-exp-log 41.5%

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

       6. +-commutative 41.5%

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

       7. associate-*r/ 41.5%

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

       8. *-commutative 41.5%

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

       9. associate-/l* 41.5%

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

       10. fma-define 41.5%

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

   13. Simplified 41.5%

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

   14. Taylor expanded in n around 0 69.8%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2200000:\\ \;\;\;\;{\left(\frac{k}{2 \cdot \left(\pi \cdot n\right)}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2200000:\\ \;\;\;\;\sqrt{2 \cdot \left(\pi \cdot \frac{n}{k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2200000.0) (sqrt (* 2.0 (* PI (/ n k)))) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2200000.0) {
		tmp = sqrt((2.0 * (((double) M_PI) * (n / k))));
	} else {
		tmp = sqrt(0.0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2200000.0) {
		tmp = Math.sqrt((2.0 * (Math.PI * (n / k))));
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2200000.0:
		tmp = math.sqrt((2.0 * (math.pi * (n / k))))
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2200000.0)
		tmp = sqrt(Float64(2.0 * Float64(pi * Float64(n / k))));
	else
		tmp = sqrt(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2200000.0)
		tmp = sqrt((2.0 * (pi * (n / k))));
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2200000.0], N[Sqrt[N[(2.0 * N[(Pi * N[(n / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.2e6

   1. Initial program 99.0%

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

      1. associate-*l/ 99.1%

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

      2. *-lft-identity 99.1%

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

      3. associate-*l* 99.1%

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

      4. div-sub 99.1%

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

      5. sub-neg 99.1%

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

      6. distribute-frac-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. neg-mul-1 99.1%

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

      9. *-commutative 99.1%

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

      10. associate-/l* 99.1%

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

      11. fma-define 99.1%

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

      12. metadata-eval 99.1%

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

      13. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in k around 0 69.6%

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

      1. associate-/l* 69.6%

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

   7. Simplified 69.6%

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

      1. pow1 69.6%

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

      2. *-commutative 69.6%

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

      3. sqrt-unprod 69.8%

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

   9. Applied egg-rr 69.8%

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

       1. unpow1 69.8%

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

       2. associate-*r/ 69.8%

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

       3. *-commutative 69.8%

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

       4. associate-/l* 69.9%

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

   11. Simplified 69.9%

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

   if 2.2e6 < k

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. div-sub 100.0%

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

      5. sub-neg 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-/l* 100.0%

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

      11. fma-define 100.0%

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

      12. metadata-eval 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 2.0%

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

      1. associate-/l* 2.0%

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

   7. Simplified 2.0%

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

      1. *-commutative 2.0%

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

      2. sqrt-unprod 2.0%

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

   9. Applied egg-rr 2.0%

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

       1. expm1-log1p-u 2.0%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)\right)}} \]

       2. expm1-undefine 41.5%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} - 1\right)}} \]

   11. Applied egg-rr 41.5%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} - 1\right)}} \]
   12. Step-by-step derivation

       1. sub-neg 41.5%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} + \left(-1\right)\right)}} \]

       2. metadata-eval 41.5%

          \[\leadsto \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} + \color{blue}{-1}\right)} \]

       3. +-commutative 41.5%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)}\right)}} \]

       4. log1p-undefine 41.5%

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

       5. rem-exp-log 41.5%

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

       6. +-commutative 41.5%

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

       7. associate-*r/ 41.5%

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

       8. *-commutative 41.5%

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

       9. associate-/l* 41.5%

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

       10. fma-define 41.5%

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

   13. Simplified 41.5%

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

   14. Taylor expanded in n around 0 69.8%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2200000:\\ \;\;\;\;\sqrt{2 \cdot \left(\pi \cdot \frac{n}{k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2200000:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2200000.0) (sqrt (* 2.0 (* n (/ PI k)))) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2200000.0) {
		tmp = sqrt((2.0 * (n * (((double) M_PI) / k))));
	} else {
		tmp = sqrt(0.0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2200000.0) {
		tmp = Math.sqrt((2.0 * (n * (Math.PI / k))));
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2200000.0:
		tmp = math.sqrt((2.0 * (n * (math.pi / k))))
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2200000.0)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(pi / k))));
	else
		tmp = sqrt(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2200000.0)
		tmp = sqrt((2.0 * (n * (pi / k))));
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2200000.0], N[Sqrt[N[(2.0 * N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.2e6

   1. Initial program 99.0%

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

      1. associate-*l/ 99.1%

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

      2. *-lft-identity 99.1%

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

      3. associate-*l* 99.1%

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

      4. div-sub 99.1%

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

      5. sub-neg 99.1%

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

      6. distribute-frac-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. neg-mul-1 99.1%

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

      9. *-commutative 99.1%

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

      10. associate-/l* 99.1%

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

      11. fma-define 99.1%

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

      12. metadata-eval 99.1%

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

      13. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in k around 0 69.6%

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

      1. associate-/l* 69.6%

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

   7. Simplified 69.6%

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

      1. *-commutative 69.6%

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

      2. sqrt-unprod 69.8%

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

   9. Applied egg-rr 69.8%

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

   if 2.2e6 < k

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. div-sub 100.0%

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

      5. sub-neg 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. neg-mul-1 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-/l* 100.0%

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

      11. fma-define 100.0%

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

      12. metadata-eval 100.0%

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

      13. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 2.0%

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

      1. associate-/l* 2.0%

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

   7. Simplified 2.0%

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

      1. *-commutative 2.0%

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

      2. sqrt-unprod 2.0%

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

   9. Applied egg-rr 2.0%

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

       1. expm1-log1p-u 2.0%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)\right)}} \]

       2. expm1-undefine 41.5%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} - 1\right)}} \]

   11. Applied egg-rr 41.5%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} - 1\right)}} \]
   12. Step-by-step derivation

       1. sub-neg 41.5%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} + \left(-1\right)\right)}} \]

       2. metadata-eval 41.5%

          \[\leadsto \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} + \color{blue}{-1}\right)} \]

       3. +-commutative 41.5%

          \[\leadsto \sqrt{2 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)}\right)}} \]

       4. log1p-undefine 41.5%

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

       5. rem-exp-log 41.5%

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

       6. +-commutative 41.5%

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

       7. associate-*r/ 41.5%

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

       8. *-commutative 41.5%

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

       9. associate-/l* 41.5%

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

       10. fma-define 41.5%

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

   13. Simplified 41.5%

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

   14. Taylor expanded in n around 0 69.8%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2200000:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{0} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt 0.0))

C

double code(double k, double n) {
	return sqrt(0.0);
}

Fortran

real(8) function code(k, n)
    real(8), intent (in) :: k
    real(8), intent (in) :: n
    code = sqrt(0.0d0)
end function

Java

public static double code(double k, double n) {
	return Math.sqrt(0.0);
}

Python

def code(k, n):
	return math.sqrt(0.0)

Julia

function code(k, n)
	return sqrt(0.0)
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt(0.0);
end

Wolfram

code[k_, n_] := N[Sqrt[0.0], $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.7%

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

   2. *-lft-identity 99.7%

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

   3. associate-*l* 99.7%

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

   4. div-sub 99.7%

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

   5. sub-neg 99.7%

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

   6. distribute-frac-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. *-commutative 99.7%

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

   10. associate-/l* 99.7%

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

   11. fma-define 99.7%

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

   12. metadata-eval 99.7%

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

   13. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in k around 0 29.2%

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

   1. associate-/l* 29.2%

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

7. Simplified 29.2%

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

   1. *-commutative 29.2%

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

   2. sqrt-unprod 29.3%

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

9. Applied egg-rr 29.3%

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

    1. expm1-log1p-u 28.0%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)\right)}} \]

    2. expm1-undefine 44.2%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} - 1\right)}} \]

11. Applied egg-rr 44.2%

    \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} - 1\right)}} \]
12. Step-by-step derivation

    1. sub-neg 44.2%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} + \left(-1\right)\right)}} \]

    2. metadata-eval 44.2%

       \[\leadsto \sqrt{2 \cdot \left(e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)} + \color{blue}{-1}\right)} \]

    3. +-commutative 44.2%

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(n \cdot \frac{\pi}{k}\right)}\right)}} \]

    4. log1p-undefine 44.2%

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

    5. rem-exp-log 45.5%

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

    6. +-commutative 45.5%

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

    7. associate-*r/ 45.5%

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

    8. *-commutative 45.5%

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

    9. associate-/l* 45.5%

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

    10. fma-define 45.5%

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

13. Simplified 45.5%

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

14. Taylor expanded in n around 0 42.7%

    \[\leadsto \sqrt{2 \cdot \left(-1 + \color{blue}{1}\right)} \]

15. Final simplification 42.7%

    \[\leadsto \sqrt{0} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(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))))