Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 13.1s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. k = 3.753645533205836e+217
  2. n = -1.4956897758213192e-14

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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7.6 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 7.6e-56)
   (/ (sqrt (* 2.0 n)) (sqrt (/ k PI)))
   (sqrt (/ (pow (* 2.0 (* PI n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 7.6e-56) {
		tmp = sqrt((2.0 * n)) / sqrt((k / ((double) M_PI)));
	} else {
		tmp = sqrt((pow((2.0 * (((double) M_PI) * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 7.6e-56) {
		tmp = Math.sqrt((2.0 * n)) / Math.sqrt((k / Math.PI));
	} else {
		tmp = Math.sqrt((Math.pow((2.0 * (Math.PI * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 7.6e-56:
		tmp = math.sqrt((2.0 * n)) / math.sqrt((k / math.pi))
	else:
		tmp = math.sqrt((math.pow((2.0 * (math.pi * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 7.6e-56)
		tmp = Float64(sqrt(Float64(2.0 * n)) / sqrt(Float64(k / pi)));
	else
		tmp = sqrt(Float64((Float64(2.0 * Float64(pi * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 7.6e-56)
		tmp = sqrt((2.0 * n)) / sqrt((k / pi));
	else
		tmp = sqrt((((2.0 * (pi * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 7.6e-56], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / Pi), $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 < 7.6000000000000004e-56

   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 72.9%

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

      1. associate-/l* 72.9%

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

   5. Simplified 72.9%

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

      1. *-commutative 72.9%

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

      2. sqrt-unprod 73.1%

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

   7. Applied egg-rr 73.1%

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

      1. *-un-lft-identity 73.1%

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

      2. clear-num 73.0%

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

      3. un-div-inv 73.0%

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

   9. Applied egg-rr 73.0%

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

       1. *-lft-identity 73.0%

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

       2. associate-/r/ 73.0%

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

   11. Simplified 73.0%

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

       1. sqrt-prod 72.8%

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

       2. div-inv 72.8%

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

       3. associate-*r* 72.8%

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

       4. *-commutative 72.8%

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

       5. sqrt-prod 99.2%

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

       6. un-div-inv 99.3%

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

       7. clear-num 99.3%

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

       8. sqrt-prod 72.8%

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

       9. div-inv 72.9%

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

       10. sqrt-prod 73.0%

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

       11. associate-*r/ 73.0%

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

       12. *-commutative 73.0%

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

       13. sqrt-div 99.5%

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

   13. Applied egg-rr 99.5%

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

   if 7.6000000000000004e-56 < k

   1. Initial program 99.7%

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

   4. Applied egg-rr 98.5%

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

      1. *-commutative 98.5%

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

      2. distribute-lft-in 98.5%

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

      3. metadata-eval 98.5%

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

      4. *-commutative 98.5%

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

      5. associate-*r* 98.5%

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

      6. metadata-eval 98.5%

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

      7. neg-mul-1 98.5%

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

      8. sub-neg 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.6 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, \pi \cdot \frac{2}{k}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.8e+68)
   (/ (sqrt (* 2.0 n)) (sqrt (/ k PI)))
   (sqrt (+ -1.0 (fma n (* PI (/ 2.0 k)) 1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.8e+68) {
		tmp = sqrt((2.0 * n)) / sqrt((k / ((double) M_PI)));
	} 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+68)
		tmp = Float64(sqrt(Float64(2.0 * n)) / sqrt(Float64(k / pi)));
	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+68], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / Pi), $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 < 3.8000000000000001e68

   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 60.5%

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

      1. associate-/l* 60.5%

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

   5. Simplified 60.5%

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

      1. *-commutative 60.5%

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

      2. sqrt-unprod 60.7%

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

   7. Applied egg-rr 60.7%

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

      1. *-un-lft-identity 60.7%

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

      2. clear-num 60.6%

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

      3. un-div-inv 60.6%

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

   9. Applied egg-rr 60.6%

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

       1. *-lft-identity 60.6%

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

       2. associate-/r/ 60.6%

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

   11. Simplified 60.6%

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

       1. sqrt-prod 60.5%

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

       2. div-inv 60.4%

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

       3. associate-*r* 60.5%

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

       4. *-commutative 60.5%

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

       5. sqrt-prod 79.3%

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

       6. un-div-inv 79.3%

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

       7. clear-num 79.3%

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

       8. sqrt-prod 60.5%

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

       9. div-inv 60.5%

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

       10. sqrt-prod 60.6%

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

       11. associate-*r/ 60.6%

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

       12. *-commutative 60.6%

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

       13. sqrt-div 79.5%

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

   13. Applied egg-rr 79.5%

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

   if 3.8000000000000001e68 < 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 2.8%

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

      1. associate-/l* 2.8%

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

   5. Simplified 2.8%

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

      1. *-commutative 2.8%

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

      2. sqrt-unprod 2.8%

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

   7. Applied egg-rr 2.8%

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

      1. *-un-lft-identity 2.8%

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

      2. clear-num 2.8%

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

      3. un-div-inv 2.8%

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

   9. Applied egg-rr 2.8%

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

       1. *-lft-identity 2.8%

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

       2. associate-/r/ 2.8%

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

   11. Simplified 2.8%

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

       1. associate-/r/ 2.8%

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

       2. expm1-log1p-u 2.8%

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

       3. expm1-undefine 34.0%

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

       4. associate-*r/ 34.0%

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

       5. *-commutative 34.0%

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

       6. div-inv 34.0%

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

       7. clear-num 34.0%

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

   13. Applied egg-rr 34.0%

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

       1. sub-neg 34.0%

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

       2. metadata-eval 34.0%

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

       3. +-commutative 34.0%

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

       4. log1p-undefine 34.0%

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

       5. rem-exp-log 34.0%

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

       6. +-commutative 34.0%

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

       7. associate-*l* 34.0%

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

       8. fma-define 34.0%

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

       9. associate-*r/ 34.0%

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

       10. *-commutative 34.0%

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

       11. associate-/l* 34.0%

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

   15. Simplified 34.0%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, \pi \cdot \frac{2}{k}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{+68}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3e+68)
   (/ (sqrt (* 2.0 n)) (sqrt (/ k PI)))
   (sqrt (* 2.0 (+ -1.0 (fma n (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3e+68) {
		tmp = sqrt((2.0 * n)) / sqrt((k / ((double) M_PI)));
	} 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 <= 3e+68)
		tmp = Float64(sqrt(Float64(2.0 * n)) / sqrt(Float64(k / pi)));
	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, 3e+68], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / Pi), $MachinePrecision]], $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 < 3.0000000000000002e68

   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 60.5%

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

      1. associate-/l* 60.5%

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

   5. Simplified 60.5%

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

      1. *-commutative 60.5%

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

      2. sqrt-unprod 60.7%

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

   7. Applied egg-rr 60.7%

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

      1. *-un-lft-identity 60.7%

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

      2. clear-num 60.6%

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

      3. un-div-inv 60.6%

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

   9. Applied egg-rr 60.6%

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

       1. *-lft-identity 60.6%

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

       2. associate-/r/ 60.6%

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

   11. Simplified 60.6%

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

       1. sqrt-prod 60.5%

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

       2. div-inv 60.4%

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

       3. associate-*r* 60.5%

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

       4. *-commutative 60.5%

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

       5. sqrt-prod 79.3%

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

       6. un-div-inv 79.3%

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

       7. clear-num 79.3%

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

       8. sqrt-prod 60.5%

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

       9. div-inv 60.5%

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

       10. sqrt-prod 60.6%

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

       11. associate-*r/ 60.6%

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

       12. *-commutative 60.6%

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

       13. sqrt-div 79.5%

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

   13. Applied egg-rr 79.5%

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

   if 3.0000000000000002e68 < 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 2.8%

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

      1. associate-/l* 2.8%

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

   5. Simplified 2.8%

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

      1. *-commutative 2.8%

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

      2. sqrt-unprod 2.8%

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

   7. Applied egg-rr 2.8%

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

      1. expm1-log1p-u 2.8%

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

      2. expm1-undefine 34.0%

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

   9. Applied egg-rr 34.0%

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

       1. sub-neg 34.0%

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

       2. metadata-eval 34.0%

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

       3. +-commutative 34.0%

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

       4. log1p-undefine 34.0%

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

       5. rem-exp-log 34.0%

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

       6. +-commutative 34.0%

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

       7. fma-define 34.0%

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

   11. Simplified 34.0%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{+68}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{+243}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \frac{n}{\frac{k}{\pi}}\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.8e+243)
   (/ (sqrt (* 2.0 n)) (sqrt (/ k PI)))
   (cbrt (pow (* 2.0 (/ n (/ k PI))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.8e+243) {
		tmp = sqrt((2.0 * n)) / sqrt((k / ((double) M_PI)));
	} else {
		tmp = cbrt(pow((2.0 * (n / (k / ((double) M_PI)))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.8e+243) {
		tmp = Math.sqrt((2.0 * n)) / Math.sqrt((k / Math.PI));
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * (n / (k / Math.PI))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.8e+243)
		tmp = Float64(sqrt(Float64(2.0 * n)) / sqrt(Float64(k / pi)));
	else
		tmp = cbrt((Float64(2.0 * Float64(n / Float64(k / pi))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.8e+243], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(n / N[(k / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.8000000000000001e243

   1. Initial program 99.5%

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

   3. Taylor expanded in k around 0 42.5%

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

      1. associate-/l* 42.5%

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

   5. Simplified 42.5%

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

      1. *-commutative 42.5%

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

      2. sqrt-unprod 42.6%

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

   7. Applied egg-rr 42.6%

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

      1. *-un-lft-identity 42.6%

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

      2. clear-num 42.5%

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

      3. un-div-inv 42.5%

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

   9. Applied egg-rr 42.5%

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

       1. *-lft-identity 42.5%

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

       2. associate-/r/ 42.6%

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

   11. Simplified 42.6%

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

       1. sqrt-prod 42.4%

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

       2. div-inv 42.4%

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

       3. associate-*r* 42.4%

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

       4. *-commutative 42.4%

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

       5. sqrt-prod 55.4%

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

       6. un-div-inv 55.4%

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

       7. clear-num 55.4%

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

       8. sqrt-prod 42.4%

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

       9. div-inv 42.5%

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

       10. sqrt-prod 42.5%

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

       11. associate-*r/ 42.5%

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

       12. *-commutative 42.5%

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

       13. sqrt-div 55.6%

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

   13. Applied egg-rr 55.6%

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

   if 4.8000000000000001e243 < 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 3.1%

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

      1. associate-/l* 3.1%

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

   5. Simplified 3.1%

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

      1. *-commutative 3.1%

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

      2. sqrt-unprod 3.1%

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

   7. Applied egg-rr 3.1%

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

      1. add-cbrt-cube 29.2%

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

      2. add-sqr-sqrt 29.2%

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

      3. pow1 29.2%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)}} \]

      4. pow1/2 29.2%

         \[\leadsto \sqrt[3]{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}}} \]

      5. pow-prod-up 29.2%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{\left(1 + 0.5\right)}}} \]

      6. clear-num 29.2%

         \[\leadsto \sqrt[3]{{\left(2 \cdot \left(n \cdot \color{blue}{\frac{1}{\frac{k}{\pi}}}\right)\right)}^{\left(1 + 0.5\right)}} \]

      7. un-div-inv 29.2%

         \[\leadsto \sqrt[3]{{\left(2 \cdot \color{blue}{\frac{n}{\frac{k}{\pi}}}\right)}^{\left(1 + 0.5\right)}} \]

      8. metadata-eval 29.2%

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

   9. Applied egg-rr 29.2%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(2 \cdot \frac{n}{\frac{k}{\pi}}\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{+243}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \frac{n}{\frac{k}{\pi}}\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.1%

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

   1. associate-/l* 39.1%

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

5. Simplified 39.1%

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

   1. *-commutative 39.1%

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

   2. sqrt-unprod 39.2%

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

7. Applied egg-rr 39.2%

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

   1. *-un-lft-identity 39.2%

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

   2. clear-num 39.1%

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

   3. un-div-inv 39.1%

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

9. Applied egg-rr 39.1%

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

    1. *-lft-identity 39.1%

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

    2. associate-/r/ 39.2%

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

11. Simplified 39.2%

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

    1. sqrt-prod 39.1%

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

    2. div-inv 39.0%

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

    3. associate-*r* 39.1%

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

    4. *-commutative 39.1%

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

    5. sqrt-prod 50.9%

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

    6. un-div-inv 50.9%

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

    7. clear-num 50.9%

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

    8. sqrt-prod 39.1%

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

    9. div-inv 39.1%

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

    10. sqrt-prod 39.1%

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

    11. associate-*r/ 39.1%

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

    12. *-commutative 39.1%

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

    13. sqrt-div 51.0%

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

13. Applied egg-rr 51.0%

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

14. Final simplification 51.0%

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

Alternative 7: 48.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 50.9%

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

   1. associate-*l/ 50.9%

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

   2. *-un-lft-identity 50.9%

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

   3. sqrt-unprod 51.0%

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

   4. *-commutative 51.0%

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

   5. *-commutative 51.0%

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

5. Applied egg-rr 51.0%

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

Alternative 8: 48.6% 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 99.5%

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

3. Taylor expanded in k around 0 39.1%

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

   1. associate-/l* 39.1%

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

5. Simplified 39.1%

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

   1. *-commutative 39.1%

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

   2. sqrt-unprod 39.2%

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

7. Applied egg-rr 39.2%

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

   1. pow1/2 39.2%

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

   2. associate-*r* 39.2%

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

   3. unpow-prod-down 51.0%

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

   4. pow1/2 51.0%

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

9. Applied egg-rr 51.0%

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

    1. unpow1/2 51.0%

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

    2. *-commutative 51.0%

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

11. Simplified 51.0%

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

12. Final simplification 51.0%

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

Alternative 9: 37.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.1%

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

   1. associate-/l* 39.1%

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

5. Simplified 39.1%

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

   1. *-commutative 39.1%

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

   2. sqrt-unprod 39.2%

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

7. Applied egg-rr 39.2%

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

   1. associate-*r/ 39.2%

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

   2. associate-*r/ 39.2%

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

   3. *-commutative 39.2%

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

   4. sqrt-undiv 51.0%

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

   5. pow1/2 51.0%

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

   6. metadata-eval 51.0%

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

   7. pow-flip 51.0%

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

   8. clear-num 50.9%

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

   9. pow-flip 50.9%

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

   10. metadata-eval 50.9%

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

   11. pow1/2 50.9%

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

   12. sqrt-undiv 39.8%

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

   13. *-commutative 39.8%

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

   14. associate-*l* 39.8%

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

9. Applied egg-rr 39.8%

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

10. Final simplification 39.8%

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

Alternative 10: 37.2% 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 99.5%

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

3. Taylor expanded in k around 0 39.1%

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

   1. associate-/l* 39.1%

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

5. Simplified 39.1%

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

   1. *-commutative 39.1%

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

   2. sqrt-unprod 39.2%

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

7. Applied egg-rr 39.2%

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

Reproduce

herbie shell --seed 2024188 
(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))))