Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 13.6s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. k = 1.395624662457704e+143
  2. n = -1.8530291640667726e-29

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / N[Power[t$95$0, N[(k * 0.5), $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-*r* 99.6%

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

   2. div-sub 99.6%

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

   3. metadata-eval 99.6%

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

   4. pow-div 99.7%

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

   5. pow1/2 99.7%

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

   6. associate-*r/ 99.7%

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

   7. pow1/2 99.7%

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

   8. pow-flip 99.8%

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

   9. metadata-eval 99.8%

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

   10. div-inv 99.8%

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

   11. metadata-eval 99.8%

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

4. Applied egg-rr 99.8%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(k, n)
	t_0 = 2.0 * (pi * n);
	tmp = sqrt(t_0) / ((t_0 ^ (k * 0.5)) * sqrt(k));
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[Power[t$95$0, N[(k * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[k], $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/ 99.6%

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

   2. *-un-lft-identity 99.6%

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

   3. associate-*r* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

   6. pow-div 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.1e-49)
   (* (sqrt (* 2.0 n)) (sqrt (/ PI k)))
   (sqrt (/ (pow (* 2.0 (* PI n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.1e-49) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / 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 <= 2.1e-49) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((Math.PI / 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 <= 2.1e-49:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((math.pi / 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 <= 2.1e-49)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / 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 <= 2.1e-49)
		tmp = sqrt((2.0 * n)) * sqrt((pi / k));
	else
		tmp = sqrt((((2.0 * (pi * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.1e-49], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $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 < 2.0999999999999999e-49

   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 71.5%

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

      1. associate-/l* 71.5%

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

   5. Simplified 71.5%

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

      1. *-commutative 71.5%

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

      2. sqrt-unprod 71.9%

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

   7. Applied egg-rr 71.9%

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

      1. pow1/2 71.9%

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

      2. associate-*r* 71.9%

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

      3. unpow-prod-down 99.6%

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

      4. pow1/2 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. unpow1/2 99.6%

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

       2. *-commutative 99.6%

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

   11. Simplified 99.6%

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

   if 2.0999999999999999e-49 < 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 99.2%

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

      1. *-commutative 99.2%

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

      2. distribute-lft-in 99.2%

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

      3. metadata-eval 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*r* 99.2%

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

      6. metadata-eval 99.2%

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

      7. neg-mul-1 99.2%

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

      8. sub-neg 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, \pi \cdot \frac{2}{k}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6.8e+28)
   (* (sqrt (* 2.0 n)) (sqrt (/ PI k)))
   (sqrt (+ -1.0 (fma n (* PI (/ 2.0 k)) 1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6.8e+28) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	} else {
		tmp = sqrt((-1.0 + fma(n, (((double) M_PI) * (2.0 / k)), 1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 6.8e+28)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	else
		tmp = sqrt(Float64(-1.0 + fma(n, Float64(pi * Float64(2.0 / k)), 1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6.8e+28], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-1.0 + N[(n * N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.8e28

   1. Initial program 99.2%

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

   3. Taylor expanded in k around 0 66.8%

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

      1. associate-/l* 66.8%

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

   5. Simplified 66.8%

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

      1. *-commutative 66.8%

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

      2. sqrt-unprod 67.1%

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

   7. Applied egg-rr 67.1%

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

      1. pow1/2 67.1%

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

      2. associate-*r* 67.1%

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

      3. unpow-prod-down 88.1%

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

      4. pow1/2 88.1%

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

   9. Applied egg-rr 88.1%

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

       1. unpow1/2 88.1%

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

       2. *-commutative 88.1%

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

   11. Simplified 88.1%

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

   if 6.8e28 < 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. associate-/l* 2.7%

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

   5. Simplified 2.7%

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

      1. *-commutative 2.7%

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

      2. sqrt-unprod 2.7%

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

   7. Applied egg-rr 2.7%

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

      1. associate-*r/ 2.7%

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

      2. *-commutative 2.7%

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

      3. associate-*r/ 2.7%

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

      4. associate-*r* 2.7%

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

      5. *-commutative 2.7%

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

      6. expm1-log1p-u 2.7%

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

      7. expm1-undefine 29.0%

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

   9. Applied egg-rr 29.0%

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

       1. sub-neg 29.0%

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

       2. metadata-eval 29.0%

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

       3. +-commutative 29.0%

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

       4. log1p-undefine 29.0%

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

       5. rem-exp-log 29.0%

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

       6. +-commutative 29.0%

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

       7. associate-*r* 29.0%

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

       8. associate-*r/ 29.0%

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

       9. associate-*l/ 29.0%

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

       10. *-commutative 29.0%

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

       11. associate-*r* 29.0%

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

       12. *-commutative 29.0%

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

       13. fma-define 29.0%

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

       14. *-commutative 29.0%

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

       15. associate-*l/ 29.0%

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

       16. associate-*r/ 29.0%

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

   11. Simplified 29.0%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, \pi \cdot \frac{2}{k}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. associate-*l* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. metadata-eval 99.6%

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

   2. div-sub 99.6%

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

   3. associate-*r* 99.6%

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

   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 99.7%

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

   15. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 6: 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/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. associate-*l* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

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

Alternative 7: 48.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $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. Taylor expanded in k around 0 35.0%

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

   1. associate-/l* 35.0%

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

5. Simplified 35.0%

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

   1. *-commutative 35.0%

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

   2. sqrt-unprod 35.2%

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

7. Applied egg-rr 35.2%

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

   1. pow1/2 35.2%

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

   2. associate-*r* 35.2%

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

   3. unpow-prod-down 45.8%

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

   4. pow1/2 45.8%

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

9. Applied egg-rr 45.8%

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

    1. unpow1/2 45.8%

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

    2. *-commutative 45.8%

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

11. Simplified 45.8%

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

12. Final simplification 45.8%

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

Alternative 8: 48.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $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. Taylor expanded in k around 0 35.0%

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

   1. associate-/l* 35.0%

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

5. Simplified 35.0%

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

   1. *-commutative 35.0%

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

   2. sqrt-unprod 35.2%

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

7. Applied egg-rr 35.2%

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

   1. *-commutative 35.2%

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

   2. associate-*r/ 35.1%

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

   3. *-commutative 35.1%

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

   4. sqrt-unprod 35.0%

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

   5. add-cbrt-cube 26.1%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{\frac{\pi \cdot n}{k}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{\pi \cdot n}{k}} \cdot \sqrt{2}\right)\right) \cdot \left(\sqrt{\frac{\pi \cdot n}{k}} \cdot \sqrt{2}\right)}} \]

   6. pow1/3 24.6%

      \[\leadsto \color{blue}{{\left(\left(\left(\sqrt{\frac{\pi \cdot n}{k}} \cdot \sqrt{2}\right) \cdot \left(\sqrt{\frac{\pi \cdot n}{k}} \cdot \sqrt{2}\right)\right) \cdot \left(\sqrt{\frac{\pi \cdot n}{k}} \cdot \sqrt{2}\right)\right)}^{0.3333333333333333}} \]

9. Applied egg-rr 24.6%

   \[\leadsto \color{blue}{{\left({\left(\pi \cdot \left(\frac{n}{k} \cdot 2\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
10. Step-by-step derivation

    1. unpow1/3 26.1%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\pi \cdot \left(\frac{n}{k} \cdot 2\right)\right)}^{1.5}}} \]

    2. *-commutative 26.1%

       \[\leadsto \sqrt[3]{{\left(\pi \cdot \color{blue}{\left(2 \cdot \frac{n}{k}\right)}\right)}^{1.5}} \]

    3. *-commutative 26.1%

       \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(2 \cdot \frac{n}{k}\right) \cdot \pi\right)}}^{1.5}} \]

    4. associate-*r/ 26.1%

       \[\leadsto \sqrt[3]{{\left(\color{blue}{\frac{2 \cdot n}{k}} \cdot \pi\right)}^{1.5}} \]

    5. associate-/r/ 26.1%

       \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{2 \cdot n}{\frac{k}{\pi}}\right)}}^{1.5}} \]

    6. *-commutative 26.1%

       \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{n \cdot 2}}{\frac{k}{\pi}}\right)}^{1.5}} \]

    7. associate-/l* 26.2%

       \[\leadsto \sqrt[3]{{\color{blue}{\left(n \cdot \frac{2}{\frac{k}{\pi}}\right)}}^{1.5}} \]

11. Simplified 26.2%

    \[\leadsto \color{blue}{\sqrt[3]{{\left(n \cdot \frac{2}{\frac{k}{\pi}}\right)}^{1.5}}} \]
12. Step-by-step derivation

    1. pow1/3 24.6%

       \[\leadsto \color{blue}{{\left({\left(n \cdot \frac{2}{\frac{k}{\pi}}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

    2. pow-pow 35.2%

       \[\leadsto \color{blue}{{\left(n \cdot \frac{2}{\frac{k}{\pi}}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

    3. metadata-eval 35.2%

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

    4. pow1/2 35.2%

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

    5. *-commutative 35.2%

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

    6. sqrt-prod 45.8%

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

    7. div-inv 45.8%

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

    8. clear-num 45.7%

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

13. Applied egg-rr 45.7%

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

Alternative 9: 37.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Power[N[(k * N[(0.5 / N[(Pi * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $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 35.0%

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

   1. associate-/l* 35.0%

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

5. Simplified 35.0%

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

   1. *-commutative 35.0%

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

   2. sqrt-unprod 35.2%

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

7. Applied egg-rr 35.2%

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

8. Taylor expanded in n around 0 35.1%

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

   1. associate-*l/ 35.1%

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

   2. associate-/r/ 35.2%

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

10. Simplified 35.2%

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

    1. associate-/r/ 35.1%

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

    2. associate-*r* 35.1%

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

    3. *-commutative 35.1%

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

    4. *-commutative 35.1%

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

    5. associate-*l/ 35.1%

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

    6. associate-*r/ 35.1%

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

    7. associate-*l* 35.1%

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

    8. *-commutative 35.1%

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

    9. clear-num 35.1%

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

    10. metadata-eval 35.1%

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

    11. add-sqr-sqrt 35.0%

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

    12. frac-times 35.1%

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

    13. sqrt-unprod 35.5%

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

    14. add-sqr-sqrt 35.6%

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

    15. inv-pow 35.6%

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

    16. sqrt-pow2 35.7%

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

12. Applied egg-rr 35.7%

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

    1. *-commutative 35.7%

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

    2. associate-*l/ 35.7%

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

    3. associate-/l* 35.7%

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

14. Simplified 35.7%

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

15. Final simplification 35.7%

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

Alternative 10: 37.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(2.0 * N[(n / N[(k / Pi), $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. Taylor expanded in k around 0 35.0%

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

   1. associate-/l* 35.0%

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

5. Simplified 35.0%

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

   1. *-commutative 35.0%

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

   2. sqrt-unprod 35.2%

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

7. Applied egg-rr 35.2%

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

8. Taylor expanded in n around 0 35.1%

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

   1. associate-*l/ 35.1%

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

   2. associate-/r/ 35.2%

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

10. Simplified 35.2%

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

Alternative 11: 37.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(2.0 * N[(n * N[(Pi / k), $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. Taylor expanded in k around 0 35.0%

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

   1. associate-/l* 35.0%

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

5. Simplified 35.0%

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

   1. *-commutative 35.0%

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

   2. sqrt-unprod 35.2%

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

7. Applied egg-rr 35.2%

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

Reproduce

herbie shell --seed 2024187 
(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))))