Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 98.7%

Time: 12.4s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. k = 6.91996897022493e+140
  2. n = -2.7896481973447925e-28

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: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6.8e-94)
   (/ (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 <= 6.8e-94) {
		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 <= 6.8e-94) {
		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 <= 6.8e-94:
		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 <= 6.8e-94)
		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 <= 6.8e-94)
		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, 6.8e-94], 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 < 6.7999999999999996e-94

   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 61.3%

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

      1. *-commutative 61.3%

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

      2. associate-/l* 61.3%

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

   5. Simplified 61.3%

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

      1. pow1 61.3%

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

      2. sqrt-unprod 61.4%

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

      3. *-commutative 61.4%

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

      4. associate-*r* 61.4%

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

   7. Applied egg-rr 61.4%

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

      1. unpow1 61.4%

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

      2. associate-*l* 61.4%

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

      3. associate-*l/ 61.4%

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

      4. associate-/l* 61.5%

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

   9. Simplified 61.5%

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

       1. associate-*r/ 61.4%

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

       2. *-commutative 61.4%

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

       3. associate-/l* 61.4%

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

       4. clear-num 61.4%

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

       5. un-div-inv 61.4%

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

   11. Applied egg-rr 61.4%

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

       1. associate-*r/ 61.4%

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

       2. sqrt-div 99.5%

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

       3. *-commutative 99.5%

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

   13. Applied egg-rr 99.5%

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

   if 6.7999999999999996e-94 < 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. distribute-rgt-in 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-*l* 99.7%

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

      4. metadata-eval 99.7%

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

      5. *-commutative 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(\pi, 2 \cdot \frac{n}{k}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.2e+16)
   (/ (sqrt (* 2.0 n)) (sqrt (/ k PI)))
   (sqrt (+ -1.0 (fma PI (* 2.0 (/ n k)) 1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.2e+16) {
		tmp = sqrt((2.0 * n)) / sqrt((k / ((double) M_PI)));
	} else {
		tmp = sqrt((-1.0 + fma(((double) M_PI), (2.0 * (n / k)), 1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.2e+16)
		tmp = Float64(sqrt(Float64(2.0 * n)) / sqrt(Float64(k / pi)));
	else
		tmp = sqrt(Float64(-1.0 + fma(pi, Float64(2.0 * Float64(n / k)), 1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.2e+16], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-1.0 + N[(Pi * N[(2.0 * N[(n / k), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.2e16

   1. Initial program 98.9%

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

   3. Taylor expanded in k around 0 67.9%

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

      1. *-commutative 67.9%

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

      2. associate-/l* 67.8%

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

   5. Simplified 67.8%

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

      1. pow1 67.8%

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

      2. sqrt-unprod 68.0%

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

      3. *-commutative 68.0%

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

      4. associate-*r* 68.0%

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

   7. Applied egg-rr 68.0%

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

      1. unpow1 68.0%

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

      2. associate-*l* 68.0%

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

      3. associate-*l/ 68.0%

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

      4. associate-/l* 68.0%

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

   9. Simplified 68.0%

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

       1. associate-*r/ 68.0%

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

       2. *-commutative 68.0%

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

       3. associate-/l* 68.0%

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

       4. clear-num 68.0%

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

       5. un-div-inv 68.0%

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

   11. Applied egg-rr 68.0%

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

       1. associate-*r/ 68.0%

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

       2. sqrt-div 91.6%

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

       3. *-commutative 91.6%

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

   13. Applied egg-rr 91.6%

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

   if 4.2e16 < 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 1.9%

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

      1. *-commutative 1.9%

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

      2. associate-/l* 1.9%

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

   5. Simplified 1.9%

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

      1. pow1 1.9%

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

      2. sqrt-unprod 1.9%

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

      3. *-commutative 1.9%

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

      4. associate-*r* 1.9%

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

   7. Applied egg-rr 1.9%

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

      1. unpow1 1.9%

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

      2. associate-*l* 1.9%

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

      3. associate-*l/ 1.9%

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

      4. associate-/l* 1.9%

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

   9. Simplified 1.9%

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

       1. associate-*r/ 1.9%

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

       2. *-commutative 1.9%

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

       3. associate-/l* 1.9%

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

       4. clear-num 1.9%

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

       5. un-div-inv 1.9%

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

   11. Applied egg-rr 1.9%

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

       1. associate-/r/ 1.9%

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

       2. clear-num 1.9%

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

       3. associate-*l/ 1.9%

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

       4. *-un-lft-identity 1.9%

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

       5. expm1-log1p-u 1.9%

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

       6. associate-/l* 1.9%

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

       7. expm1-undefine 34.9%

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

       8. associate-/r/ 34.9%

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

       9. *-commutative 34.9%

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

       10. *-commutative 34.9%

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

       11. associate-/l* 34.9%

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

   13. Applied egg-rr 34.9%

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

       1. sub-neg 34.9%

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

       2. metadata-eval 34.9%

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

       3. +-commutative 34.9%

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

       4. log1p-undefine 34.9%

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

       5. rem-exp-log 34.9%

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

       6. +-commutative 34.9%

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

       7. *-commutative 34.9%

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

       8. associate-*l* 34.9%

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

       9. associate-*l/ 34.9%

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

       10. *-commutative 34.9%

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

       11. fma-define 34.9%

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

       12. *-commutative 34.9%

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

       13. associate-*r/ 34.9%

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

   15. Simplified 34.9%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(\pi, 2 \cdot \frac{n}{k}, 1\right)}\\ \end{array} \]
5. 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/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. associate-*l* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

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

Alternative 4: 40.2% 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 31.3%

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

   1. *-commutative 31.3%

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

   2. associate-/l* 31.3%

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

5. Simplified 31.3%

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

   1. pow1 31.3%

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

   2. sqrt-unprod 31.4%

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

   3. *-commutative 31.4%

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

   4. associate-*r* 31.4%

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

7. Applied egg-rr 31.4%

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

   1. unpow1 31.4%

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

   2. associate-*l* 31.4%

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

   3. associate-*l/ 31.4%

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

   4. associate-/l* 31.4%

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

9. Simplified 31.4%

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

    1. associate-*r/ 31.4%

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

    2. *-commutative 31.4%

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

    3. associate-/l* 31.4%

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

    4. clear-num 31.3%

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

    5. un-div-inv 31.4%

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

11. Applied egg-rr 31.4%

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

    1. associate-*r/ 31.4%

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

    2. sqrt-div 41.9%

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

    3. *-commutative 41.9%

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

13. Applied egg-rr 41.9%

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

14. Final simplification 41.9%

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

Alternative 5: 40.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $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 31.3%

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

   1. *-commutative 31.3%

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

   2. associate-/l* 31.3%

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

5. Simplified 31.3%

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

   1. pow1 31.3%

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

   2. sqrt-unprod 31.4%

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

   3. *-commutative 31.4%

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

   4. associate-*r* 31.4%

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

7. Applied egg-rr 31.4%

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

   1. unpow1 31.4%

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

   2. associate-*l* 31.4%

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

   3. associate-*l/ 31.4%

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

   4. associate-/l* 31.4%

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

9. Simplified 31.4%

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

    1. associate-*r/ 31.4%

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

    2. *-commutative 31.4%

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

    3. sqrt-prod 31.3%

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

    4. associate-/l* 31.3%

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

    5. sqrt-prod 41.8%

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

    6. *-commutative 41.8%

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

    7. associate-*r* 41.8%

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

    8. sqrt-unprod 41.9%

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

11. Applied egg-rr 41.9%

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

Alternative 6: 31.1% accurate, 1.0× 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 (fabs (* n (/ PI k))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(2.0 * N[Abs[N[(n * N[(Pi / k), $MachinePrecision]), $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 31.3%

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

   1. *-commutative 31.3%

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

   2. associate-/l* 31.3%

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

5. Simplified 31.3%

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

   1. pow1 31.3%

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

   2. sqrt-unprod 31.4%

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

   3. *-commutative 31.4%

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

   4. associate-*r* 31.4%

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

7. Applied egg-rr 31.4%

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

   1. unpow1 31.4%

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

   2. associate-*l* 31.4%

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

   3. associate-*l/ 31.4%

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

   4. associate-/l* 31.4%

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

9. Simplified 31.4%

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

    1. associate-*r/ 31.4%

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

    2. *-commutative 31.4%

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

    3. add-sqr-sqrt 31.3%

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

    4. pow1/2 31.3%

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

    5. pow1/2 31.3%

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

    6. pow-prod-down 27.3%

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

    7. pow2 27.3%

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

    8. associate-/l* 27.3%

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

11. Applied egg-rr 27.3%

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

    1. unpow1/2 27.3%

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

    2. unpow2 27.3%

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

    3. rem-sqrt-square 31.9%

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

13. Simplified 31.9%

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

Alternative 7: 31.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[Abs[N[(n * N[(Pi * N[(2.0 / k), $MachinePrecision]), $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 31.3%

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

   1. *-commutative 31.3%

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

   2. associate-/l* 31.3%

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

5. Simplified 31.3%

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

   1. pow1 31.3%

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

   2. sqrt-unprod 31.4%

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

   3. *-commutative 31.4%

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

   4. associate-*r* 31.4%

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

7. Applied egg-rr 31.4%

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

   1. unpow1 31.4%

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

   2. associate-*l* 31.4%

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

   3. associate-*l/ 31.4%

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

   4. associate-/l* 31.4%

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

9. Simplified 31.4%

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

    1. add-sqr-sqrt 31.4%

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

    2. pow1/2 31.4%

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

    3. pow1/2 31.4%

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

    4. pow-prod-down 27.3%

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

    5. pow2 27.3%

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

    6. associate-*r/ 27.3%

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

    7. *-commutative 27.3%

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

    8. associate-/l* 27.3%

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

11. Applied egg-rr 27.3%

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

    1. unpow1/2 27.3%

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

    2. unpow2 27.3%

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

    3. rem-sqrt-square 31.9%

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

    4. *-commutative 31.9%

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

    5. associate-*l* 31.9%

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

    6. associate-*l/ 31.9%

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

    7. associate-/l* 31.8%

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

13. Simplified 31.8%

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

Alternative 8: 31.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\frac{\frac{k}{\pi}}{2 \cdot n}}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(1.0 / N[Sqrt[N[(N[(k / Pi), $MachinePrecision] / N[(2.0 * n), $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 31.3%

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

   1. *-commutative 31.3%

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

   2. associate-/l* 31.3%

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

5. Simplified 31.3%

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

   1. pow1 31.3%

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

   2. sqrt-unprod 31.4%

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

   3. *-commutative 31.4%

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

   4. associate-*r* 31.4%

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

7. Applied egg-rr 31.4%

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

   1. unpow1 31.4%

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

   2. associate-*l* 31.4%

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

   3. associate-*l/ 31.4%

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

   4. associate-/l* 31.4%

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

9. Simplified 31.4%

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

    1. associate-*r/ 31.4%

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

    2. *-commutative 31.4%

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

    3. associate-/l* 31.4%

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

    4. clear-num 31.3%

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

    5. un-div-inv 31.4%

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

11. Applied egg-rr 31.4%

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

    1. sqrt-prod 31.3%

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

    2. associate-/r/ 31.3%

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

    3. clear-num 31.3%

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

    4. associate-*l/ 31.3%

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

    5. *-un-lft-identity 31.3%

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

    6. sqrt-prod 31.3%

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

    7. associate-/l* 31.3%

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

    8. clear-num 31.3%

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

    9. sqrt-div 31.5%

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

    10. metadata-eval 31.5%

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

    11. *-commutative 31.5%

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

    12. associate-/l/ 31.5%

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

    13. *-commutative 31.5%

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

13. Applied egg-rr 31.5%

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

    1. associate-*r* 31.5%

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

    2. associate-/r* 31.5%

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

    3. associate-/l/ 31.5%

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

    4. associate-/l/ 31.5%

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

15. Simplified 31.5%

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

Alternative 9: 30.5% 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 31.3%

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

   1. *-commutative 31.3%

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

   2. associate-/l* 31.3%

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

5. Simplified 31.3%

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

   1. pow1 31.3%

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

   2. sqrt-unprod 31.4%

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

   3. *-commutative 31.4%

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

   4. associate-*r* 31.4%

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

7. Applied egg-rr 31.4%

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

   1. unpow1 31.4%

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

   2. associate-*l* 31.4%

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

   3. associate-*l/ 31.4%

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

   4. associate-/l* 31.4%

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

9. Simplified 31.4%

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

Reproduce

herbie shell --seed 2024182 
(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))))