Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 14.0s

Alternatives: 11

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, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. div-sub 99.5%

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

   2. metadata-eval 99.5%

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

   3. pow-sub 99.6%

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

   4. pow1/2 99.6%

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

   5. associate-*l* 99.6%

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

   6. associate-*l* 99.6%

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

   7. div-inv 99.6%

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

   8. metadata-eval 99.6%

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

3. Applied egg-rr 99.6%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}} \]
4. Step-by-step derivation

   1. associate-*r* 99.6%

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

   2. associate-*r* 99.6%

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

5. Simplified 99.6%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(k \cdot 0.5\right)}}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 96.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{k}} \cdot \frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(k \cdot 0.5\right)}}\right)\right)} \]

   2. expm1-udef 84.1%

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

   3. frac-times 84.1%

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

   4. *-un-lft-identity 84.1%

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

   5. associate-*r* 84.1%

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

   6. associate-*r* 84.1%

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

7. Applied egg-rr 84.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k} \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 96.7%

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

   2. expm1-log1p 99.7%

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

   3. associate-*r* 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*r* 99.7%

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

   6. *-commutative 99.7%

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

9. Simplified 99.7%

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

    1. expm1-log1p-u 96.7%

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

    2. expm1-udef 84.1%

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

    3. pow1/2 84.1%

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

    4. pow-unpow 84.1%

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

    5. pow-prod-down 84.1%

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

11. Applied egg-rr 84.1%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{{\left(k \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{k}\right)}^{0.5}}\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 96.8%

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

    2. expm1-log1p 99.7%

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

    3. associate-*r* 99.7%

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

    4. *-commutative 99.7%

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

    5. unpow1/2 99.7%

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

    6. associate-*r* 99.7%

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

    7. *-commutative 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-unprod 99.5%

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

   3. frac-times 99.5%

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

   4. metadata-eval 99.5%

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

   5. add-sqr-sqrt 99.6%

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

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {k}^{-0.5} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (pow (* n (* 2.0 PI)) (/ (- 1.0 k) 2.0)) (pow k -0.5)))

C

double code(double k, double n) {
	return pow((n * (2.0 * ((double) M_PI))), ((1.0 - k) / 2.0)) * pow(k, -0.5);
}

Java

public static double code(double k, double n) {
	return Math.pow((n * (2.0 * Math.PI)), ((1.0 - k) / 2.0)) * Math.pow(k, -0.5);
}

Python

def code(k, n):
	return math.pow((n * (2.0 * math.pi)), ((1.0 - k) / 2.0)) * math.pow(k, -0.5)

Julia

function code(k, n)
	return Float64((Float64(n * Float64(2.0 * pi)) ^ Float64(Float64(1.0 - k) / 2.0)) * (k ^ -0.5))
end

MATLAB

function tmp = code(k, n)
	tmp = ((n * (2.0 * pi)) ^ ((1.0 - k) / 2.0)) * (k ^ -0.5);
end

Wolfram

code[k_, n_] := N[(N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[k, -0.5], $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. expm1-log1p-u 95.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{k}}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. expm1-udef 70.7%

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

   3. inv-pow 70.7%

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

   4. sqrt-pow2 70.7%

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

   5. metadata-eval 70.7%

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

3. Applied egg-rr 70.7%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-0.5}\right)} - 1\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
4. Step-by-step derivation

   1. expm1-def 95.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. expm1-log1p 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $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. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 5: 38.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{+226}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{n \cdot \pi}}} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6.2e+226)
   (* (/ 1.0 (sqrt (/ k (* n PI)))) (sqrt 2.0))
   (cbrt (pow (* 2.0 (* PI (/ n k))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6.2e+226) {
		tmp = (1.0 / sqrt((k / (n * ((double) M_PI))))) * sqrt(2.0);
	} else {
		tmp = cbrt(pow((2.0 * (((double) M_PI) * (n / k))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 6.2e+226) {
		tmp = (1.0 / Math.sqrt((k / (n * Math.PI)))) * Math.sqrt(2.0);
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * (Math.PI * (n / k))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 6.2e+226)
		tmp = Float64(Float64(1.0 / sqrt(Float64(k / Float64(n * pi)))) * sqrt(2.0));
	else
		tmp = cbrt((Float64(2.0 * Float64(pi * Float64(n / k))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6.2e+226], N[(N[(1.0 / N[Sqrt[N[(k / N[(n * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(Pi * N[(n / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.19999999999999952e226

   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. *-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*r* 99.5%

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

      4. div-inv 99.5%

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

      5. expm1-log1p-u 96.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

      6. expm1-udef 81.7%

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

   3. Applied egg-rr 67.6%

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

      1. expm1-def 82.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

      2. expm1-log1p 84.0%

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

      3. associate-*r* 84.0%

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

   5. Simplified 84.0%

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

   6. Taylor expanded in k around 0 47.2%

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

      1. associate-/l* 47.2%

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

      2. associate-/r/ 47.2%

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

   8. Simplified 47.2%

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

      1. pow1/2 47.2%

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

      2. *-commutative 47.2%

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

      3. unpow-prod-down 47.1%

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

      4. pow1/2 47.1%

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

      5. *-commutative 47.1%

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

      6. pow1/2 47.1%

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

   10. Applied egg-rr 47.1%

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

       1. clear-num 47.1%

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

       2. div-inv 47.1%

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

       3. clear-num 47.1%

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

       4. sqrt-div 47.8%

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

       5. metadata-eval 47.8%

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

   12. Applied egg-rr 47.8%

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

       1. associate-/r* 47.8%

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

   14. Simplified 47.8%

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

   if 6.19999999999999952e226 < 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. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. div-inv 100.0%

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

      5. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

      6. expm1-udef 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in k around 0 2.8%

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

      1. associate-/l* 2.8%

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

      2. associate-/r/ 2.8%

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

   8. Simplified 2.8%

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

      1. add-cbrt-cube 17.2%

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

      2. pow1/3 17.2%

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

      3. add-sqr-sqrt 17.2%

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

      4. pow1 17.2%

         \[\leadsto {\left(\color{blue}{{\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(\frac{n}{k} \cdot \pi\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 17.2%

         \[\leadsto {\left({\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 17.2%

         \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. *-commutative 17.2%

         \[\leadsto {\left({\left(2 \cdot \color{blue}{\left(\pi \cdot \frac{n}{k}\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 17.2%

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

   10. Applied egg-rr 17.2%

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

       1. unpow1/3 17.2%

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

   12. Simplified 17.2%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{+226}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{n \cdot \pi}}} \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}}\\ \end{array} \]

Alternative 6: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{k}{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(1 - k\right)}}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (pow (/ k (pow (* (* 2.0 n) PI) (- 1.0 k))) -0.5))

C

double code(double k, double n) {
	return pow((k / pow(((2.0 * n) * ((double) M_PI)), (1.0 - k))), -0.5);
}

Java

public static double code(double k, double n) {
	return Math.pow((k / Math.pow(((2.0 * n) * Math.PI), (1.0 - k))), -0.5);
}

Python

def code(k, n):
	return math.pow((k / math.pow(((2.0 * n) * math.pi), (1.0 - k))), -0.5)

Julia

function code(k, n)
	return Float64(k / (Float64(Float64(2.0 * n) * pi) ^ Float64(1.0 - k))) ^ -0.5
end

MATLAB

function tmp = code(k, n)
	tmp = (k / (((2.0 * n) * pi) ^ (1.0 - k))) ^ -0.5;
end

Wolfram

code[k_, n_] := N[Power[N[(k / N[Power[N[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], $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. *-commutative 99.5%

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

   2. associate-*r* 99.5%

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

   3. associate-/r/ 99.5%

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

   4. add-sqr-sqrt 99.3%

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

   5. sqrt-unprod 99.5%

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

   6. associate-*r* 99.5%

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

   7. *-commutative 99.5%

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

   8. associate-*r* 99.5%

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

   9. *-commutative 99.5%

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

   10. pow-prod-up 99.5%

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

3. Applied egg-rr 99.5%

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

4. Taylor expanded in n around 0 83.7%

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

   1. +-commutative 83.7%

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

   2. exp-prod 83.4%

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

   3. log-prod 83.8%

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

   4. associate-*r* 83.8%

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

   5. *-commutative 83.8%

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

   6. exp-prod 83.9%

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

   7. *-commutative 83.9%

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

   8. exp-to-pow 86.7%

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

   9. *-commutative 86.7%

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

   10. associate-*l* 86.7%

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

   11. *-commutative 86.7%

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

6. Simplified 86.7%

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

   1. expm1-log1p-u 85.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\frac{k}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}}}\right)\right)} \]

   2. expm1-udef 72.4%

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

   3. pow1/2 72.4%

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

   4. pow-flip 72.4%

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

   5. metadata-eval 72.4%

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

8. Applied egg-rr 72.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\frac{k}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 85.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{k}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\right)\right)} \]

   2. expm1-log1p 86.7%

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

10. Simplified 86.7%

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

11. Final simplification 86.7%

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

Alternative 7: 38.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 10^{+234}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \pi}}{\sqrt{\frac{k}{n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1e+234)
   (/ (sqrt (* 2.0 PI)) (sqrt (/ k n)))
   (cbrt (pow (* 2.0 (* PI (/ n k))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1e+234) {
		tmp = sqrt((2.0 * ((double) M_PI))) / sqrt((k / n));
	} else {
		tmp = cbrt(pow((2.0 * (((double) M_PI) * (n / k))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1e+234) {
		tmp = Math.sqrt((2.0 * Math.PI)) / Math.sqrt((k / n));
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * (Math.PI * (n / k))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1e+234)
		tmp = Float64(sqrt(Float64(2.0 * pi)) / sqrt(Float64(k / n)));
	else
		tmp = cbrt((Float64(2.0 * Float64(pi * Float64(n / k))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1e+234], N[(N[Sqrt[N[(2.0 * Pi), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(Pi * N[(n / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000002e234

   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. *-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*r* 99.5%

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

      4. div-inv 99.5%

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

      5. expm1-log1p-u 96.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

      6. expm1-udef 81.9%

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

   3. Applied egg-rr 67.8%

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

      1. expm1-def 82.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

      2. expm1-log1p 84.2%

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

      3. associate-*r* 84.2%

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

   5. Simplified 84.2%

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

   6. Taylor expanded in k around 0 46.8%

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

      1. associate-/l* 46.8%

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

      2. associate-/r/ 46.8%

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

   8. Simplified 46.8%

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

   9. Taylor expanded in n around 0 46.8%

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

       1. *-commutative 46.8%

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

       2. associate-/l* 46.8%

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

   11. Simplified 46.8%

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

       1. associate-*r/ 46.8%

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

       2. sqrt-div 47.2%

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

   13. Applied egg-rr 47.2%

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

   if 1.00000000000000002e234 < 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. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. div-inv 100.0%

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

      5. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

      6. expm1-udef 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in k around 0 2.9%

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

      1. associate-/l* 2.9%

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

      2. associate-/r/ 2.9%

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

   8. Simplified 2.9%

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

      1. add-cbrt-cube 18.2%

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

      2. pow1/3 18.2%

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

      3. add-sqr-sqrt 18.2%

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

      4. pow1 18.2%

         \[\leadsto {\left(\color{blue}{{\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(\frac{n}{k} \cdot \pi\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 18.2%

         \[\leadsto {\left({\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 18.2%

         \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. *-commutative 18.2%

         \[\leadsto {\left({\left(2 \cdot \color{blue}{\left(\pi \cdot \frac{n}{k}\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 18.2%

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

   10. Applied egg-rr 18.2%

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

       1. unpow1/3 18.2%

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

   12. Simplified 18.2%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{+234}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \pi}}{\sqrt{\frac{k}{n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}}\\ \end{array} \]

Alternative 8: 87.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / 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. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*r* 99.5%

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

   4. div-inv 99.6%

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

   5. expm1-log1p-u 96.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

   6. expm1-udef 84.1%

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

3. Applied egg-rr 71.7%

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

   1. expm1-def 84.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

   2. expm1-log1p 86.1%

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

   3. associate-*r* 86.1%

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

5. Simplified 86.1%

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

6. Final simplification 86.1%

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

Alternative 9: 37.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*r* 99.5%

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

   4. div-inv 99.6%

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

   5. expm1-log1p-u 96.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

   6. expm1-udef 84.1%

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

3. Applied egg-rr 71.7%

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

   1. expm1-def 84.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

   2. expm1-log1p 86.1%

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

   3. associate-*r* 86.1%

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

5. Simplified 86.1%

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

6. Taylor expanded in k around 0 41.5%

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

   1. associate-/l* 41.5%

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

   2. associate-/r/ 41.5%

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

8. Simplified 41.5%

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

9. Taylor expanded in n around 0 41.5%

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

    1. *-commutative 41.5%

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

    2. associate-/l* 41.5%

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

11. Simplified 41.5%

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

    1. associate-*r/ 41.5%

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

    2. sqrt-div 41.8%

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

13. Applied egg-rr 41.8%

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

14. Final simplification 41.8%

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

Alternative 10: 37.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(2.0 * N[(Pi * N[(n / 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. Step-by-step derivation

   1. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*r* 99.5%

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

   4. div-inv 99.6%

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

   5. expm1-log1p-u 96.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

   6. expm1-udef 84.1%

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

3. Applied egg-rr 71.7%

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

   1. expm1-def 84.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

   2. expm1-log1p 86.1%

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

   3. associate-*r* 86.1%

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

5. Simplified 86.1%

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

6. Taylor expanded in k around 0 41.5%

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

   1. associate-/l* 41.5%

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

   2. associate-/r/ 41.5%

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

8. Simplified 41.5%

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

9. Final simplification 41.5%

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

Alternative 11: 37.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(2.0 * Pi), $MachinePrecision] * N[(n / 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. Step-by-step derivation

   1. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*r* 99.5%

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

   4. div-inv 99.6%

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

   5. expm1-log1p-u 96.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

   6. expm1-udef 84.1%

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

3. Applied egg-rr 71.7%

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

   1. expm1-def 84.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]

   2. expm1-log1p 86.1%

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

   3. associate-*r* 86.1%

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

5. Simplified 86.1%

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

6. Taylor expanded in k around 0 41.5%

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

   1. associate-/l* 41.5%

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

   2. associate-/r/ 41.5%

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

8. Simplified 41.5%

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

9. Taylor expanded in n around 0 41.5%

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

    1. *-commutative 41.5%

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

    2. associate-/l* 41.5%

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

11. Simplified 41.5%

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

    1. expm1-log1p-u 39.8%

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

    2. expm1-udef 37.4%

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

    3. div-inv 37.4%

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

    4. clear-num 37.4%

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

13. Applied egg-rr 37.4%

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

    1. expm1-def 39.8%

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

    2. expm1-log1p 41.5%

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

    3. associate-*r* 41.5%

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

15. Simplified 41.5%

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

16. Final simplification 41.5%

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

Reproduce

herbie shell --seed 2023214 
(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))))