Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.4%

Time: 21.6s

Alternatives: 8

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: 99.5% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. div-sub 99.4%

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

   3. metadata-eval 99.4%

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

   4. div-inv 99.5%

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

   5. expm1-log1p-u 96.2%

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

   6. expm1-udef 85.6%

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

3. Applied egg-rr 74.6%

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

   1. expm1-def 85.3%

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

   2. expm1-log1p 87.2%

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

   3. *-commutative 87.2%

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

   4. associate-*r* 87.2%

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

5. Simplified 87.2%

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

   1. sqrt-div 99.5%

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

   2. associate-*r* 99.5%

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

   3. sqrt-pow1 99.5%

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

   4. *-commutative 99.5%

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

   5. sqrt-pow1 99.5%

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

   6. add-sqr-sqrt 99.2%

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

   7. *-un-lft-identity 99.2%

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

   8. times-frac 99.3%

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

7. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}\right)}^{0.25}}{1} \cdot \frac{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}\right)}^{0.25}}{\sqrt{k}}} \]
8. Step-by-step derivation

   1. /-rgt-identity 99.3%

      \[\leadsto \color{blue}{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}\right)}^{0.25}} \cdot \frac{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}\right)}^{0.25}}{\sqrt{k}} \]

   2. associate-*r/ 99.3%

      \[\leadsto \color{blue}{\frac{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}\right)}^{0.25} \cdot {\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}\right)}^{0.25}}{\sqrt{k}}} \]

   3. pow-sqr 99.5%

      \[\leadsto \frac{\color{blue}{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}\right)}^{\left(2 \cdot 0.25\right)}}}{\sqrt{k}} \]

   4. metadata-eval 99.5%

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

   5. unpow1/2 99.5%

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

   6. associate-*r* 99.5%

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

   7. *-commutative 99.5%

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

   8. associate-*l* 99.5%

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

9. Simplified 99.5%

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

    1. pow-sub 99.7%

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

    2. pow1 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. div-sub 99.4%

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

   3. metadata-eval 99.4%

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

   4. div-inv 99.5%

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

   5. expm1-log1p-u 96.2%

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

   6. expm1-udef 85.6%

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

3. Applied egg-rr 74.6%

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

   1. expm1-def 85.3%

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

   2. expm1-log1p 87.2%

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

   3. *-commutative 87.2%

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

   4. associate-*r* 87.2%

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

5. Simplified 87.2%

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

   1. sqrt-div 99.5%

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

   2. associate-*r* 99.5%

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

   3. sqrt-pow1 99.5%

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

   4. *-commutative 99.5%

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

   5. sqrt-pow1 99.5%

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

   6. add-sqr-sqrt 99.2%

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

   7. *-un-lft-identity 99.2%

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

   8. times-frac 99.3%

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

7. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}\right)}^{0.25}}{1} \cdot \frac{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}\right)}^{0.25}}{\sqrt{k}}} \]
8. Step-by-step derivation

   1. /-rgt-identity 99.3%

      \[\leadsto \color{blue}{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}\right)}^{0.25}} \cdot \frac{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}\right)}^{0.25}}{\sqrt{k}} \]

   2. associate-*r/ 99.3%

      \[\leadsto \color{blue}{\frac{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}\right)}^{0.25} \cdot {\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}\right)}^{0.25}}{\sqrt{k}}} \]

   3. pow-sqr 99.5%

      \[\leadsto \frac{\color{blue}{{\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}\right)}^{\left(2 \cdot 0.25\right)}}}{\sqrt{k}} \]

   4. metadata-eval 99.5%

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

   5. unpow1/2 99.5%

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

   6. associate-*r* 99.5%

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

   7. *-commutative 99.5%

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

   8. associate-*l* 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;k \leq 1.3 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{t_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* n 2.0))))
   (if (<= k 1.3e-47)
     (/ (sqrt t_0) (sqrt k))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (n * 2.0);
	double tmp;
	if (k <= 1.3e-47) {
		tmp = sqrt(t_0) / sqrt(k);
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (n * 2.0);
	double tmp;
	if (k <= 1.3e-47) {
		tmp = Math.sqrt(t_0) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.pi * (n * 2.0)
	tmp = 0
	if k <= 1.3e-47:
		tmp = math.sqrt(t_0) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(n * 2.0))
	tmp = 0.0
	if (k <= 1.3e-47)
		tmp = Float64(sqrt(t_0) / sqrt(k));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = pi * (n * 2.0);
	tmp = 0.0;
	if (k <= 1.3e-47)
		tmp = sqrt(t_0) / sqrt(k);
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.3e-47], N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3e-47

   1. Initial program 99.2%

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

   2. Taylor expanded in k around 0 99.1%

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

      1. expm1-log1p-u 93.1%

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

      2. expm1-udef 76.6%

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

      3. associate-*l/ 76.6%

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

      4. *-un-lft-identity 76.6%

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

      5. sqrt-unprod 76.6%

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

   4. Applied egg-rr 76.6%

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

      1. expm1-def 93.2%

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

      2. expm1-log1p 99.4%

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

      3. associate-*r* 99.4%

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

   6. Simplified 99.4%

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

   if 1.3e-47 < k

   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. div-sub 99.5%

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

      3. metadata-eval 99.5%

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

      4. div-inv 99.6%

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

      5. expm1-log1p-u 99.2%

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

      6. expm1-udef 94.3%

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

   3. Applied egg-rr 94.3%

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

      1. expm1-def 99.2%

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

      2. expm1-log1p 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-*r* 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-47}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. sqr-pow 99.3%

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

   4. pow-sqr 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-*l/ 99.5%

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

   7. associate-/l* 99.5%

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

   8. metadata-eval 99.5%

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

   9. /-rgt-identity 99.5%

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

   10. div-sub 99.5%

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

   11. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 5: 50.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. div-sub 99.4%

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

   3. metadata-eval 99.4%

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

   4. div-inv 99.5%

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

   5. add-cube-cbrt 98.7%

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

   6. pow3 98.7%

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

3. Applied egg-rr 86.5%

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

4. Taylor expanded in k around 0 49.9%

   \[\leadsto \color{blue}{{\left(e^{0.16666666666666666 \cdot \left(-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)\right)}\right)}^{3}} \]
5. Step-by-step derivation

6. Simplified 39.5%

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

   1. pow-pow 42.6%

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

   2. metadata-eval 42.6%

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

   3. pow-to-exp 40.0%

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

   4. associate-/l* 40.0%

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

   5. *-commutative 40.0%

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

8. Applied egg-rr 40.0%

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

   1. exp-to-pow 42.6%

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

   2. associate-/r/ 42.6%

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

   3. unpow-prod-down 54.9%

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

   4. pow1/2 54.9%

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

   5. *-commutative 54.9%

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

10. Applied egg-rr 54.9%

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

    1. unpow1/2 54.9%

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

12. Simplified 54.9%

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

13. Final simplification 54.9%

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

Alternative 6: 50.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0 54.7%

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

   1. expm1-log1p-u 51.7%

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

   2. expm1-udef 50.4%

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

   3. associate-*l/ 50.4%

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

   4. *-un-lft-identity 50.4%

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

   5. sqrt-unprod 50.4%

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

4. Applied egg-rr 50.4%

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

   1. expm1-def 51.7%

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

   2. expm1-log1p 54.9%

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

   3. associate-*r* 54.9%

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

6. Simplified 54.9%

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

7. Final simplification 54.9%

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

Alternative 7: 38.5% 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 99.4%

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

   1. *-commutative 99.4%

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

   2. div-sub 99.4%

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

   3. metadata-eval 99.4%

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

   4. div-inv 99.5%

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

   5. add-cube-cbrt 98.7%

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

   6. pow3 98.7%

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

3. Applied egg-rr 86.5%

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

4. Taylor expanded in k around 0 49.9%

   \[\leadsto \color{blue}{{\left(e^{0.16666666666666666 \cdot \left(-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)\right)}\right)}^{3}} \]
5. Step-by-step derivation

6. Simplified 39.5%

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

   1. expm1-log1p-u 39.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{0.16666666666666666}\right)}^{3}\right)\right)} \]

   2. expm1-udef 39.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left({\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{0.16666666666666666}\right)}^{3}\right)} - 1} \]

   3. pow-pow 39.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{\left(0.16666666666666666 \cdot 3\right)}}\right)} - 1 \]

   4. associate-/l* 39.4%

      \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{\pi}{\frac{k}{n \cdot 2}}\right)}}^{\left(0.16666666666666666 \cdot 3\right)}\right)} - 1 \]

   5. *-commutative 39.4%

      \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\pi}{\frac{k}{\color{blue}{2 \cdot n}}}\right)}^{\left(0.16666666666666666 \cdot 3\right)}\right)} - 1 \]

   6. metadata-eval 39.4%

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

8. Applied egg-rr 39.4%

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

   1. expm1-def 40.7%

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

   2. expm1-log1p 42.6%

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

   3. unpow1/2 42.6%

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

   4. associate-/r/ 42.6%

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

   5. associate-*l/ 42.6%

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

   6. *-commutative 42.6%

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

   7. associate-*r* 42.6%

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

   8. associate-*r/ 42.6%

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

   9. associate-/l* 42.6%

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

   10. associate-/r/ 42.6%

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

10. Simplified 42.6%

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

11. Final simplification 42.6%

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

Alternative 8: 38.5% accurate, 1.5× speedup

Math

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

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(2.0 * Float64(Float64(pi * 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[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. div-sub 99.4%

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

   3. metadata-eval 99.4%

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

   4. div-inv 99.5%

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

   5. add-cube-cbrt 98.7%

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

   6. pow3 98.7%

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

3. Applied egg-rr 86.5%

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

4. Taylor expanded in k around 0 49.9%

   \[\leadsto \color{blue}{{\left(e^{0.16666666666666666 \cdot \left(-1 \cdot \log k + \log \left(2 \cdot \left(n \cdot \pi\right)\right)\right)}\right)}^{3}} \]
5. Step-by-step derivation

6. Simplified 39.5%

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

   1. expm1-log1p-u 39.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{0.16666666666666666}\right)}^{3}\right)\right)} \]

   2. expm1-udef 39.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left({\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{0.16666666666666666}\right)}^{3}\right)} - 1} \]

   3. pow-pow 39.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{\pi \cdot \left(n \cdot 2\right)}{k}\right)}^{\left(0.16666666666666666 \cdot 3\right)}}\right)} - 1 \]

   4. associate-/l* 39.4%

      \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\frac{\pi}{\frac{k}{n \cdot 2}}\right)}}^{\left(0.16666666666666666 \cdot 3\right)}\right)} - 1 \]

   5. *-commutative 39.4%

      \[\leadsto e^{\mathsf{log1p}\left({\left(\frac{\pi}{\frac{k}{\color{blue}{2 \cdot n}}}\right)}^{\left(0.16666666666666666 \cdot 3\right)}\right)} - 1 \]

   6. metadata-eval 39.4%

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

8. Applied egg-rr 39.4%

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

   1. expm1-def 40.7%

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

   2. expm1-log1p 42.6%

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

   3. unpow1/2 42.6%

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

   4. associate-/r/ 42.6%

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

   5. associate-*l/ 42.6%

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

   6. *-commutative 42.6%

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

   7. associate-*r* 42.6%

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

   8. associate-*r/ 42.6%

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

   9. associate-/l* 42.6%

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

   10. associate-/r/ 42.6%

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

10. Simplified 42.6%

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

11. Taylor expanded in n around 0 42.6%

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

12. Final simplification 42.6%

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

Reproduce

herbie shell --seed 2023274 
(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))))