Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 14.2s

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, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*r* 99.5%

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

   2. div-sub 99.5%

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

   3. metadata-eval 99.5%

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

   4. pow-div 99.6%

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

   5. pow1/2 99.6%

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

   6. associate-*r/ 99.7%

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

   7. pow1/2 99.7%

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

   8. pow-flip 99.7%

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

   9. metadata-eval 99.7%

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

   10. div-inv 99.7%

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

   11. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-/r/ 99.5%

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

   2. associate-*r* 99.5%

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

   3. div-sub 99.5%

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

   4. metadata-eval 99.5%

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

   5. sub-neg 99.5%

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

   6. div-inv 99.5%

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

   7. metadata-eval 99.5%

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

   8. distribute-rgt-neg-in 99.5%

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

   9. metadata-eval 99.5%

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

4. Applied egg-rr 99.5%

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

   1. inv-pow 99.5%

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

   2. div-inv 99.5%

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

   3. unpow-prod-down 99.5%

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

   4. inv-pow 99.5%

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

   5. pow1/2 99.5%

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

   6. pow-flip 99.5%

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

   7. metadata-eval 99.5%

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

   8. pow-flip 99.5%

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

   9. *-commutative 99.5%

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

   10. *-commutative 99.5%

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

   11. associate-*r* 99.5%

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

   12. +-commutative 99.5%

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

   13. fma-define 99.5%

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

6. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{{k}^{-0.5} \cdot {\left({\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(-\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}\right)}^{-1}} \]
7. Step-by-step derivation

   1. *-commutative 99.5%

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

   2. unpow-1 99.5%

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

   3. associate-*l/ 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.2e-125)
   (/ (sqrt (* 2.0 n)) (sqrt (/ k PI)))
   (sqrt (/ (pow (* 2.0 (* PI n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.2e-125) {
		tmp = sqrt((2.0 * n)) / sqrt((k / ((double) M_PI)));
	} else {
		tmp = sqrt((pow((2.0 * (((double) M_PI) * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.2e-125) {
		tmp = Math.sqrt((2.0 * n)) / Math.sqrt((k / Math.PI));
	} else {
		tmp = Math.sqrt((Math.pow((2.0 * (Math.PI * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.2e-125:
		tmp = math.sqrt((2.0 * n)) / math.sqrt((k / math.pi))
	else:
		tmp = math.sqrt((math.pow((2.0 * (math.pi * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.2e-125)
		tmp = Float64(sqrt(Float64(2.0 * n)) / sqrt(Float64(k / pi)));
	else
		tmp = sqrt(Float64((Float64(2.0 * Float64(pi * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.2e-125)
		tmp = sqrt((2.0 * n)) / sqrt((k / pi));
	else
		tmp = sqrt((((2.0 * (pi * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.2e-125], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.2000000000000001e-125

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 68.6%

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

      1. associate-/l* 68.5%

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

   5. Simplified 68.5%

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

      1. *-commutative 68.5%

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

      2. pow1/2 68.5%

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

      3. pow1/2 68.5%

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

      4. pow-prod-down 68.8%

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

   7. Applied egg-rr 68.8%

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

      1. associate-*r* 68.8%

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

      2. clear-num 68.8%

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

      3. un-div-inv 68.8%

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

   9. Applied egg-rr 68.8%

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

       1. unpow1/2 68.8%

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

       2. sqrt-div 99.4%

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

       3. *-commutative 99.4%

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

   11. Applied egg-rr 99.4%

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

   if 1.2000000000000001e-125 < k

   1. Initial program 99.6%

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

   4. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. metadata-eval 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.7%

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

      6. metadata-eval 99.7%

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

      7. neg-mul-1 99.7%

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

      8. sub-neg 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.45 \cdot 10^{+205}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{\frac{k}{n}}{2 \cdot \pi}\right)}^{-2}\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.45e+205)
   (/ (sqrt (* 2.0 n)) (sqrt (/ k PI)))
   (pow (pow (/ (/ k n) (* 2.0 PI)) -2.0) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.45e+205) {
		tmp = sqrt((2.0 * n)) / sqrt((k / ((double) M_PI)));
	} else {
		tmp = pow(pow(((k / n) / (2.0 * ((double) M_PI))), -2.0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.45e+205) {
		tmp = Math.sqrt((2.0 * n)) / Math.sqrt((k / Math.PI));
	} else {
		tmp = Math.pow(Math.pow(((k / n) / (2.0 * Math.PI)), -2.0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.45e+205:
		tmp = math.sqrt((2.0 * n)) / math.sqrt((k / math.pi))
	else:
		tmp = math.pow(math.pow(((k / n) / (2.0 * math.pi)), -2.0), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.45e+205)
		tmp = Float64(sqrt(Float64(2.0 * n)) / sqrt(Float64(k / pi)));
	else
		tmp = (Float64(Float64(k / n) / Float64(2.0 * pi)) ^ -2.0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.45e+205)
		tmp = sqrt((2.0 * n)) / sqrt((k / pi));
	else
		tmp = (((k / n) / (2.0 * pi)) ^ -2.0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.45e+205], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(k / n), $MachinePrecision] / N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.4499999999999999e205

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0 46.2%

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

      1. associate-/l* 46.2%

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

   5. Simplified 46.2%

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

      1. *-commutative 46.2%

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

      2. pow1/2 46.2%

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

      3. pow1/2 46.2%

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

      4. pow-prod-down 46.3%

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

   7. Applied egg-rr 46.3%

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

      1. associate-*r* 46.3%

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

      2. clear-num 46.2%

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

      3. un-div-inv 46.3%

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

   9. Applied egg-rr 46.3%

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

       1. unpow1/2 46.3%

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

       2. sqrt-div 57.4%

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

       3. *-commutative 57.4%

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

   11. Applied egg-rr 57.4%

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

   if 3.4499999999999999e205 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.8%

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

      1. associate-/l* 2.8%

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

   5. Simplified 2.8%

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

      1. *-commutative 2.8%

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

      2. pow1/2 2.8%

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

      3. pow1/2 2.8%

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

      4. pow-prod-down 2.8%

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

   7. Applied egg-rr 2.8%

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

      1. metadata-eval 2.8%

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

      2. associate-*r/ 2.8%

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

      3. *-commutative 2.8%

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

      4. times-frac 2.8%

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

      5. *-commutative 2.8%

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

      6. *-commutative 2.8%

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

      7. associate-*r* 2.8%

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

      8. *-un-lft-identity 2.8%

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

      9. pow1/2 2.8%

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

      10. sqrt-undiv 3.0%

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

      11. frac-2neg 3.0%

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

   9. Applied egg-rr 3.0%

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

       1. add-sqr-sqrt 3.0%

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

       2. sqrt-unprod 2.8%

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

       3. frac-times 2.8%

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

       4. sqr-neg 2.8%

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

       5. add-sqr-sqrt 2.8%

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

       6. sqr-neg 2.8%

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

       7. add-sqr-sqrt 2.8%

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

       8. clear-num 2.8%

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

       9. unpow1/2 2.8%

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

       10. sqr-pow 2.8%

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

       11. pow-prod-down 15.1%

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

   11. Applied egg-rr 15.1%

       \[\leadsto \color{blue}{{\left({\left(\frac{\frac{k}{n}}{\pi \cdot 2}\right)}^{-2}\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.45 \cdot 10^{+205}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{\frac{k}{n}}{2 \cdot \pi}\right)}^{-2}\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. metadata-eval 99.5%

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

   2. div-sub 99.5%

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

   3. associate-*r* 99.5%

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

   4. div-inv 99.5%

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

   5. associate-*r* 99.5%

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

   6. div-sub 99.5%

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

   7. metadata-eval 99.5%

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

   8. sub-neg 99.5%

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

   9. div-inv 99.5%

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

   10. metadata-eval 99.5%

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

   11. distribute-rgt-neg-in 99.5%

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

   12. metadata-eval 99.5%

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

   13. pow1/2 99.5%

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

   14. pow-flip 99.6%

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

   15. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.5%

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

   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. associate-*l* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

3. Simplified 99.5%

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

Alternative 7: 49.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.6%

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

   1. associate-/l* 39.6%

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

5. Simplified 39.6%

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

   1. *-commutative 39.6%

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

   2. pow1/2 39.6%

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

   3. pow1/2 39.6%

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

   4. pow-prod-down 39.7%

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

7. Applied egg-rr 39.7%

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

   1. associate-*r* 39.7%

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

   2. clear-num 39.6%

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

   3. un-div-inv 39.7%

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

9. Applied egg-rr 39.7%

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

    1. unpow1/2 39.7%

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

    2. sqrt-div 49.1%

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

    3. *-commutative 49.1%

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

11. Applied egg-rr 49.1%

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

12. Final simplification 49.1%

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

Alternative 8: 49.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.6%

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

   1. associate-/l* 39.6%

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

5. Simplified 39.6%

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

   1. *-commutative 39.6%

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

   2. pow1/2 39.6%

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

   3. pow1/2 39.6%

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

   4. pow-prod-down 39.7%

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

7. Applied egg-rr 39.7%

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

   1. associate-*r* 39.7%

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

   2. clear-num 39.6%

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

   3. un-div-inv 39.7%

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

9. Applied egg-rr 39.7%

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

    1. div-inv 39.6%

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

    2. unpow-prod-down 49.1%

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

    3. pow1/2 49.1%

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

    4. *-commutative 49.1%

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

    5. clear-num 49.1%

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

11. Applied egg-rr 49.1%

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

    1. unpow1/2 49.1%

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

13. Simplified 49.1%

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

14. Final simplification 49.1%

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

Alternative 9: 38.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.6%

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

   1. associate-/l* 39.6%

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

5. Simplified 39.6%

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

   1. *-commutative 39.6%

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

   2. pow1/2 39.6%

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

   3. pow1/2 39.6%

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

   4. pow-prod-down 39.7%

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

7. Applied egg-rr 39.7%

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

   1. metadata-eval 39.7%

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

   2. associate-*r/ 39.6%

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

   3. *-commutative 39.6%

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

   4. times-frac 39.6%

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

   5. *-un-lft-identity 39.6%

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

   6. clear-num 39.6%

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

   7. *-commutative 39.6%

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

   8. *-commutative 39.6%

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

   9. associate-*r* 39.6%

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

9. Applied egg-rr 39.6%

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

    1. sqr-pow 39.4%

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

    2. inv-pow 39.4%

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

    3. pow-pow 39.5%

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

    4. associate-/r* 39.5%

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

    5. metadata-eval 39.5%

       \[\leadsto {\left(\frac{\frac{k}{n}}{\pi \cdot 2}\right)}^{\left(-1 \cdot \color{blue}{0.25}\right)} \cdot {\left(\frac{1}{\frac{k}{n \cdot \left(\pi \cdot 2\right)}}\right)}^{\left(\frac{0.5}{2}\right)} \]

    6. metadata-eval 39.5%

       \[\leadsto {\left(\frac{\frac{k}{n}}{\pi \cdot 2}\right)}^{\color{blue}{-0.25}} \cdot {\left(\frac{1}{\frac{k}{n \cdot \left(\pi \cdot 2\right)}}\right)}^{\left(\frac{0.5}{2}\right)} \]

    7. inv-pow 39.5%

       \[\leadsto {\left(\frac{\frac{k}{n}}{\pi \cdot 2}\right)}^{-0.25} \cdot {\color{blue}{\left({\left(\frac{k}{n \cdot \left(\pi \cdot 2\right)}\right)}^{-1}\right)}}^{\left(\frac{0.5}{2}\right)} \]

    8. pow-pow 39.9%

       \[\leadsto {\left(\frac{\frac{k}{n}}{\pi \cdot 2}\right)}^{-0.25} \cdot \color{blue}{{\left(\frac{k}{n \cdot \left(\pi \cdot 2\right)}\right)}^{\left(-1 \cdot \frac{0.5}{2}\right)}} \]

    9. associate-/r* 39.9%

       \[\leadsto {\left(\frac{\frac{k}{n}}{\pi \cdot 2}\right)}^{-0.25} \cdot {\color{blue}{\left(\frac{\frac{k}{n}}{\pi \cdot 2}\right)}}^{\left(-1 \cdot \frac{0.5}{2}\right)} \]

    10. metadata-eval 39.9%

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

    11. metadata-eval 39.9%

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

11. Applied egg-rr 39.9%

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

    1. pow-sqr 40.1%

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

    2. metadata-eval 40.1%

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

    3. associate-/r* 40.1%

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

13. Simplified 40.1%

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

14. Final simplification 40.1%

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

Alternative 10: 38.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Power[N[(k * N[(N[(0.5 / Pi), $MachinePrecision] / n), $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. Add Preprocessing

3. Taylor expanded in k around 0 39.6%

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

   1. associate-/l* 39.6%

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

5. Simplified 39.6%

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

   1. *-commutative 39.6%

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

   2. pow1/2 39.6%

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

   3. pow1/2 39.6%

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

   4. pow-prod-down 39.7%

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

7. Applied egg-rr 39.7%

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

   1. metadata-eval 39.7%

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

   2. associate-*r/ 39.6%

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

   3. *-commutative 39.6%

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

   4. times-frac 39.6%

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

   5. *-un-lft-identity 39.6%

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

   6. clear-num 39.6%

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

   7. *-commutative 39.6%

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

   8. *-commutative 39.6%

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

   9. associate-*r* 39.6%

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

9. Applied egg-rr 39.6%

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

    1. sqr-pow 39.4%

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

    2. inv-pow 39.4%

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

    3. pow-pow 39.5%

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

    4. associate-/r* 39.5%

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

    5. metadata-eval 39.5%

       \[\leadsto {\left(\frac{\frac{k}{n}}{\pi \cdot 2}\right)}^{\left(-1 \cdot \color{blue}{0.25}\right)} \cdot {\left(\frac{1}{\frac{k}{n \cdot \left(\pi \cdot 2\right)}}\right)}^{\left(\frac{0.5}{2}\right)} \]

    6. metadata-eval 39.5%

       \[\leadsto {\left(\frac{\frac{k}{n}}{\pi \cdot 2}\right)}^{\color{blue}{-0.25}} \cdot {\left(\frac{1}{\frac{k}{n \cdot \left(\pi \cdot 2\right)}}\right)}^{\left(\frac{0.5}{2}\right)} \]

    7. inv-pow 39.5%

       \[\leadsto {\left(\frac{\frac{k}{n}}{\pi \cdot 2}\right)}^{-0.25} \cdot {\color{blue}{\left({\left(\frac{k}{n \cdot \left(\pi \cdot 2\right)}\right)}^{-1}\right)}}^{\left(\frac{0.5}{2}\right)} \]

    8. pow-pow 39.9%

       \[\leadsto {\left(\frac{\frac{k}{n}}{\pi \cdot 2}\right)}^{-0.25} \cdot \color{blue}{{\left(\frac{k}{n \cdot \left(\pi \cdot 2\right)}\right)}^{\left(-1 \cdot \frac{0.5}{2}\right)}} \]

    9. associate-/r* 39.9%

       \[\leadsto {\left(\frac{\frac{k}{n}}{\pi \cdot 2}\right)}^{-0.25} \cdot {\color{blue}{\left(\frac{\frac{k}{n}}{\pi \cdot 2}\right)}}^{\left(-1 \cdot \frac{0.5}{2}\right)} \]

    10. metadata-eval 39.9%

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

    11. metadata-eval 39.9%

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

11. Applied egg-rr 39.9%

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

    1. pow-sqr 40.1%

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

    2. metadata-eval 40.1%

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

    3. *-rgt-identity 40.1%

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

    4. associate-*r/ 40.0%

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

    5. associate-*l/ 40.0%

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

    6. associate-/l* 40.0%

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

    7. *-commutative 40.0%

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

    8. associate-/r* 40.0%

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

    9. metadata-eval 40.0%

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

13. Simplified 40.0%

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

Alternative 11: 37.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.6%

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

   1. associate-/l* 39.6%

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

5. Simplified 39.6%

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

   1. *-commutative 39.6%

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

   2. pow1/2 39.6%

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

   3. pow1/2 39.6%

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

   4. pow-prod-down 39.7%

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

7. Applied egg-rr 39.7%

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

   1. associate-*r* 39.7%

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

   2. clear-num 39.6%

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

   3. un-div-inv 39.7%

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

9. Applied egg-rr 39.7%

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

    1. unpow1/2 39.7%

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

    2. associate-/l* 39.7%

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

    3. div-inv 39.6%

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

    4. clear-num 39.7%

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

11. Applied egg-rr 39.7%

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

Reproduce

herbie shell --seed 2024151 
(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))))