Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 18.4s

Alternatives: 13

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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. sqr-pow 99.2%

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

   4. pow-sqr 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*l* 99.4%

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

   7. associate-*r/ 99.4%

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

   8. *-commutative 99.4%

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

   9. associate-/l* 99.4%

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

   10. metadata-eval 99.4%

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

   11. /-rgt-identity 99.4%

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

   12. div-sub 99.4%

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

   13. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. div-inv 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*l* 99.3%

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

   5. sub-neg 99.3%

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

   6. div-inv 99.3%

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

   7. metadata-eval 99.3%

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

   8. distribute-rgt-neg-in 99.3%

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

   9. metadata-eval 99.3%

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

   10. inv-pow 99.3%

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

   11. sqrt-pow2 99.4%

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

   12. metadata-eval 99.4%

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

6. Applied egg-rr 99.4%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 54.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8.6 \cdot 10^{+151}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{\pi}{\frac{k}{\frac{n}{0.5}}}\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 8.6e+151)
   (/ (sqrt (* 2.0 n)) (sqrt (/ k PI)))
   (pow (pow (/ PI (/ k (/ n 0.5))) 3.0) 0.16666666666666666)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 8.6e+151) {
		tmp = sqrt((2.0 * n)) / sqrt((k / ((double) M_PI)));
	} else {
		tmp = pow(pow((((double) M_PI) / (k / (n / 0.5))), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 8.6e+151) {
		tmp = Math.sqrt((2.0 * n)) / Math.sqrt((k / Math.PI));
	} else {
		tmp = Math.pow(Math.pow((Math.PI / (k / (n / 0.5))), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 8.6e+151:
		tmp = math.sqrt((2.0 * n)) / math.sqrt((k / math.pi))
	else:
		tmp = math.pow(math.pow((math.pi / (k / (n / 0.5))), 3.0), 0.16666666666666666)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 8.6e+151)
		tmp = Float64(sqrt(Float64(2.0 * n)) / sqrt(Float64(k / pi)));
	else
		tmp = (Float64(pi / Float64(k / Float64(n / 0.5))) ^ 3.0) ^ 0.16666666666666666;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 8.6e+151)
		tmp = sqrt((2.0 * n)) / sqrt((k / pi));
	else
		tmp = ((pi / (k / (n / 0.5))) ^ 3.0) ^ 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 8.6e+151], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(Pi / N[(k / N[(n / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.59999999999999965e151

   1. Initial program 99.1%

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

   3. Taylor expanded in k around 0 67.4%

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

      1. pow1/2 67.4%

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

      2. pow-flip 67.4%

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

      3. metadata-eval 67.4%

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

      4. *-commutative 67.4%

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

      5. *-commutative 67.4%

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

      6. sqrt-prod 67.6%

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

      7. expm1-log1p-u 63.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)\right)} \]

      8. expm1-udef 53.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)} - 1} \]

   5. Applied egg-rr 32.4%

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

      1. expm1-def 42.7%

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

      2. expm1-log1p 44.7%

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

      3. associate-/l* 44.7%

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

      4. associate-/r/ 44.7%

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

      5. *-commutative 44.7%

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

   7. Simplified 44.7%

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

   8. Taylor expanded in k around 0 44.7%

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

      1. associate-/l* 44.7%

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

   10. Simplified 44.7%

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

       1. associate-*r/ 44.7%

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

       2. sqrt-div 67.6%

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

   12. Applied egg-rr 67.6%

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

       1. *-commutative 67.6%

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

   14. Simplified 67.6%

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

   if 8.59999999999999965e151 < 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%

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

      1. pow1/2 2.8%

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

      2. pow-flip 2.8%

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

      3. metadata-eval 2.8%

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

      4. *-commutative 2.8%

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

      5. *-commutative 2.8%

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

      6. sqrt-prod 2.8%

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

      7. expm1-log1p-u 2.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)\right)} \]

      8. expm1-udef 30.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)} - 1} \]

   5. Applied egg-rr 30.4%

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

      1. expm1-def 2.7%

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

      2. expm1-log1p 2.7%

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

      3. associate-/l* 2.7%

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

      4. associate-/r/ 2.7%

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

      5. *-commutative 2.7%

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

   7. Simplified 2.7%

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

      1. associate-*l/ 2.7%

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

      2. *-commutative 2.7%

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

      3. rem-cbrt-cube 6.3%

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

      4. unpow1/3 6.3%

         \[\leadsto \color{blue}{{\left({\left(\sqrt{\frac{2 \cdot \left(\pi \cdot n\right)}{k}}\right)}^{3}\right)}^{0.3333333333333333}} \]

      5. sqr-pow 6.3%

         \[\leadsto \color{blue}{{\left({\left(\sqrt{\frac{2 \cdot \left(\pi \cdot n\right)}{k}}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left({\left(\sqrt{\frac{2 \cdot \left(\pi \cdot n\right)}{k}}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

      6. pow-prod-down 19.2%

         \[\leadsto \color{blue}{{\left({\left(\sqrt{\frac{2 \cdot \left(\pi \cdot n\right)}{k}}\right)}^{3} \cdot {\left(\sqrt{\frac{2 \cdot \left(\pi \cdot n\right)}{k}}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

      7. pow-prod-down 19.2%

         \[\leadsto {\color{blue}{\left({\left(\sqrt{\frac{2 \cdot \left(\pi \cdot n\right)}{k}} \cdot \sqrt{\frac{2 \cdot \left(\pi \cdot n\right)}{k}}\right)}^{3}\right)}}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      8. add-sqr-sqrt 19.2%

         \[\leadsto {\left({\color{blue}{\left(\frac{2 \cdot \left(\pi \cdot n\right)}{k}\right)}}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      9. *-commutative 19.2%

         \[\leadsto {\left({\left(\frac{\color{blue}{\left(\pi \cdot n\right) \cdot 2}}{k}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      10. associate-*r/ 19.2%

          \[\leadsto {\left({\color{blue}{\left(\left(\pi \cdot n\right) \cdot \frac{2}{k}\right)}}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      11. associate-*l* 19.2%

          \[\leadsto {\left({\color{blue}{\left(\pi \cdot \left(n \cdot \frac{2}{k}\right)\right)}}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      12. metadata-eval 19.2%

          \[\leadsto {\left({\left(\pi \cdot \left(n \cdot \frac{2}{k}\right)\right)}^{3}\right)}^{\color{blue}{0.16666666666666666}} \]

   9. Applied egg-rr 19.2%

      \[\leadsto \color{blue}{{\left({\left(\pi \cdot \left(n \cdot \frac{2}{k}\right)\right)}^{3}\right)}^{0.16666666666666666}} \]
   10. Step-by-step derivation

       1. associate-*r/ 19.2%

          \[\leadsto {\left({\left(\pi \cdot \color{blue}{\frac{n \cdot 2}{k}}\right)}^{3}\right)}^{0.16666666666666666} \]

       2. associate-*r/ 19.2%

          \[\leadsto {\left({\color{blue}{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}}^{3}\right)}^{0.16666666666666666} \]

       3. /-rgt-identity 19.2%

          \[\leadsto {\left({\left(\frac{\color{blue}{\frac{\pi \cdot \left(n \cdot 2\right)}{1}}}{k}\right)}^{3}\right)}^{0.16666666666666666} \]

       4. associate-*r* 19.2%

          \[\leadsto {\left({\left(\frac{\frac{\color{blue}{\left(\pi \cdot n\right) \cdot 2}}{1}}{k}\right)}^{3}\right)}^{0.16666666666666666} \]

       5. associate-/l* 19.2%

          \[\leadsto {\left({\left(\frac{\color{blue}{\frac{\pi \cdot n}{\frac{1}{2}}}}{k}\right)}^{3}\right)}^{0.16666666666666666} \]

       6. metadata-eval 19.2%

          \[\leadsto {\left({\left(\frac{\frac{\pi \cdot n}{\color{blue}{0.5}}}{k}\right)}^{3}\right)}^{0.16666666666666666} \]

       7. associate-/r* 19.2%

          \[\leadsto {\left({\color{blue}{\left(\frac{\pi \cdot n}{0.5 \cdot k}\right)}}^{3}\right)}^{0.16666666666666666} \]

       8. *-commutative 19.2%

          \[\leadsto {\left({\left(\frac{\pi \cdot n}{\color{blue}{k \cdot 0.5}}\right)}^{3}\right)}^{0.16666666666666666} \]

       9. associate-/l* 19.2%

          \[\leadsto {\left({\color{blue}{\left(\frac{\pi}{\frac{k \cdot 0.5}{n}}\right)}}^{3}\right)}^{0.16666666666666666} \]

       10. associate-/l* 19.2%

           \[\leadsto {\left({\left(\frac{\pi}{\color{blue}{\frac{k}{\frac{n}{0.5}}}}\right)}^{3}\right)}^{0.16666666666666666} \]

   11. Simplified 19.2%

       \[\leadsto \color{blue}{{\left({\left(\frac{\pi}{\frac{k}{\frac{n}{0.5}}}\right)}^{3}\right)}^{0.16666666666666666}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.6 \cdot 10^{+151}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{\pi}{\frac{k}{\frac{n}{0.5}}}\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. sqr-pow 99.2%

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

   4. pow-sqr 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*l* 99.4%

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

   7. associate-*r/ 99.4%

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

   8. *-commutative 99.4%

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

   9. associate-/l* 99.4%

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

   10. metadata-eval 99.4%

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

   11. /-rgt-identity 99.4%

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

   12. div-sub 99.4%

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

   13. metadata-eval 99.4%

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

3. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 4: 49.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 54.3%

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

   1. pow1/2 54.3%

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

   2. pow-flip 54.3%

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

   3. metadata-eval 54.3%

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

   4. *-commutative 54.3%

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

   5. *-commutative 54.3%

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

   6. sqrt-prod 54.4%

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

   7. expm1-log1p-u 51.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)\right)} \]

   8. expm1-udef 48.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)} - 1} \]

5. Applied egg-rr 32.0%

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

   1. expm1-def 34.5%

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

   2. expm1-log1p 36.2%

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

   3. associate-/l* 36.2%

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

   4. associate-/r/ 36.2%

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

   5. *-commutative 36.2%

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

7. Simplified 36.2%

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

   1. sqrt-prod 54.4%

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

   2. *-commutative 54.4%

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

9. Applied egg-rr 54.4%

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

10. Final simplification 54.4%

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

Alternative 5: 49.1% 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 99.3%

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

3. Taylor expanded in k around 0 54.3%

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

   1. pow1/2 54.3%

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

   2. pow-flip 54.3%

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

   3. metadata-eval 54.3%

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

   4. *-commutative 54.3%

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

   5. *-commutative 54.3%

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

   6. sqrt-prod 54.4%

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

   7. expm1-log1p-u 51.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)\right)} \]

   8. expm1-udef 48.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)} - 1} \]

5. Applied egg-rr 32.0%

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

   1. expm1-def 34.5%

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

   2. expm1-log1p 36.2%

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

   3. associate-/l* 36.2%

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

   4. associate-/r/ 36.2%

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

   5. *-commutative 36.2%

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

7. Simplified 36.2%

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

8. Taylor expanded in k around 0 36.2%

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

   1. associate-/l* 36.2%

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

10. Simplified 36.2%

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

    1. pow1/2 36.2%

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

    2. *-commutative 36.2%

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

    3. clear-num 36.2%

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

    4. associate-/r/ 36.2%

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

    5. clear-num 36.2%

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

    6. associate-*r* 36.2%

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

    7. metadata-eval 36.2%

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

    8. div-inv 36.2%

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

    9. unpow-prod-down 54.4%

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

    10. pow1/2 54.4%

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

    11. clear-num 54.4%

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

    12. associate-/r/ 54.4%

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

    13. metadata-eval 54.4%

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

12. Applied egg-rr 54.4%

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

    1. unpow1/2 54.4%

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

    2. *-commutative 54.4%

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

14. Simplified 54.4%

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

15. Final simplification 54.4%

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

Alternative 6: 49.1% 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 99.3%

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

3. Taylor expanded in k around 0 54.3%

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

   1. associate-*l/ 54.3%

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

   2. *-un-lft-identity 54.3%

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

   3. *-commutative 54.3%

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

   4. *-commutative 54.3%

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

   5. sqrt-prod 54.5%

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

5. Applied egg-rr 54.5%

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

6. Final simplification 54.5%

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

Alternative 7: 49.1% 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 99.3%

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

3. Taylor expanded in k around 0 54.3%

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

   1. pow1/2 54.3%

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

   2. pow-flip 54.3%

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

   3. metadata-eval 54.3%

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

   4. *-commutative 54.3%

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

   5. *-commutative 54.3%

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

   6. sqrt-prod 54.4%

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

   7. expm1-log1p-u 51.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)\right)} \]

   8. expm1-udef 48.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)} - 1} \]

5. Applied egg-rr 32.0%

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

   1. expm1-def 34.5%

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

   2. expm1-log1p 36.2%

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

   3. associate-/l* 36.2%

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

   4. associate-/r/ 36.2%

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

   5. *-commutative 36.2%

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

7. Simplified 36.2%

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

8. Taylor expanded in k around 0 36.2%

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

   1. associate-/l* 36.2%

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

10. Simplified 36.2%

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

    1. associate-*r/ 36.2%

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

    2. sqrt-div 54.5%

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

12. Applied egg-rr 54.5%

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

    1. *-commutative 54.5%

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

14. Simplified 54.5%

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

15. Final simplification 54.5%

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

Alternative 8: 38.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\frac{k}{\pi} \cdot \frac{0.5}{n}}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (/ 1.0 (sqrt (* (/ k PI) (/ 0.5 n)))))

C

double code(double k, double n) {
	return 1.0 / sqrt(((k / ((double) M_PI)) * (0.5 / n)));
}

Java

public static double code(double k, double n) {
	return 1.0 / Math.sqrt(((k / Math.PI) * (0.5 / n)));
}

Python

def code(k, n):
	return 1.0 / math.sqrt(((k / math.pi) * (0.5 / n)))

Julia

function code(k, n)
	return Float64(1.0 / sqrt(Float64(Float64(k / pi) * Float64(0.5 / n))))
end

MATLAB

function tmp = code(k, n)
	tmp = 1.0 / sqrt(((k / pi) * (0.5 / n)));
end

Wolfram

code[k_, n_] := N[(1.0 / N[Sqrt[N[(N[(k / Pi), $MachinePrecision] * N[(0.5 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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 54.3%

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

   1. pow1/2 54.3%

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

   2. pow-flip 54.3%

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

   3. metadata-eval 54.3%

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

   4. *-commutative 54.3%

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

   5. *-commutative 54.3%

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

   6. sqrt-prod 54.4%

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

   7. expm1-log1p-u 51.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)\right)} \]

   8. expm1-udef 48.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)} - 1} \]

5. Applied egg-rr 32.0%

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

   1. expm1-def 34.5%

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

   2. expm1-log1p 36.2%

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

   3. associate-/l* 36.2%

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

   4. associate-/r/ 36.2%

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

   5. *-commutative 36.2%

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

7. Simplified 36.2%

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

8. Taylor expanded in k around 0 36.2%

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

   1. associate-/l* 36.2%

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

10. Simplified 36.2%

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

    1. associate-*r/ 36.2%

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

    2. metadata-eval 36.2%

       \[\leadsto \sqrt{\frac{\color{blue}{\frac{1}{0.5}} \cdot n}{\frac{k}{\pi}}} \]

    3. associate-/r/ 36.2%

       \[\leadsto \sqrt{\frac{\color{blue}{\frac{1}{\frac{0.5}{n}}}}{\frac{k}{\pi}}} \]

    4. clear-num 36.2%

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

    5. associate-/r/ 36.2%

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

    6. clear-num 36.2%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{k}{\frac{n}{0.5}}}} \cdot \pi} \]

    7. associate-/r/ 36.2%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\frac{k}{\frac{n}{0.5}}}{\pi}}}} \]

    8. sqrt-div 37.8%

       \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\frac{k}{\frac{n}{0.5}}}{\pi}}}} \]

    9. metadata-eval 37.8%

       \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{\frac{k}{\frac{n}{0.5}}}{\pi}}} \]

    10. div-inv 37.8%

        \[\leadsto \frac{1}{\sqrt{\frac{\color{blue}{k \cdot \frac{1}{\frac{n}{0.5}}}}{\pi}}} \]

    11. clear-num 37.8%

        \[\leadsto \frac{1}{\sqrt{\frac{k \cdot \color{blue}{\frac{0.5}{n}}}{\pi}}} \]

    12. associate-/l* 37.8%

        \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{k}{\frac{\pi}{\frac{0.5}{n}}}}}} \]

12. Applied egg-rr 37.8%

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{k}{\frac{\pi}{\frac{0.5}{n}}}}}} \]
13. Step-by-step derivation

    1. associate-/r/ 37.8%

       \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{k}{\pi} \cdot \frac{0.5}{n}}}} \]

14. Simplified 37.8%

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{k}{\pi} \cdot \frac{0.5}{n}}}} \]

15. Final simplification 37.8%

    \[\leadsto \frac{1}{\sqrt{\frac{k}{\pi} \cdot \frac{0.5}{n}}} \]
16. Add Preprocessing

Alternative 9: 38.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\frac{k \cdot 0.5}{\pi \cdot n}}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (/ 1.0 (sqrt (/ (* k 0.5) (* PI n)))))

C

double code(double k, double n) {
	return 1.0 / sqrt(((k * 0.5) / (((double) M_PI) * n)));
}

Java

public static double code(double k, double n) {
	return 1.0 / Math.sqrt(((k * 0.5) / (Math.PI * n)));
}

Python

def code(k, n):
	return 1.0 / math.sqrt(((k * 0.5) / (math.pi * n)))

Julia

function code(k, n)
	return Float64(1.0 / sqrt(Float64(Float64(k * 0.5) / Float64(pi * n))))
end

MATLAB

function tmp = code(k, n)
	tmp = 1.0 / sqrt(((k * 0.5) / (pi * n)));
end

Wolfram

code[k_, n_] := N[(1.0 / N[Sqrt[N[(N[(k * 0.5), $MachinePrecision] / N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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 54.3%

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

   1. pow1/2 54.3%

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

   2. pow-flip 54.3%

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

   3. metadata-eval 54.3%

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

   4. *-commutative 54.3%

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

   5. *-commutative 54.3%

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

   6. sqrt-prod 54.4%

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

   7. expm1-log1p-u 51.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)\right)} \]

   8. expm1-udef 48.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)} - 1} \]

5. Applied egg-rr 32.0%

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

   1. expm1-def 34.5%

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

   2. expm1-log1p 36.2%

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

   3. associate-/l* 36.2%

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

   4. associate-/r/ 36.2%

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

   5. *-commutative 36.2%

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

7. Simplified 36.2%

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

   1. associate-*l/ 36.2%

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

   2. *-commutative 36.2%

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

   3. clear-num 36.2%

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

   4. sqrt-div 37.8%

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

   5. metadata-eval 37.8%

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

   6. *-un-lft-identity 37.8%

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

   7. times-frac 37.8%

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

   8. metadata-eval 37.8%

      \[\leadsto \frac{1}{\sqrt{\color{blue}{0.5} \cdot \frac{k}{\pi \cdot n}}} \]

9. Applied egg-rr 37.8%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{0.5 \cdot \frac{k}{\pi \cdot n}}}} \]
10. Step-by-step derivation

    1. associate-*r/ 37.8%

       \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{0.5 \cdot k}{\pi \cdot n}}}} \]

    2. *-commutative 37.8%

       \[\leadsto \frac{1}{\sqrt{\frac{\color{blue}{k \cdot 0.5}}{\pi \cdot n}}} \]

    3. *-commutative 37.8%

       \[\leadsto \frac{1}{\sqrt{\frac{k \cdot 0.5}{\color{blue}{n \cdot \pi}}}} \]

11. Simplified 37.8%

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{k \cdot 0.5}{n \cdot \pi}}}} \]

12. Final simplification 37.8%

    \[\leadsto \frac{1}{\sqrt{\frac{k \cdot 0.5}{\pi \cdot n}}} \]
13. Add Preprocessing

Alternative 10: 38.3% 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 99.3%

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

3. Taylor expanded in k around 0 54.3%

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

   1. pow1/2 54.3%

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

   2. pow-flip 54.3%

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

   3. metadata-eval 54.3%

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

   4. *-commutative 54.3%

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

   5. *-commutative 54.3%

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

   6. sqrt-prod 54.4%

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

   7. expm1-log1p-u 51.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)\right)} \]

   8. expm1-udef 48.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)} - 1} \]

5. Applied egg-rr 32.0%

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

   1. expm1-def 34.5%

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

   2. expm1-log1p 36.2%

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

   3. associate-/l* 36.2%

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

   4. associate-/r/ 36.2%

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

   5. *-commutative 36.2%

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

7. Simplified 36.2%

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

8. Taylor expanded in k around 0 36.2%

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

   1. associate-/l* 36.2%

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

10. Simplified 36.2%

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

11. Final simplification 36.2%

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

Alternative 11: 38.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 54.3%

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

   1. pow1/2 54.3%

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

   2. pow-flip 54.3%

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

   3. metadata-eval 54.3%

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

   4. *-commutative 54.3%

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

   5. *-commutative 54.3%

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

   6. sqrt-prod 54.4%

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

   7. expm1-log1p-u 51.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)\right)} \]

   8. expm1-udef 48.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)} - 1} \]

5. Applied egg-rr 32.0%

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

   1. expm1-def 34.5%

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

   2. expm1-log1p 36.2%

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

   3. associate-/l* 36.2%

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

   4. *-commutative 36.2%

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

7. Simplified 36.2%

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

8. Final simplification 36.2%

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

Alternative 12: 38.3% accurate, 2.0× speedup

Math

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

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(k * Float64(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[(k * N[(0.5 / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

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 54.3%

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

   1. pow1/2 54.3%

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

   2. pow-flip 54.3%

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

   3. metadata-eval 54.3%

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

   4. *-commutative 54.3%

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

   5. *-commutative 54.3%

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

   6. sqrt-prod 54.4%

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

   7. expm1-log1p-u 51.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)\right)} \]

   8. expm1-udef 48.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)} - 1} \]

5. Applied egg-rr 32.0%

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

   1. expm1-def 34.5%

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

   2. expm1-log1p 36.2%

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

   3. associate-/l* 36.2%

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

   4. associate-/r/ 36.2%

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

   5. *-commutative 36.2%

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

7. Simplified 36.2%

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

8. Taylor expanded in k around 0 36.2%

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

   1. associate-/l* 36.2%

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

10. Simplified 36.2%

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

    1. *-commutative 36.2%

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

    2. clear-num 36.2%

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

    3. associate-/r/ 36.2%

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

    4. clear-num 36.2%

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

    5. associate-*r* 36.2%

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

    6. metadata-eval 36.2%

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

    7. div-inv 36.2%

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

    8. clear-num 36.2%

       \[\leadsto \sqrt{\frac{\pi}{k} \cdot \color{blue}{\frac{1}{\frac{0.5}{n}}}} \]

    9. un-div-inv 36.2%

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

    10. associate-/r* 36.2%

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

12. Applied egg-rr 36.2%

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

13. Final simplification 36.2%

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

Alternative 13: 38.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 54.3%

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

   1. pow1/2 54.3%

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

   2. pow-flip 54.3%

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

   3. metadata-eval 54.3%

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

   4. *-commutative 54.3%

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

   5. *-commutative 54.3%

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

   6. sqrt-prod 54.4%

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

   7. expm1-log1p-u 51.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)\right)} \]

   8. expm1-udef 48.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({k}^{-0.5} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}\right)} - 1} \]

5. Applied egg-rr 32.0%

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

   1. expm1-def 34.5%

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

   2. expm1-log1p 36.2%

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

   3. associate-/l* 36.2%

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

   4. associate-/r/ 36.2%

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

   5. *-commutative 36.2%

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

7. Simplified 36.2%

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

   1. associate-*l/ 36.2%

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

   2. *-commutative 36.2%

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

   3. associate-*r* 36.2%

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

   4. associate-/l* 36.2%

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

9. Applied egg-rr 36.2%

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

10. Final simplification 36.2%

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

Reproduce

herbie shell --seed 2024020 
(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))))