Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 16.9s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.6%

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

   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. Step-by-step derivation

   1. remove-double-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. distribute-neg-frac 99.7%

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

   4. distribute-frac-neg2 99.7%

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

   5. distribute-lft-neg-in 99.7%

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

   6. associate-/r* 99.7%

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

6. Simplified 99.7%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. div-inv 99.6%

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

   3. metadata-eval 99.6%

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

   4. inv-pow 99.6%

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

   5. sqrt-pow2 99.6%

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

   6. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot n\right) \cdot \pi\\ \mathbf{if}\;k \leq 4.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt{t\_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t\_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* 2.0 n) PI)))
   (if (<= k 4.2e-39)
     (/ (sqrt t_0) (sqrt k))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = (2.0 * n) * ((double) M_PI);
	double tmp;
	if (k <= 4.2e-39) {
		tmp = sqrt(t_0) / sqrt(k);
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = (2.0 * n) * Math.PI;
	double tmp;
	if (k <= 4.2e-39) {
		tmp = Math.sqrt(t_0) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = (2.0 * n) * math.pi
	tmp = 0
	if k <= 4.2e-39:
		tmp = math.sqrt(t_0) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(Float64(2.0 * n) * pi)
	tmp = 0.0
	if (k <= 4.2e-39)
		tmp = Float64(sqrt(t_0) / sqrt(k));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = (2.0 * n) * pi;
	tmp = 0.0;
	if (k <= 4.2e-39)
		tmp = sqrt(t_0) / sqrt(k);
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[k, 4.2e-39], N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.19999999999999987e-39

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0 72.2%

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

      1. associate-/l* 72.1%

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

   5. Simplified 72.1%

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

      1. *-commutative 72.1%

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

      2. sqrt-prod 98.3%

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

      3. associate-*r* 98.2%

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

      4. sqrt-prod 98.6%

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

      5. div-inv 98.6%

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

      6. sqrt-prod 98.2%

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

      7. metadata-eval 98.2%

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

      8. add-sqr-sqrt 98.2%

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

      9. frac-times 98.2%

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

      10. sqrt-unprod 98.9%

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

      11. add-sqr-sqrt 99.0%

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

      12. associate-*l* 99.0%

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

      13. sqrt-prod 99.4%

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

      14. div-inv 99.5%

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

      15. *-commutative 99.5%

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

   7. Applied egg-rr 99.5%

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

   if 4.19999999999999987e-39 < k

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 99.6%

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

      2. sqrt-unprod 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r* 99.7%

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

      5. div-sub 99.7%

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

      6. metadata-eval 99.7%

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

      7. div-inv 99.7%

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

      8. *-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot n\right) \cdot \pi}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot n\right) \cdot \pi}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, \pi \cdot \frac{2}{k}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.8e+39)
   (/ (sqrt (* (* 2.0 n) PI)) (sqrt k))
   (sqrt (+ -1.0 (fma n (* PI (/ 2.0 k)) 1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.8e+39) {
		tmp = sqrt(((2.0 * n) * ((double) M_PI))) / sqrt(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 <= 3.8e+39)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * pi)) / sqrt(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, 3.8e+39], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $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 < 3.7999999999999998e39

   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 68.0%

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

      1. associate-/l* 67.9%

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

   5. Simplified 67.9%

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

      1. *-commutative 67.9%

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

      2. sqrt-prod 89.1%

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

      3. associate-*r* 89.0%

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

      4. sqrt-prod 89.3%

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

      5. div-inv 89.3%

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

      6. sqrt-prod 88.9%

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

      7. metadata-eval 88.9%

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

      8. add-sqr-sqrt 88.9%

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

      9. frac-times 88.9%

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

      10. sqrt-unprod 89.5%

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

      11. add-sqr-sqrt 89.6%

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

      12. associate-*l* 89.6%

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

      13. sqrt-prod 90.0%

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

      14. div-inv 90.0%

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

      15. *-commutative 90.0%

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

   7. Applied egg-rr 90.0%

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

   if 3.7999999999999998e39 < 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. sqrt-unprod 2.6%

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

      3. *-commutative 2.6%

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

      4. associate-*l* 2.6%

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

      5. *-commutative 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. unpow1 2.6%

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

      2. associate-*l/ 2.6%

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

      3. associate-/l* 2.6%

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

   9. Simplified 2.6%

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

       1. expm1-log1p-u 2.6%

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

       2. expm1-undefine 29.8%

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

       3. associate-/l* 29.8%

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

   11. Applied egg-rr 29.8%

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

       1. sub-neg 29.8%

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

       2. metadata-eval 29.8%

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

       3. +-commutative 29.8%

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

       4. log1p-undefine 29.8%

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

       5. rem-exp-log 29.8%

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

       6. +-commutative 29.8%

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

       7. associate-*r* 29.8%

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

       8. associate-*r/ 29.8%

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

       9. *-commutative 29.8%

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

       10. associate-/l* 29.8%

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

       11. fma-define 29.8%

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

       12. associate-/l* 29.8%

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

   13. Simplified 29.8%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot n\right) \cdot \pi}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, \pi \cdot \frac{2}{k}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot n\right) \cdot \pi}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.15e+39)
   (/ (sqrt (* (* 2.0 n) PI)) (sqrt k))
   (sqrt (* 2.0 (+ -1.0 (fma n (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.15e+39) {
		tmp = sqrt(((2.0 * n) * ((double) M_PI))) / sqrt(k);
	} else {
		tmp = sqrt((2.0 * (-1.0 + fma(n, (((double) M_PI) / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.15e+39)
		tmp = Float64(sqrt(Float64(Float64(2.0 * n) * pi)) / sqrt(k));
	else
		tmp = sqrt(Float64(2.0 * Float64(-1.0 + fma(n, Float64(pi / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.15e+39], N[(N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(-1.0 + N[(n * N[(Pi / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.15000000000000006e39

   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 68.0%

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

      1. associate-/l* 67.9%

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

   5. Simplified 67.9%

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

      1. *-commutative 67.9%

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

      2. sqrt-prod 89.1%

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

      3. associate-*r* 89.0%

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

      4. sqrt-prod 89.3%

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

      5. div-inv 89.3%

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

      6. sqrt-prod 88.9%

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

      7. metadata-eval 88.9%

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

      8. add-sqr-sqrt 88.9%

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

      9. frac-times 88.9%

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

      10. sqrt-unprod 89.5%

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

      11. add-sqr-sqrt 89.6%

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

      12. associate-*l* 89.6%

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

      13. sqrt-prod 90.0%

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

      14. div-inv 90.0%

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

      15. *-commutative 90.0%

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

   7. Applied egg-rr 90.0%

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

   if 1.15000000000000006e39 < 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. sqrt-unprod 2.6%

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

      3. *-commutative 2.6%

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

      4. associate-*l* 2.6%

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

      5. *-commutative 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. unpow1 2.6%

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

      2. associate-*l/ 2.6%

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

      3. associate-/l* 2.6%

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

   9. Simplified 2.6%

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

   10. Taylor expanded in n around 0 2.6%

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

       1. associate-/l* 2.6%

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

   12. Simplified 2.6%

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

       1. associate-*r/ 2.6%

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

       2. expm1-log1p-u 2.6%

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

       3. expm1-undefine 29.8%

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

       4. *-commutative 29.8%

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

       5. associate-/l* 29.8%

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

   14. Applied egg-rr 29.8%

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

       1. sub-neg 29.8%

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

       2. metadata-eval 29.8%

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

       3. +-commutative 29.8%

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

       4. log1p-undefine 29.8%

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

       5. rem-exp-log 29.8%

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

       6. +-commutative 29.8%

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

       7. associate-*r/ 29.8%

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

       8. *-commutative 29.8%

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

       9. associate-/l* 29.8%

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

       10. fma-define 29.8%

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

   16. Simplified 29.8%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot n\right) \cdot \pi}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(2.0 * N[(n * Pi), $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. Final simplification 99.6%

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

Alternative 7: 50.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0 37.3%

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

   1. associate-/l* 37.3%

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

5. Simplified 37.3%

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

   1. *-commutative 37.3%

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

   2. sqrt-prod 48.6%

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

   3. associate-*r* 48.6%

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

   4. sqrt-prod 48.7%

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

   5. div-inv 48.7%

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

   6. sqrt-prod 48.5%

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

   7. metadata-eval 48.5%

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

   8. add-sqr-sqrt 48.5%

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

   9. frac-times 48.5%

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

   10. sqrt-unprod 48.8%

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

   11. add-sqr-sqrt 48.9%

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

   12. associate-*l* 48.9%

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

   13. sqrt-prod 49.1%

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

   14. div-inv 49.1%

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

   15. *-commutative 49.1%

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

7. Applied egg-rr 49.1%

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

8. Final simplification 49.1%

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

Alternative 8: 50.6% 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 37.3%

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

   1. associate-/l* 37.3%

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

5. Simplified 37.3%

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

   1. pow1 37.3%

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

   2. sqrt-unprod 37.4%

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

   3. *-commutative 37.4%

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

   4. associate-*l* 37.4%

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

   5. *-commutative 37.4%

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

7. Applied egg-rr 37.4%

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

   1. unpow1 37.4%

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

   2. associate-*l/ 37.4%

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

   3. associate-/l* 37.4%

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

9. Simplified 37.4%

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

10. Taylor expanded in n around 0 37.4%

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

    1. associate-/l* 37.4%

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

12. Simplified 37.4%

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

    1. associate-*r* 37.4%

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

    2. sqrt-prod 48.7%

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

14. Applied egg-rr 48.7%

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

    1. *-commutative 48.7%

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

16. Simplified 48.7%

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

17. Final simplification 48.7%

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

Alternative 9: 39.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Power[N[(k / N[(n * N[(2.0 * Pi), $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 37.3%

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

   1. associate-/l* 37.3%

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

5. Simplified 37.3%

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

   1. pow1 37.3%

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

   2. sqrt-unprod 37.4%

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

   3. *-commutative 37.4%

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

   4. associate-*l* 37.4%

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

   5. *-commutative 37.4%

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

7. Applied egg-rr 37.4%

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

   1. unpow1 37.4%

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

   2. associate-*l/ 37.4%

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

   3. associate-/l* 37.4%

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

9. Simplified 37.4%

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

    1. associate-*r/ 37.4%

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

    2. sqrt-undiv 49.1%

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

    3. clear-num 49.1%

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

    4. inv-pow 49.1%

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

    5. sqrt-undiv 38.1%

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

    6. sqrt-pow2 38.2%

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

    7. associate-*r* 38.2%

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

    8. *-commutative 38.2%

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

    9. metadata-eval 38.2%

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

11. Applied egg-rr 38.2%

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

12. Final simplification 38.2%

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

Alternative 10: 38.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{\pi}{\frac{k \cdot 0.5}{n}}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(Pi / N[(N[(k * 0.5), $MachinePrecision] / n), $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 37.3%

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

   1. associate-/l* 37.3%

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

5. Simplified 37.3%

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

   1. pow1 37.3%

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

   2. sqrt-unprod 37.4%

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

   3. *-commutative 37.4%

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

   4. associate-*l* 37.4%

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

   5. *-commutative 37.4%

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

7. Applied egg-rr 37.4%

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

   1. unpow1 37.4%

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

   2. associate-*l/ 37.4%

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

   3. associate-/l* 37.4%

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

9. Simplified 37.4%

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

    1. clear-num 37.4%

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

    2. un-div-inv 37.4%

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

    3. *-un-lft-identity 37.4%

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

    4. times-frac 37.4%

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

    5. metadata-eval 37.4%

       \[\leadsto \sqrt{\frac{\pi}{\color{blue}{0.5} \cdot \frac{k}{n}}} \]

11. Applied egg-rr 37.4%

    \[\leadsto \sqrt{\color{blue}{\frac{\pi}{0.5 \cdot \frac{k}{n}}}} \]
12. Step-by-step derivation

    1. associate-*r/ 37.4%

       \[\leadsto \sqrt{\frac{\pi}{\color{blue}{\frac{0.5 \cdot k}{n}}}} \]

13. Simplified 37.4%

    \[\leadsto \sqrt{\color{blue}{\frac{\pi}{\frac{0.5 \cdot k}{n}}}} \]

14. Final simplification 37.4%

    \[\leadsto \sqrt{\frac{\pi}{\frac{k \cdot 0.5}{n}}} \]
15. Add Preprocessing

Alternative 11: 38.5% 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 37.3%

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

   1. associate-/l* 37.3%

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

5. Simplified 37.3%

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

   1. pow1 37.3%

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

   2. sqrt-unprod 37.4%

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

   3. *-commutative 37.4%

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

   4. associate-*l* 37.4%

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

   5. *-commutative 37.4%

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

7. Applied egg-rr 37.4%

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

   1. unpow1 37.4%

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

   2. associate-*l/ 37.4%

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

   3. associate-/l* 37.4%

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

9. Simplified 37.4%

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

10. Taylor expanded in n around 0 37.4%

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

    1. associate-/l* 37.4%

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

12. Simplified 37.4%

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

    1. clear-num 37.4%

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

    2. un-div-inv 37.4%

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

14. Applied egg-rr 37.4%

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

Alternative 12: 38.5% 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 37.3%

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

   1. associate-/l* 37.3%

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

5. Simplified 37.3%

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

   1. pow1 37.3%

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

   2. sqrt-unprod 37.4%

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

   3. *-commutative 37.4%

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

   4. associate-*l* 37.4%

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

   5. *-commutative 37.4%

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

7. Applied egg-rr 37.4%

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

   1. unpow1 37.4%

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

   2. associate-*l/ 37.4%

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

   3. associate-/l* 37.4%

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

9. Simplified 37.4%

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

10. Taylor expanded in n around 0 37.4%

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

    1. associate-/l* 37.4%

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

12. Simplified 37.4%

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

Reproduce

herbie shell --seed 2024108 
(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))))