Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.6%

Time: 13.4s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. k = 3.9705379456882177e+170
  2. n = -0.008518371145348362

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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. pow-div 99.8%

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

   7. pow1/2 99.8%

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

   8. associate-/l/ 99.8%

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

   9. div-inv 99.8%

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

   10. metadata-eval 99.8%

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

4. Applied egg-rr 99.8%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.06 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.06e-16)
   (/ (sqrt (* n (* 2.0 PI))) (sqrt k))
   (sqrt (/ (pow (* 2.0 (* PI n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.06e-16) {
		tmp = sqrt((n * (2.0 * ((double) M_PI)))) / sqrt(k);
	} 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 <= 1.06e-16) {
		tmp = Math.sqrt((n * (2.0 * Math.PI))) / Math.sqrt(k);
	} 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 <= 1.06e-16:
		tmp = math.sqrt((n * (2.0 * math.pi))) / math.sqrt(k)
	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 <= 1.06e-16)
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * pi))) / sqrt(k));
	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 <= 1.06e-16)
		tmp = sqrt((n * (2.0 * pi))) / sqrt(k);
	else
		tmp = sqrt((((2.0 * (pi * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.06e-16], N[(N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $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 < 1.06e-16

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 99.0%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\left(\sqrt{n \cdot \pi} \cdot \sqrt{2}\right)} \]
   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. *-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. sqrt-prod 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. associate-*r* 99.5%

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

      2. *-commutative 99.5%

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

   7. Simplified 99.5%

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

   if 1.06e-16 < 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. Add Preprocessing

   4. Applied egg-rr 99.8%

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

      1. distribute-lft-in 99.8%

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

      2. metadata-eval 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r* 99.8%

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

      5. metadata-eval 99.8%

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

      6. neg-mul-1 99.8%

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

      7. sub-neg 99.8%

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

      8. *-commutative 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.06 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.1) {
		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 <= 3.1) {
		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 <= 3.1:
		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 <= 3.1)
		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 <= 3.1)
		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, 3.1], 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 < 3.10000000000000009

   1. Initial program 99.1%

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

   3. Taylor expanded in k around 0 96.9%

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

      1. associate-*l/ 96.9%

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

      2. *-un-lft-identity 96.9%

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

      3. *-commutative 96.9%

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

      4. *-commutative 96.9%

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

      5. sqrt-prod 97.3%

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

   5. Applied egg-rr 97.3%

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

      1. associate-*r* 97.3%

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

      2. *-commutative 97.3%

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

   7. Simplified 97.3%

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

   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 2.6%

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

      1. associate-/l* 2.6%

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

   5. Simplified 2.6%

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

      1. pow1 2.6%

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

      2. *-commutative 2.6%

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

      3. sqrt-unprod 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. unpow1 2.6%

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

      2. *-commutative 2.6%

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

      3. *-commutative 2.6%

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

      4. associate-*l* 2.6%

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

   9. Simplified 2.6%

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

       1. expm1-log1p-u 2.6%

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

       2. expm1-undefine 24.5%

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

       3. associate-*l/ 24.5%

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

       4. associate-/l* 24.5%

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

   11. Applied egg-rr 24.5%

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

       1. sub-neg 24.5%

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

       2. metadata-eval 24.5%

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

       3. +-commutative 24.5%

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

       4. log1p-undefine 24.5%

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

       5. rem-exp-log 24.5%

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

       6. +-commutative 24.5%

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

       7. associate-*r/ 24.5%

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

       8. *-rgt-identity 24.5%

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

       9. times-frac 24.5%

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

       10. /-rgt-identity 24.5%

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

       11. associate-*l* 24.5%

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

       12. *-commutative 24.5%

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

       13. associate-*l* 24.5%

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

       14. *-commutative 24.5%

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

       15. fma-define 24.5%

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

       16. associate-*r/ 24.5%

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

       17. *-commutative 24.5%

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

       18. associate-/l* 24.5%

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

   13. Simplified 24.5%

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

   14. Taylor expanded in n around 0 48.2%

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

4. Final simplification 71.4%

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

Alternative 4: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.2)
		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, 3.2], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.2000000000000002

   1. Initial program 99.1%

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

   3. Taylor expanded in k around 0 73.9%

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

      1. associate-/l* 73.9%

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

   5. Simplified 73.9%

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

      1. pow1 73.9%

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

      2. *-commutative 73.9%

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

      3. sqrt-unprod 74.1%

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

   7. Applied egg-rr 74.1%

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

      1. unpow1 74.1%

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

      2. *-commutative 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-*l* 74.1%

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

   9. Simplified 74.1%

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

   10. Taylor expanded in k around 0 74.0%

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

       1. associate-*r/ 74.0%

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

       2. *-commutative 74.0%

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

       3. associate-*l/ 74.0%

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

       4. associate-/l* 74.1%

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

       5. associate-*l* 74.1%

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

       6. *-commutative 74.1%

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

       7. associate-*r/ 74.1%

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

       8. *-commutative 74.1%

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

       9. associate-/l* 74.1%

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

   12. Simplified 74.1%

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

       1. sqrt-prod 97.0%

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

   14. Applied egg-rr 97.0%

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

       1. associate-*r/ 97.1%

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

       2. *-commutative 97.1%

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

       3. associate-*r/ 97.1%

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

   16. Simplified 97.1%

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

   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 2.6%

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

      1. associate-/l* 2.6%

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

   5. Simplified 2.6%

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

      1. pow1 2.6%

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

      2. *-commutative 2.6%

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

      3. sqrt-unprod 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. unpow1 2.6%

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

      2. *-commutative 2.6%

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

      3. *-commutative 2.6%

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

      4. associate-*l* 2.6%

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

   9. Simplified 2.6%

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

       1. expm1-log1p-u 2.6%

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

       2. expm1-undefine 24.5%

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

       3. associate-*l/ 24.5%

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

       4. associate-/l* 24.5%

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

   11. Applied egg-rr 24.5%

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

       1. sub-neg 24.5%

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

       2. metadata-eval 24.5%

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

       3. +-commutative 24.5%

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

       4. log1p-undefine 24.5%

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

       5. rem-exp-log 24.5%

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

       6. +-commutative 24.5%

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

       7. associate-*r/ 24.5%

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

       8. *-rgt-identity 24.5%

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

       9. times-frac 24.5%

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

       10. /-rgt-identity 24.5%

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

       11. associate-*l* 24.5%

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

       12. *-commutative 24.5%

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

       13. associate-*l* 24.5%

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

       14. *-commutative 24.5%

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

       15. fma-define 24.5%

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

       16. associate-*r/ 24.5%

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

       17. *-commutative 24.5%

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

       18. associate-/l* 24.5%

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

   13. Simplified 24.5%

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

   14. Taylor expanded in n around 0 48.2%

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

4. Final simplification 71.3%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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. metadata-eval 99.7%

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

3. Simplified 99.7%

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

Alternative 6: 62.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.6:\\ \;\;\;\;{\left(\frac{\frac{k}{\pi}}{2 \cdot n}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.6) (pow (/ (/ k PI) (* 2.0 n)) -0.5) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.6) {
		tmp = pow(((k / ((double) M_PI)) / (2.0 * n)), -0.5);
	} else {
		tmp = sqrt(0.0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.6) {
		tmp = Math.pow(((k / Math.PI) / (2.0 * n)), -0.5);
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.6:
		tmp = math.pow(((k / math.pi) / (2.0 * n)), -0.5)
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.6)
		tmp = Float64(Float64(k / pi) / Float64(2.0 * n)) ^ -0.5;
	else
		tmp = sqrt(0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.6)
		tmp = ((k / pi) / (2.0 * n)) ^ -0.5;
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.6], N[Power[N[(N[(k / Pi), $MachinePrecision] / N[(2.0 * n), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.60000000000000009

   1. Initial program 99.1%

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

   3. Taylor expanded in k around 0 73.9%

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

      1. associate-/l* 73.9%

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

   5. Simplified 73.9%

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

      1. pow1 73.9%

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

      2. *-commutative 73.9%

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

      3. sqrt-unprod 74.1%

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

   7. Applied egg-rr 74.1%

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

      1. unpow1 74.1%

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

      2. *-commutative 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-*l* 74.1%

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

   9. Simplified 74.1%

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

   10. Taylor expanded in k around 0 74.0%

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

       1. associate-*r/ 74.0%

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

       2. *-commutative 74.0%

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

       3. associate-*l/ 74.0%

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

       4. associate-/l* 74.1%

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

       5. associate-*l* 74.1%

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

       6. *-commutative 74.1%

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

       7. associate-*r/ 74.1%

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

       8. *-commutative 74.1%

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

       9. associate-/l* 74.1%

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

   12. Simplified 74.1%

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

       1. associate-*r* 74.1%

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

       2. associate-*r/ 74.0%

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

       3. *-commutative 74.0%

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

       4. associate-*r* 74.0%

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

       5. *-commutative 74.0%

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

       6. *-commutative 74.0%

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

       7. clear-num 74.0%

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

       8. inv-pow 74.0%

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

       9. sqrt-pow1 75.5%

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

       10. associate-/r* 75.5%

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

       11. *-commutative 75.5%

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

       12. metadata-eval 75.5%

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

   14. Applied egg-rr 75.5%

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

   if 3.60000000000000009 < 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 2.6%

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

      1. associate-/l* 2.6%

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

   5. Simplified 2.6%

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

      1. pow1 2.6%

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

      2. *-commutative 2.6%

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

      3. sqrt-unprod 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. unpow1 2.6%

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

      2. *-commutative 2.6%

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

      3. *-commutative 2.6%

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

      4. associate-*l* 2.6%

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

   9. Simplified 2.6%

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

       1. expm1-log1p-u 2.6%

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

       2. expm1-undefine 24.5%

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

       3. associate-*l/ 24.5%

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

       4. associate-/l* 24.5%

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

   11. Applied egg-rr 24.5%

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

       1. sub-neg 24.5%

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

       2. metadata-eval 24.5%

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

       3. +-commutative 24.5%

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

       4. log1p-undefine 24.5%

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

       5. rem-exp-log 24.5%

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

       6. +-commutative 24.5%

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

       7. associate-*r/ 24.5%

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

       8. *-rgt-identity 24.5%

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

       9. times-frac 24.5%

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

       10. /-rgt-identity 24.5%

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

       11. associate-*l* 24.5%

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

       12. *-commutative 24.5%

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

       13. associate-*l* 24.5%

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

       14. *-commutative 24.5%

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

       15. fma-define 24.5%

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

       16. associate-*r/ 24.5%

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

       17. *-commutative 24.5%

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

       18. associate-/l* 24.5%

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

   13. Simplified 24.5%

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

   14. Taylor expanded in n around 0 48.2%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.6:\\ \;\;\;\;{\left(\frac{\frac{k}{\pi}}{2 \cdot n}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.15:\\ \;\;\;\;{\left(k \cdot \frac{0.5}{\pi \cdot n}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.15) (pow (* k (/ 0.5 (* PI n))) -0.5) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.15) {
		tmp = pow((k * (0.5 / (((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 <= 3.15) {
		tmp = Math.pow((k * (0.5 / (Math.PI * n))), -0.5);
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.15:
		tmp = math.pow((k * (0.5 / (math.pi * n))), -0.5)
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.15)
		tmp = Float64(k * Float64(0.5 / 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 <= 3.15)
		tmp = (k * (0.5 / (pi * n))) ^ -0.5;
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.15], N[Power[N[(k * N[(0.5 / N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.14999999999999991

   1. Initial program 99.1%

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

   3. Taylor expanded in k around 0 73.9%

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

      1. associate-/l* 73.9%

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

   5. Simplified 73.9%

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

      1. pow1 73.9%

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

      2. *-commutative 73.9%

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

      3. sqrt-unprod 74.1%

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

   7. Applied egg-rr 74.1%

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

      1. unpow1 74.1%

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

      2. *-commutative 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-*l* 74.1%

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

   9. Simplified 74.1%

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

   10. Taylor expanded in k around 0 74.0%

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

       1. associate-*r/ 74.0%

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

       2. *-commutative 74.0%

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

       3. associate-*l/ 74.0%

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

       4. associate-/l* 74.1%

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

       5. associate-*l* 74.1%

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

       6. *-commutative 74.1%

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

       7. associate-*r/ 74.1%

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

       8. *-commutative 74.1%

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

       9. associate-/l* 74.1%

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

   12. Simplified 74.1%

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

       1. associate-*r* 74.1%

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

       2. associate-*r/ 74.0%

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

       3. *-commutative 74.0%

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

       4. associate-*r* 74.0%

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

       5. *-commutative 74.0%

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

       6. *-commutative 74.0%

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

       7. clear-num 74.0%

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

       8. inv-pow 74.0%

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

       9. sqrt-pow1 75.5%

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

       10. associate-/r* 75.5%

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

       11. *-commutative 75.5%

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

       12. metadata-eval 75.5%

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

   14. Applied egg-rr 75.5%

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

       1. *-lft-identity 75.5%

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

       2. associate-*l/ 75.5%

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

       3. associate-/r* 75.5%

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

       4. metadata-eval 75.5%

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

       5. times-frac 75.5%

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

       6. associate-*l/ 75.5%

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

       7. *-commutative 75.5%

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

   16. Simplified 75.5%

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

   if 3.14999999999999991 < 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 2.6%

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

      1. associate-/l* 2.6%

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

   5. Simplified 2.6%

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

      1. pow1 2.6%

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

      2. *-commutative 2.6%

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

      3. sqrt-unprod 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. unpow1 2.6%

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

      2. *-commutative 2.6%

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

      3. *-commutative 2.6%

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

      4. associate-*l* 2.6%

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

   9. Simplified 2.6%

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

       1. expm1-log1p-u 2.6%

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

       2. expm1-undefine 24.5%

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

       3. associate-*l/ 24.5%

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

       4. associate-/l* 24.5%

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

   11. Applied egg-rr 24.5%

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

       1. sub-neg 24.5%

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

       2. metadata-eval 24.5%

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

       3. +-commutative 24.5%

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

       4. log1p-undefine 24.5%

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

       5. rem-exp-log 24.5%

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

       6. +-commutative 24.5%

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

       7. associate-*r/ 24.5%

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

       8. *-rgt-identity 24.5%

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

       9. times-frac 24.5%

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

       10. /-rgt-identity 24.5%

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

       11. associate-*l* 24.5%

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

       12. *-commutative 24.5%

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

       13. associate-*l* 24.5%

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

       14. *-commutative 24.5%

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

       15. fma-define 24.5%

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

       16. associate-*r/ 24.5%

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

       17. *-commutative 24.5%

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

       18. associate-/l* 24.5%

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

   13. Simplified 24.5%

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

   14. Taylor expanded in n around 0 48.2%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.15:\\ \;\;\;\;{\left(k \cdot \frac{0.5}{\pi \cdot n}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.15:\\ \;\;\;\;\sqrt{\frac{\pi}{k} \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.15) (sqrt (* (/ PI k) (* 2.0 n))) (sqrt 0.0)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.15) {
		tmp = sqrt(((((double) M_PI) / k) * (2.0 * n)));
	} else {
		tmp = sqrt(0.0);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.15) {
		tmp = Math.sqrt(((Math.PI / k) * (2.0 * n)));
	} else {
		tmp = Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.15:
		tmp = math.sqrt(((math.pi / k) * (2.0 * n)))
	else:
		tmp = math.sqrt(0.0)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.15)
		tmp = sqrt(Float64(Float64(pi / k) * 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 <= 3.15)
		tmp = sqrt(((pi / k) * (2.0 * n)));
	else
		tmp = sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.15], N[Sqrt[N[(N[(Pi / k), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.14999999999999991

   1. Initial program 99.1%

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

   3. Taylor expanded in k around 0 73.9%

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

      1. associate-/l* 73.9%

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

   5. Simplified 73.9%

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

      1. pow1 73.9%

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

      2. *-commutative 73.9%

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

      3. sqrt-unprod 74.1%

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

   7. Applied egg-rr 74.1%

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

      1. unpow1 74.1%

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

      2. *-commutative 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-*l* 74.1%

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

   9. Simplified 74.1%

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

   if 3.14999999999999991 < 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 2.6%

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

      1. associate-/l* 2.6%

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

   5. Simplified 2.6%

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

      1. pow1 2.6%

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

      2. *-commutative 2.6%

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

      3. sqrt-unprod 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. unpow1 2.6%

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

      2. *-commutative 2.6%

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

      3. *-commutative 2.6%

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

      4. associate-*l* 2.6%

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

   9. Simplified 2.6%

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

       1. expm1-log1p-u 2.6%

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

       2. expm1-undefine 24.5%

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

       3. associate-*l/ 24.5%

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

       4. associate-/l* 24.5%

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

   11. Applied egg-rr 24.5%

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

       1. sub-neg 24.5%

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

       2. metadata-eval 24.5%

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

       3. +-commutative 24.5%

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

       4. log1p-undefine 24.5%

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

       5. rem-exp-log 24.5%

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

       6. +-commutative 24.5%

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

       7. associate-*r/ 24.5%

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

       8. *-rgt-identity 24.5%

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

       9. times-frac 24.5%

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

       10. /-rgt-identity 24.5%

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

       11. associate-*l* 24.5%

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

       12. *-commutative 24.5%

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

       13. associate-*l* 24.5%

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

       14. *-commutative 24.5%

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

       15. fma-define 24.5%

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

       16. associate-*r/ 24.5%

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

       17. *-commutative 24.5%

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

       18. associate-/l* 24.5%

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

   13. Simplified 24.5%

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

   14. Taylor expanded in n around 0 48.2%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.15:\\ \;\;\;\;\sqrt{\frac{\pi}{k} \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 3.5

   1. Initial program 99.1%

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

   3. Taylor expanded in k around 0 73.9%

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

      1. associate-/l* 73.9%

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

   5. Simplified 73.9%

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

      1. pow1 73.9%

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

      2. *-commutative 73.9%

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

      3. sqrt-unprod 74.1%

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

   7. Applied egg-rr 74.1%

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

      1. unpow1 74.1%

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

      2. *-commutative 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-*l* 74.1%

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

   9. Simplified 74.1%

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

   10. Taylor expanded in k around 0 74.0%

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

       1. associate-*r/ 74.0%

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

       2. *-commutative 74.0%

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

       3. associate-*l/ 74.0%

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

       4. associate-/l* 74.1%

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

       5. associate-*l* 74.1%

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

       6. *-commutative 74.1%

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

       7. associate-*r/ 74.1%

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

       8. *-commutative 74.1%

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

       9. associate-/l* 74.1%

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

   12. Simplified 74.1%

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

   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 2.6%

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

      1. associate-/l* 2.6%

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

   5. Simplified 2.6%

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

      1. pow1 2.6%

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

      2. *-commutative 2.6%

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

      3. sqrt-unprod 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. unpow1 2.6%

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

      2. *-commutative 2.6%

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

      3. *-commutative 2.6%

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

      4. associate-*l* 2.6%

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

   9. Simplified 2.6%

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

       1. expm1-log1p-u 2.6%

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

       2. expm1-undefine 24.5%

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

       3. associate-*l/ 24.5%

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

       4. associate-/l* 24.5%

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

   11. Applied egg-rr 24.5%

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

       1. sub-neg 24.5%

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

       2. metadata-eval 24.5%

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

       3. +-commutative 24.5%

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

       4. log1p-undefine 24.5%

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

       5. rem-exp-log 24.5%

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

       6. +-commutative 24.5%

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

       7. associate-*r/ 24.5%

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

       8. *-rgt-identity 24.5%

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

       9. times-frac 24.5%

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

       10. /-rgt-identity 24.5%

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

       11. associate-*l* 24.5%

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

       12. *-commutative 24.5%

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

       13. associate-*l* 24.5%

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

       14. *-commutative 24.5%

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

       15. fma-define 24.5%

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

       16. associate-*r/ 24.5%

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

       17. *-commutative 24.5%

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

       18. associate-/l* 24.5%

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

   13. Simplified 24.5%

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

   14. Taylor expanded in n around 0 48.2%

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

4. Final simplification 60.5%

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

Alternative 10: 25.8% 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 36.3%

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

   1. associate-/l* 36.3%

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

5. Simplified 36.3%

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

   1. pow1 36.3%

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

   2. *-commutative 36.3%

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

   3. sqrt-unprod 36.4%

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

7. Applied egg-rr 36.4%

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

   1. unpow1 36.4%

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

   2. *-commutative 36.4%

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

   3. *-commutative 36.4%

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

   4. associate-*l* 36.4%

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

9. Simplified 36.4%

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

    1. expm1-log1p-u 34.6%

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

    2. expm1-undefine 37.3%

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

    3. associate-*l/ 37.3%

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

    4. associate-/l* 37.3%

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

11. Applied egg-rr 37.3%

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

    1. sub-neg 37.3%

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

    2. metadata-eval 37.3%

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

    3. +-commutative 37.3%

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

    4. log1p-undefine 37.3%

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

    5. rem-exp-log 39.1%

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

    6. +-commutative 39.1%

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

    7. associate-*r/ 39.1%

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

    8. *-rgt-identity 39.1%

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

    9. times-frac 39.1%

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

    10. /-rgt-identity 39.1%

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

    11. associate-*l* 39.1%

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

    12. *-commutative 39.1%

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

    13. associate-*l* 39.1%

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

    14. *-commutative 39.1%

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

    15. fma-define 39.1%

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

    16. associate-*r/ 39.1%

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

    17. *-commutative 39.1%

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

    18. associate-/l* 39.1%

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

13. Simplified 39.1%

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

14. Taylor expanded in n around 0 26.7%

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

15. Final simplification 26.7%

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

Reproduce

herbie shell --seed 2024186 
(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))))