Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.3%

Time: 13.9s

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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-48}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(2 \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.5e-48)
   (* (pow k -0.5) (sqrt (* n (* 2.0 PI))))
   (sqrt (/ (pow (* 2.0 (* PI n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.5e-48) {
		tmp = pow(k, -0.5) * sqrt((n * (2.0 * ((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 <= 2.5e-48) {
		tmp = Math.pow(k, -0.5) * Math.sqrt((n * (2.0 * 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 <= 2.5e-48:
		tmp = math.pow(k, -0.5) * math.sqrt((n * (2.0 * 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 <= 2.5e-48)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(n * Float64(2.0 * 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 <= 2.5e-48)
		tmp = (k ^ -0.5) * sqrt((n * (2.0 * pi)));
	else
		tmp = sqrt((((2.0 * (pi * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.5e-48], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $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 < 2.4999999999999999e-48

   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 74.0%

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

      1. associate-/l* 74.0%

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

   5. Simplified 74.0%

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

      1. pow1 74.0%

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

      2. sqrt-unprod 74.2%

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

   7. Applied egg-rr 74.2%

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

      1. unpow1 74.2%

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

      2. *-commutative 74.2%

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

      3. associate-*r/ 74.3%

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

      4. *-commutative 74.3%

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

      5. associate-/l* 74.3%

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

   9. Simplified 74.3%

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

       1. associate-*r/ 74.3%

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

       2. *-commutative 74.3%

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

       3. associate-/l* 74.3%

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

       4. div-inv 74.2%

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

       5. *-commutative 74.2%

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

       6. sqrt-prod 99.3%

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

       7. inv-pow 99.3%

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

       8. sqrt-pow1 99.4%

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

       9. metadata-eval 99.4%

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

   11. Applied egg-rr 99.4%

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

       1. *-commutative 99.4%

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

       2. associate-*r* 99.4%

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

       3. rem-exp-log 99.1%

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

       4. rem-exp-log 92.0%

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

       5. exp-sum 91.8%

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

       6. +-commutative 91.8%

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

       7. exp-sum 92.0%

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

       8. rem-exp-log 99.1%

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

       9. rem-exp-log 99.4%

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

       10. *-commutative 99.4%

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

   13. Simplified 99.4%

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

   if 2.4999999999999999e-48 < k

   1. Initial program 99.7%

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

   4. Applied egg-rr 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. distribute-lft-in 99.7%

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

      2. metadata-eval 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r* 99.7%

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

      5. metadata-eval 99.7%

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

      6. neg-mul-1 99.7%

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

      7. sub-neg 99.7%

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

      8. *-commutative 99.7%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{-48}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(2 \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. 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. Add Preprocessing

Alternative 3: 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 4: 50.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(n * N[(2.0 * Pi), $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 37.7%

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

   1. associate-/l* 37.7%

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

5. Simplified 37.7%

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

   1. pow1 37.7%

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

   2. sqrt-unprod 37.8%

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

7. Applied egg-rr 37.8%

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

   1. unpow1 37.8%

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

   2. *-commutative 37.8%

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

   3. associate-*r/ 37.8%

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

   4. *-commutative 37.8%

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

   5. associate-/l* 37.8%

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

9. Simplified 37.8%

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

    1. associate-*r/ 37.8%

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

    2. *-commutative 37.8%

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

    3. associate-/l* 37.8%

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

    4. div-inv 37.8%

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

    5. *-commutative 37.8%

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

    6. sqrt-prod 48.0%

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

    7. inv-pow 48.0%

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

    8. sqrt-pow1 48.0%

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

    9. metadata-eval 48.0%

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

11. Applied egg-rr 48.0%

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

    1. *-commutative 48.0%

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

    2. associate-*r* 48.0%

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

    3. rem-exp-log 47.8%

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

    4. rem-exp-log 44.6%

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

    5. exp-sum 44.5%

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

    6. +-commutative 44.5%

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

    7. exp-sum 44.6%

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

    8. rem-exp-log 47.8%

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

    9. rem-exp-log 48.0%

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

    10. *-commutative 48.0%

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

13. Simplified 48.0%

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

14. Final simplification 48.0%

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

Alternative 5: 50.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 37.7%

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

   1. associate-/l* 37.7%

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

5. Simplified 37.7%

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

   1. pow1 37.7%

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

   2. sqrt-unprod 37.8%

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

7. Applied egg-rr 37.8%

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

   1. unpow1 37.8%

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

   2. *-commutative 37.8%

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

   3. associate-*r/ 37.8%

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

   4. *-commutative 37.8%

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

   5. associate-/l* 37.8%

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

9. Simplified 37.8%

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

    1. associate-*r/ 37.8%

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

    2. *-commutative 37.8%

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

    3. associate-/l* 37.8%

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

    4. *-commutative 37.8%

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

    5. sqrt-div 47.9%

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

11. Applied egg-rr 47.9%

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

    1. associate-*r* 47.9%

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

    2. rem-exp-log 47.8%

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

    3. rem-exp-log 44.7%

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

    4. exp-sum 44.5%

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

    5. +-commutative 44.5%

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

    6. exp-sum 44.7%

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

    7. rem-exp-log 47.8%

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

    8. rem-exp-log 47.9%

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

    9. *-commutative 47.9%

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

13. Simplified 47.9%

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

14. Final simplification 47.9%

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

Alternative 6: 38.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Power[N[(N[(k / Pi), $MachinePrecision] * N[(0.5 / 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 37.7%

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

   1. associate-/l* 37.7%

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

5. Simplified 37.7%

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

   1. pow1 37.7%

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

   2. sqrt-unprod 37.8%

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

7. Applied egg-rr 37.8%

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

   1. unpow1 37.8%

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

   2. *-commutative 37.8%

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

   3. associate-*r/ 37.8%

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

   4. *-commutative 37.8%

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

   5. associate-/l* 37.8%

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

9. Simplified 37.8%

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

    1. clear-num 37.8%

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

    2. un-div-inv 37.8%

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

11. Applied egg-rr 37.8%

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

    1. div-inv 37.8%

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

    2. clear-num 37.8%

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

    3. associate-/l* 37.8%

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

    4. clear-num 37.8%

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

    5. *-commutative 37.8%

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

    6. associate-/l/ 37.8%

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

    7. un-div-inv 37.8%

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

    8. sqrt-undiv 37.9%

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

    9. clear-num 37.9%

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

    10. inv-pow 37.9%

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

    11. sqrt-undiv 37.9%

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

    12. sqrt-pow2 38.0%

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

    13. div-inv 38.0%

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

    14. associate-/l/ 38.0%

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

    15. *-commutative 38.0%

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

    16. metadata-eval 38.0%

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

    17. metadata-eval 38.0%

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

13. Applied egg-rr 38.0%

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

    1. associate-*l/ 38.0%

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

    2. times-frac 38.0%

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

15. Simplified 38.0%

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

Alternative 7: 38.2% accurate, 2.0× 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 99.5%

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

3. Taylor expanded in k around 0 37.7%

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

   1. associate-/l* 37.7%

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

5. Simplified 37.7%

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

   1. pow1 37.7%

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

   2. sqrt-unprod 37.8%

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

7. Applied egg-rr 37.8%

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

   1. unpow1 37.8%

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

   2. *-commutative 37.8%

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

   3. associate-*r/ 37.8%

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

   4. *-commutative 37.8%

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

   5. associate-/l* 37.8%

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

9. Simplified 37.8%

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

    1. associate-*r/ 37.8%

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

11. Applied egg-rr 37.8%

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

Alternative 8: 38.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 37.7%

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

   1. associate-/l* 37.7%

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

5. Simplified 37.7%

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

   1. pow1 37.7%

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

   2. sqrt-unprod 37.8%

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

7. Applied egg-rr 37.8%

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

   1. unpow1 37.8%

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

   2. *-commutative 37.8%

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

   3. associate-*r/ 37.8%

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

   4. *-commutative 37.8%

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

   5. associate-/l* 37.8%

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

9. Simplified 37.8%

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

Reproduce

herbie shell --seed 2024136 
(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))))