Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.2%

Time: 15.0s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{k}}{\sqrt{n \cdot \left(2 \cdot \pi\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 9.8e-28)
   (/ 1.0 (/ (sqrt k) (sqrt (* n (* 2.0 PI)))))
   (sqrt (/ (pow (* 2.0 (* PI n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 9.8e-28) {
		tmp = 1.0 / (sqrt(k) / sqrt((n * (2.0 * ((double) M_PI)))));
	} else {
		tmp = sqrt((pow((2.0 * (((double) M_PI) * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 9.8e-28) {
		tmp = 1.0 / (Math.sqrt(k) / Math.sqrt((n * (2.0 * Math.PI))));
	} else {
		tmp = Math.sqrt((Math.pow((2.0 * (Math.PI * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 9.8e-28:
		tmp = 1.0 / (math.sqrt(k) / math.sqrt((n * (2.0 * math.pi))))
	else:
		tmp = math.sqrt((math.pow((2.0 * (math.pi * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 9.8e-28)
		tmp = Float64(1.0 / Float64(sqrt(k) / sqrt(Float64(n * Float64(2.0 * pi)))));
	else
		tmp = sqrt(Float64((Float64(2.0 * Float64(pi * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 9.8e-28)
		tmp = 1.0 / (sqrt(k) / sqrt((n * (2.0 * pi))));
	else
		tmp = sqrt((((2.0 * (pi * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 9.8e-28], N[(1.0 / N[(N[Sqrt[k], $MachinePrecision] / N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.80000000000000059e-28

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0 99.2%

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

      1. associate-*l/ 99.1%

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

      2. *-un-lft-identity 99.1%

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

      3. clear-num 99.1%

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

      4. sqrt-unprod 99.4%

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

      5. *-commutative 99.4%

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

      6. *-commutative 99.4%

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

      7. associate-*r* 99.4%

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

   5. Applied egg-rr 99.4%

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

   if 9.80000000000000059e-28 < k

   1. Initial program 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. metadata-eval 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r* 99.7%

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

      5. metadata-eval 99.7%

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

      6. neg-mul-1 99.7%

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

      7. sub-neg 99.7%

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

      8. *-commutative 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{k}}{\sqrt{n \cdot \left(2 \cdot \pi\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.5:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.5) (* (pow k -0.5) (sqrt (* PI (* 2.0 n)))) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.5) {
		tmp = pow(k, -0.5) * sqrt((((double) M_PI) * (2.0 * n)));
	} else {
		tmp = sqrt(0.0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.5) {
		tmp = Math.pow(k, -0.5) * Math.sqrt((Math.PI * (2.0 * n)));
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.5:
		tmp = math.pow(k, -0.5) * math.sqrt((math.pi * (2.0 * n)))
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.5)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(pi * Float64(2.0 * n))));
	else
		tmp = sqrt(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.5)
		tmp = (k ^ -0.5) * sqrt((pi * (2.0 * n)));
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.5], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.5

   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. Add Preprocessing

   3. Taylor expanded in k around 0 71.8%

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

      1. *-commutative 71.8%

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

      2. associate-/l* 71.7%

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

   5. Simplified 71.7%

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

      1. pow1 71.7%

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

      2. sqrt-unprod 72.0%

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

      3. associate-*r* 72.0%

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

   7. Applied egg-rr 72.0%

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

      1. unpow1 72.0%

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

      2. associate-*l* 72.0%

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

      3. associate-*r/ 72.0%

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

      4. *-commutative 72.0%

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

      5. associate-/l* 72.0%

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

   9. Simplified 72.0%

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

       1. associate-*r* 72.0%

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

       2. sqrt-prod 71.5%

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

       3. sqrt-div 96.3%

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

       4. associate-/l* 96.3%

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

       5. sqrt-prod 96.7%

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

       6. div-inv 96.7%

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

       7. *-commutative 96.7%

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

       8. inv-pow 96.7%

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

       9. sqrt-pow2 96.8%

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

       10. metadata-eval 96.8%

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

       11. associate-*r* 96.8%

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

   11. Applied egg-rr 96.8%

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

       1. *-commutative 96.8%

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

       2. associate-*r* 96.8%

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

       3. *-commutative 96.8%

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

   13. Simplified 96.8%

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

   if 3.5 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.7%

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

      1. *-commutative 2.7%

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

      2. associate-/l* 2.7%

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

   5. Simplified 2.7%

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

      1. pow1 2.7%

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

      2. sqrt-unprod 2.7%

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

      3. associate-*r* 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. associate-*l* 2.7%

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

      3. associate-*r/ 2.7%

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

      4. *-commutative 2.7%

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

      5. associate-/l* 2.7%

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

   9. Simplified 2.7%

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

       1. expm1-log1p-u 2.7%

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

       2. expm1-undefine 23.2%

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

       3. associate-*r/ 23.2%

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

       4. *-commutative 23.2%

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

       5. associate-/l* 23.2%

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

   11. Applied egg-rr 23.2%

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

       1. sub-neg 23.2%

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

       2. metadata-eval 23.2%

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

       3. +-commutative 23.2%

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

       4. log1p-undefine 23.2%

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

       5. rem-exp-log 23.2%

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

       6. +-commutative 23.2%

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

       7. associate-*r* 23.2%

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

       8. *-commutative 23.2%

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

       9. associate-*l* 23.2%

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

       10. associate-/l* 23.2%

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

       11. fma-define 23.2%

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

       12. *-commutative 23.2%

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

       13. associate-/l* 23.2%

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

   13. Simplified 23.2%

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

   14. Taylor expanded in n around 0 54.9%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(k * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[k, -0.5], $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. Step-by-step derivation

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. associate-*l* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. metadata-eval 99.5%

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

   2. div-sub 99.5%

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

   3. associate-*r* 99.5%

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

   4. div-inv 99.6%

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

   5. associate-*r* 99.6%

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

   6. div-sub 99.6%

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

   7. metadata-eval 99.6%

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

   8. sub-neg 99.6%

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

   9. div-inv 99.6%

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

   10. metadata-eval 99.6%

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

   11. distribute-rgt-neg-in 99.6%

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

   12. metadata-eval 99.6%

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

   13. pow1/2 99.6%

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

   14. pow-flip 99.6%

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

   15. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

Alternative 4: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.8:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.8) (/ (sqrt (* n (* 2.0 PI))) (sqrt k)) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.8) {
		tmp = sqrt((n * (2.0 * ((double) M_PI)))) / sqrt(k);
	} else {
		tmp = sqrt(0.0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.8) {
		tmp = Math.sqrt((n * (2.0 * Math.PI))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.8:
		tmp = math.sqrt((n * (2.0 * math.pi))) / math.sqrt(k)
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.8)
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * pi))) / sqrt(k));
	else
		tmp = sqrt(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.8)
		tmp = sqrt((n * (2.0 * pi))) / sqrt(k);
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.8], N[(N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.7999999999999998

   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. Add Preprocessing

   3. Taylor expanded in k around 0 96.6%

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

      1. *-commutative 96.6%

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

      2. div-inv 96.5%

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

      3. sqrt-unprod 96.7%

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

      4. *-commutative 96.7%

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

      5. *-commutative 96.7%

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

      6. associate-*r* 96.7%

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

   5. Applied egg-rr 96.7%

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

   if 2.7999999999999998 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.7%

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

      1. *-commutative 2.7%

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

      2. associate-/l* 2.7%

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

   5. Simplified 2.7%

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

      1. pow1 2.7%

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

      2. sqrt-unprod 2.7%

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

      3. associate-*r* 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. associate-*l* 2.7%

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

      3. associate-*r/ 2.7%

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

      4. *-commutative 2.7%

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

      5. associate-/l* 2.7%

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

   9. Simplified 2.7%

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

       1. expm1-log1p-u 2.7%

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

       2. expm1-undefine 23.2%

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

       3. associate-*r/ 23.2%

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

       4. *-commutative 23.2%

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

       5. associate-/l* 23.2%

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

   11. Applied egg-rr 23.2%

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

       1. sub-neg 23.2%

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

       2. metadata-eval 23.2%

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

       3. +-commutative 23.2%

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

       4. log1p-undefine 23.2%

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

       5. rem-exp-log 23.2%

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

       6. +-commutative 23.2%

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

       7. associate-*r* 23.2%

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

       8. *-commutative 23.2%

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

       9. associate-*l* 23.2%

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

       10. associate-/l* 23.2%

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

       11. fma-define 23.2%

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

       12. *-commutative 23.2%

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

       13. associate-/l* 23.2%

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

   13. Simplified 23.2%

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

   14. Taylor expanded in n around 0 54.9%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.8:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.8) (* (sqrt (* 2.0 (/ PI k))) (sqrt n)) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.8) {
		tmp = sqrt((2.0 * (((double) M_PI) / k))) * sqrt(n);
	} else {
		tmp = sqrt(0.0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.8) {
		tmp = Math.sqrt((2.0 * (Math.PI / k))) * Math.sqrt(n);
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.8:
		tmp = math.sqrt((2.0 * (math.pi / k))) * math.sqrt(n)
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.8)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi / k))) * sqrt(n));
	else
		tmp = sqrt(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.8)
		tmp = sqrt((2.0 * (pi / k))) * sqrt(n);
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.8], N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.7999999999999998

   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. Add Preprocessing

   3. Taylor expanded in k around 0 71.8%

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

      1. *-commutative 71.8%

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

      2. associate-/l* 71.7%

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

   5. Simplified 71.7%

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

   6. Taylor expanded in n around 0 71.8%

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

      1. *-commutative 71.8%

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

      2. *-commutative 71.8%

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

      3. associate-/l* 71.8%

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

   8. Simplified 71.8%

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

      1. sqrt-prod 72.0%

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

      2. associate-*r/ 72.0%

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

      3. *-commutative 72.0%

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

      4. associate-/l* 72.0%

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

   10. Applied egg-rr 72.0%

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

       1. associate-*r/ 72.0%

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

       2. *-commutative 72.0%

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

       3. associate-*r/ 72.0%

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

       4. *-commutative 72.0%

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

       5. associate-*r/ 72.0%

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

       6. *-commutative 72.0%

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

       7. associate-*r* 71.9%

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

       8. *-commutative 71.9%

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

       9. sqrt-prod 95.9%

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

   12. Applied egg-rr 95.9%

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

       1. associate-*r/ 96.0%

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

       2. *-commutative 96.0%

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

       3. associate-*r/ 96.0%

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

   14. Simplified 96.0%

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

   if 2.7999999999999998 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.7%

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

      1. *-commutative 2.7%

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

      2. associate-/l* 2.7%

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

   5. Simplified 2.7%

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

      1. pow1 2.7%

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

      2. sqrt-unprod 2.7%

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

      3. associate-*r* 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. associate-*l* 2.7%

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

      3. associate-*r/ 2.7%

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

      4. *-commutative 2.7%

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

      5. associate-/l* 2.7%

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

   9. Simplified 2.7%

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

       1. expm1-log1p-u 2.7%

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

       2. expm1-undefine 23.2%

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

       3. associate-*r/ 23.2%

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

       4. *-commutative 23.2%

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

       5. associate-/l* 23.2%

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

   11. Applied egg-rr 23.2%

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

       1. sub-neg 23.2%

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

       2. metadata-eval 23.2%

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

       3. +-commutative 23.2%

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

       4. log1p-undefine 23.2%

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

       5. rem-exp-log 23.2%

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

       6. +-commutative 23.2%

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

       7. associate-*r* 23.2%

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

       8. *-commutative 23.2%

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

       9. associate-*l* 23.2%

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

       10. associate-/l* 23.2%

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

       11. fma-define 23.2%

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

       12. *-commutative 23.2%

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

       13. associate-/l* 23.2%

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

   13. Simplified 23.2%

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

   14. Taylor expanded in n around 0 54.9%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. associate-*l* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

3. Simplified 99.5%

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

Alternative 7: 61.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.1:\\ \;\;\;\;\sqrt{2 \cdot \left(\pi \cdot \left(n \cdot \frac{1}{k}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.1) (sqrt (* 2.0 (* PI (* n (/ 1.0 k))))) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.1) {
		tmp = sqrt((2.0 * (((double) M_PI) * (n * (1.0 / k)))));
	} else {
		tmp = sqrt(0.0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.1) {
		tmp = Math.sqrt((2.0 * (Math.PI * (n * (1.0 / k)))));
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.1:
		tmp = math.sqrt((2.0 * (math.pi * (n * (1.0 / k)))))
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.1)
		tmp = sqrt(Float64(2.0 * Float64(pi * Float64(n * Float64(1.0 / k)))));
	else
		tmp = sqrt(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.1)
		tmp = sqrt((2.0 * (pi * (n * (1.0 / k)))));
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.1], N[Sqrt[N[(2.0 * N[(Pi * N[(n * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.10000000000000009

   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. Add Preprocessing

   3. Taylor expanded in k around 0 71.8%

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

      1. *-commutative 71.8%

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

      2. associate-/l* 71.7%

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

   5. Simplified 71.7%

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

      1. pow1 71.7%

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

      2. sqrt-unprod 72.0%

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

      3. associate-*r* 72.0%

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

   7. Applied egg-rr 72.0%

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

      1. unpow1 72.0%

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

      2. associate-*l* 72.0%

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

      3. associate-*r/ 72.0%

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

      4. *-commutative 72.0%

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

      5. associate-/l* 72.0%

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

   9. Simplified 72.0%

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

       1. associate-*r/ 72.0%

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

       2. clear-num 72.0%

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

   11. Applied egg-rr 72.0%

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

       1. associate-/r/ 72.0%

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

       2. *-commutative 72.0%

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

       3. associate-*r* 72.0%

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

   13. Applied egg-rr 72.0%

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

   if 3.10000000000000009 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.7%

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

      1. *-commutative 2.7%

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

      2. associate-/l* 2.7%

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

   5. Simplified 2.7%

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

      1. pow1 2.7%

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

      2. sqrt-unprod 2.7%

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

      3. associate-*r* 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. associate-*l* 2.7%

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

      3. associate-*r/ 2.7%

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

      4. *-commutative 2.7%

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

      5. associate-/l* 2.7%

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

   9. Simplified 2.7%

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

       1. expm1-log1p-u 2.7%

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

       2. expm1-undefine 23.2%

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

       3. associate-*r/ 23.2%

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

       4. *-commutative 23.2%

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

       5. associate-/l* 23.2%

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

   11. Applied egg-rr 23.2%

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

       1. sub-neg 23.2%

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

       2. metadata-eval 23.2%

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

       3. +-commutative 23.2%

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

       4. log1p-undefine 23.2%

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

       5. rem-exp-log 23.2%

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

       6. +-commutative 23.2%

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

       7. associate-*r* 23.2%

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

       8. *-commutative 23.2%

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

       9. associate-*l* 23.2%

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

       10. associate-/l* 23.2%

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

       11. fma-define 23.2%

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

       12. *-commutative 23.2%

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

       13. associate-/l* 23.2%

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

   13. Simplified 23.2%

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

   14. Taylor expanded in n around 0 54.9%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.1:\\ \;\;\;\;\sqrt{2 \cdot \left(\pi \cdot \left(n \cdot \frac{1}{k}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.2:\\ \;\;\;\;\sqrt{\frac{2}{\frac{k}{\pi \cdot n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.2) (sqrt (/ 2.0 (/ k (* PI n)))) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.2) {
		tmp = sqrt((2.0 / (k / (((double) M_PI) * n))));
	} else {
		tmp = sqrt(0.0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.2) {
		tmp = Math.sqrt((2.0 / (k / (Math.PI * n))));
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.2:
		tmp = math.sqrt((2.0 / (k / (math.pi * n))))
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.2)
		tmp = sqrt(Float64(2.0 / Float64(k / Float64(pi * n))));
	else
		tmp = sqrt(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.2)
		tmp = sqrt((2.0 / (k / (pi * n))));
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.2], N[Sqrt[N[(2.0 / N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.2000000000000002

   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. Add Preprocessing

   3. Taylor expanded in k around 0 71.8%

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

      1. *-commutative 71.8%

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

      2. associate-/l* 71.7%

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

   5. Simplified 71.7%

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

   6. Taylor expanded in n around 0 71.8%

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

      1. *-commutative 71.8%

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

      2. *-commutative 71.8%

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

      3. associate-/l* 71.8%

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

   8. Simplified 71.8%

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

      1. sqrt-prod 72.0%

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

      2. associate-*r/ 72.0%

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

      3. *-commutative 72.0%

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

      4. associate-/l* 72.0%

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

   10. Applied egg-rr 72.0%

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

       1. associate-*r/ 72.0%

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

       2. *-commutative 72.0%

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

       3. clear-num 72.0%

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

       4. un-div-inv 72.0%

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

       5. *-commutative 72.0%

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

   12. Applied egg-rr 72.0%

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

   if 3.2000000000000002 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.7%

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

      1. *-commutative 2.7%

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

      2. associate-/l* 2.7%

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

   5. Simplified 2.7%

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

      1. pow1 2.7%

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

      2. sqrt-unprod 2.7%

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

      3. associate-*r* 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. associate-*l* 2.7%

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

      3. associate-*r/ 2.7%

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

      4. *-commutative 2.7%

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

      5. associate-/l* 2.7%

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

   9. Simplified 2.7%

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

       1. expm1-log1p-u 2.7%

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

       2. expm1-undefine 23.2%

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

       3. associate-*r/ 23.2%

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

       4. *-commutative 23.2%

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

       5. associate-/l* 23.2%

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

   11. Applied egg-rr 23.2%

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

       1. sub-neg 23.2%

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

       2. metadata-eval 23.2%

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

       3. +-commutative 23.2%

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

       4. log1p-undefine 23.2%

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

       5. rem-exp-log 23.2%

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

       6. +-commutative 23.2%

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

       7. associate-*r* 23.2%

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

       8. *-commutative 23.2%

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

       9. associate-*l* 23.2%

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

       10. associate-/l* 23.2%

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

       11. fma-define 23.2%

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

       12. *-commutative 23.2%

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

       13. associate-/l* 23.2%

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

   13. Simplified 23.2%

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

   14. Taylor expanded in n around 0 54.9%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2:\\ \;\;\;\;\sqrt{\frac{2}{\frac{k}{\pi \cdot n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.0) (sqrt (* 2.0 (* n (/ PI k)))) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.0) {
		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 <= 3.0) {
		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 <= 3.0:
		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 <= 3.0)
		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 <= 3.0)
		tmp = sqrt((2.0 * (n * (pi / k))));
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.0], 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 < 3

   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. Add Preprocessing

   3. Taylor expanded in k around 0 71.8%

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

      1. *-commutative 71.8%

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

      2. associate-/l* 71.7%

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

   5. Simplified 71.7%

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

   6. Taylor expanded in n around 0 71.8%

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

      1. *-commutative 71.8%

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

      2. *-commutative 71.8%

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

      3. associate-/l* 71.8%

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

   8. Simplified 71.8%

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

      1. sqrt-prod 72.0%

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

      2. associate-*r/ 72.0%

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

      3. *-commutative 72.0%

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

      4. associate-/l* 72.0%

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

   10. Applied egg-rr 72.0%

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

   if 3 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.7%

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

      1. *-commutative 2.7%

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

      2. associate-/l* 2.7%

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

   5. Simplified 2.7%

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

      1. pow1 2.7%

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

      2. sqrt-unprod 2.7%

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

      3. associate-*r* 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. associate-*l* 2.7%

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

      3. associate-*r/ 2.7%

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

      4. *-commutative 2.7%

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

      5. associate-/l* 2.7%

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

   9. Simplified 2.7%

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

       1. expm1-log1p-u 2.7%

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

       2. expm1-undefine 23.2%

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

       3. associate-*r/ 23.2%

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

       4. *-commutative 23.2%

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

       5. associate-/l* 23.2%

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

   11. Applied egg-rr 23.2%

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

       1. sub-neg 23.2%

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

       2. metadata-eval 23.2%

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

       3. +-commutative 23.2%

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

       4. log1p-undefine 23.2%

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

       5. rem-exp-log 23.2%

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

       6. +-commutative 23.2%

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

       7. associate-*r* 23.2%

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

       8. *-commutative 23.2%

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

       9. associate-*l* 23.2%

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

       10. associate-/l* 23.2%

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

       11. fma-define 23.2%

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

       12. *-commutative 23.2%

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

       13. associate-/l* 23.2%

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

   13. Simplified 23.2%

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

   14. Taylor expanded in n around 0 54.9%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 26.6% 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. Add Preprocessing

3. Taylor expanded in k around 0 32.9%

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

   1. *-commutative 32.9%

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

   2. associate-/l* 32.9%

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

5. Simplified 32.9%

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

   1. pow1 32.9%

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

   2. sqrt-unprod 33.0%

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

   3. associate-*r* 33.0%

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

7. Applied egg-rr 33.0%

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

   1. unpow1 33.0%

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

   2. associate-*l* 33.0%

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

   3. associate-*r/ 33.0%

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

   4. *-commutative 33.0%

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

   5. associate-/l* 33.0%

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

9. Simplified 33.0%

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

    1. expm1-log1p-u 31.5%

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

    2. expm1-undefine 33.7%

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

    3. associate-*r/ 33.7%

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

    4. *-commutative 33.7%

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

    5. associate-/l* 33.7%

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

11. Applied egg-rr 33.7%

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

    1. sub-neg 33.7%

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

    2. metadata-eval 33.7%

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

    3. +-commutative 33.7%

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

    4. log1p-undefine 33.7%

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

    5. rem-exp-log 35.2%

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

    6. +-commutative 35.2%

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

    7. associate-*r* 35.2%

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

    8. *-commutative 35.2%

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

    9. associate-*l* 35.2%

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

    10. associate-/l* 35.2%

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

    11. fma-define 35.2%

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

    12. *-commutative 35.2%

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

    13. associate-/l* 35.2%

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

13. Simplified 35.2%

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

14. Taylor expanded in n around 0 32.0%

    \[\leadsto \sqrt{-1 + \color{blue}{1}} \]

15. Final simplification 32.0%

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

Reproduce

herbie shell --seed 2024144 
(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))))