Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 14.4s

Alternatives: 10

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: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

   5. div-sub 99.6%

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

   6. sub-neg 99.6%

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

   7. distribute-frac-neg 99.6%

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

   8. metadata-eval 99.6%

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

   9. neg-mul-1 99.6%

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

   10. associate-/l* 99.6%

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

   11. associate-/r/ 99.6%

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

   12. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.6%

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

   2. unpow-prod-up 99.7%

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

   3. pow1/2 99.7%

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

   4. associate-*l* 99.7%

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

   5. associate-*r* 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.7%

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

   8. associate-*r* 99.7%

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

   9. *-commutative 99.7%

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

   10. associate-*l* 99.7%

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

   11. *-commutative 99.7%

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

   12. inv-pow 99.7%

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

   13. sqrt-pow2 99.8%

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

   14. metadata-eval 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

   5. div-sub 99.6%

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

   6. sub-neg 99.6%

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

   7. distribute-frac-neg 99.6%

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

   8. metadata-eval 99.6%

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

   9. neg-mul-1 99.6%

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

   10. associate-/l* 99.6%

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

   11. associate-/r/ 99.6%

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

   12. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. +-commutative 99.6%

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

   2. unpow-prod-up 99.7%

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

   3. associate-*r* 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*l* 99.7%

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

   6. *-commutative 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-*r* 99.7%

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

   9. *-commutative 99.7%

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

   10. associate-*l* 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{2 \cdot \left(\pi \cdot n\right)} \cdot {k}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.6e-79)
   (* (sqrt (* 2.0 (* PI n))) (pow k -0.5))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.6e-79) {
		tmp = sqrt((2.0 * (((double) M_PI) * n))) * pow(k, -0.5);
	} else {
		tmp = sqrt((pow((((double) M_PI) * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.6e-79) {
		tmp = Math.sqrt((2.0 * (Math.PI * n))) * Math.pow(k, -0.5);
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 4.6e-79:
		tmp = math.sqrt((2.0 * (math.pi * n))) * math.pow(k, -0.5)
	else:
		tmp = math.sqrt((math.pow((math.pi * (2.0 * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.6e-79)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi * n))) * (k ^ -0.5));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(2.0 * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 4.6e-79)
		tmp = sqrt((2.0 * (pi * n))) * (k ^ -0.5);
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.6e-79], N[(N[Sqrt[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[k, -0.5], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.60000000000000023e-79

   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. Step-by-step derivation

      1. add-sqr-sqrt 99.0%

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

      2. sqrt-unprod 82.3%

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

      3. associate-*l/ 82.3%

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

      4. *-un-lft-identity 82.3%

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

      5. associate-*l/ 82.3%

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

      6. *-un-lft-identity 82.3%

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

      7. frac-times 82.3%

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

   3. Applied egg-rr 82.2%

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

   5. Simplified 82.4%

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

   6. Taylor expanded in k around 0 82.4%

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

      1. associate-/l* 82.5%

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

   8. Simplified 82.5%

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

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. associate-/l* 82.4%

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

      4. *-commutative 82.4%

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

      5. sqrt-undiv 99.5%

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

      6. clear-num 99.4%

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

      7. associate-/r/ 99.3%

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

      8. pow1/2 99.3%

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

      9. pow-flip 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-*r* 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-*l* 99.5%

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

   10. Applied egg-rr 99.5%

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

   if 4.60000000000000023e-79 < 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. Step-by-step derivation

      1. add-sqr-sqrt 99.6%

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

      2. sqrt-unprod 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-un-lft-identity 99.7%

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

      5. associate-*l/ 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. frac-times 99.7%

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

   3. Applied egg-rr 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{2 \cdot \left(\pi \cdot n\right)} \cdot {k}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

   5. div-sub 99.6%

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

   6. sub-neg 99.6%

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

   7. distribute-frac-neg 99.6%

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

   8. metadata-eval 99.6%

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

   9. neg-mul-1 99.6%

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

   10. associate-/l* 99.6%

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

   11. associate-/r/ 99.6%

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

   12. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 5: 53.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.85 \cdot 10^{+115}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{\pi}{\frac{k}{n}}\right)}^{3} \cdot 8\right)}^{0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.85e+115)
   (/ (sqrt (* PI (* 2.0 n))) (sqrt k))
   (pow (* (pow (/ PI (/ k n)) 3.0) 8.0) 0.16666666666666666)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.85e+115) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) / sqrt(k);
	} else {
		tmp = pow((pow((((double) M_PI) / (k / n)), 3.0) * 8.0), 0.16666666666666666);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.85e+115) {
		tmp = Math.sqrt((Math.PI * (2.0 * n))) / Math.sqrt(k);
	} else {
		tmp = Math.pow((Math.pow((Math.PI / (k / n)), 3.0) * 8.0), 0.16666666666666666);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.85e+115:
		tmp = math.sqrt((math.pi * (2.0 * n))) / math.sqrt(k)
	else:
		tmp = math.pow((math.pow((math.pi / (k / n)), 3.0) * 8.0), 0.16666666666666666)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.85e+115)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) / sqrt(k));
	else
		tmp = Float64((Float64(pi / Float64(k / n)) ^ 3.0) * 8.0) ^ 0.16666666666666666;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.85e+115)
		tmp = sqrt((pi * (2.0 * n))) / sqrt(k);
	else
		tmp = (((pi / (k / n)) ^ 3.0) * 8.0) ^ 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.85e+115], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(Pi / N[(k / n), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * 8.0), $MachinePrecision], 0.16666666666666666], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.84999999999999984e115

   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. Taylor expanded in k around 0 69.6%

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

      1. associate-*l/ 69.6%

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

      2. *-un-lft-identity 69.6%

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

      3. sqrt-unprod 69.8%

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

      4. *-commutative 69.8%

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

      5. *-commutative 69.8%

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

      6. associate-*r* 69.8%

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

      7. *-commutative 69.8%

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

      8. associate-*r* 69.8%

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

      9. *-commutative 69.8%

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

   4. Applied egg-rr 69.8%

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

   if 3.84999999999999984e115 < k

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. associate-*l/ 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. frac-times 100.0%

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

   3. Applied egg-rr 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in k around 0 2.6%

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

      1. associate-/l* 2.6%

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

   8. Simplified 2.6%

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

      1. associate-/r/ 2.6%

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

   10. Applied egg-rr 2.6%

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

       1. pow1/2 2.6%

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

       2. add-sqr-sqrt 2.6%

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

       3. metadata-eval 2.6%

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

       4. add-sqr-sqrt 2.6%

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

       5. frac-2neg 2.6%

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

       6. div-inv 2.6%

          \[\leadsto {\left(2 \cdot \left(\color{blue}{\left(\left(-n\right) \cdot \frac{1}{-k}\right)} \cdot \pi\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       7. associate-*r* 2.6%

          \[\leadsto {\left(2 \cdot \color{blue}{\left(\left(-n\right) \cdot \left(\frac{1}{-k} \cdot \pi\right)\right)}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       8. associate-/r/ 2.6%

          \[\leadsto {\left(2 \cdot \left(\left(-n\right) \cdot \color{blue}{\frac{1}{\frac{-k}{\pi}}}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       9. unpow-prod-down 2.6%

          \[\leadsto \color{blue}{{2}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(\left(-n\right) \cdot \frac{1}{\frac{-k}{\pi}}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       10. associate-/r/ 2.6%

           \[\leadsto {2}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(\left(-n\right) \cdot \color{blue}{\left(\frac{1}{-k} \cdot \pi\right)}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       11. associate-*r* 2.6%

           \[\leadsto {2}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\color{blue}{\left(\left(\left(-n\right) \cdot \frac{1}{-k}\right) \cdot \pi\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

       12. div-inv 2.6%

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

       13. frac-2neg 2.6%

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

       14. associate-*l/ 2.6%

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

       15. clear-num 2.6%

           \[\leadsto {2}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\color{blue}{\left(\frac{1}{\frac{k}{n \cdot \pi}}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)} \]

   12. Applied egg-rr 18.6%

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

       1. *-commutative 18.6%

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

       2. associate-*l/ 18.6%

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

       3. associate-*r/ 18.6%

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

       4. *-commutative 18.6%

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

       5. associate-*l* 18.6%

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

       6. cube-prod 19.7%

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

       7. *-commutative 19.7%

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

       8. associate-/r/ 19.7%

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

       9. metadata-eval 19.7%

          \[\leadsto {\left({\left(\frac{\pi}{\frac{k}{n}}\right)}^{3} \cdot \color{blue}{8}\right)}^{0.16666666666666666} \]

   14. Simplified 19.7%

       \[\leadsto \color{blue}{{\left({\left(\frac{\pi}{\frac{k}{n}}\right)}^{3} \cdot 8\right)}^{0.16666666666666666}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.85 \cdot 10^{+115}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{\pi}{\frac{k}{n}}\right)}^{3} \cdot 8\right)}^{0.16666666666666666}\\ \end{array} \]

Alternative 6: 49.2% 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 99.6%

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

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-unprod 94.1%

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

   3. associate-*l/ 94.1%

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

   4. *-un-lft-identity 94.1%

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

   5. associate-*l/ 94.1%

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

   6. *-un-lft-identity 94.1%

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

   7. frac-times 94.1%

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

3. Applied egg-rr 94.1%

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

5. Simplified 94.2%

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

6. Taylor expanded in k around 0 41.0%

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

   1. associate-/l* 41.0%

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

8. Simplified 41.0%

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

   1. associate-/r/ 41.0%

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

10. Applied egg-rr 41.0%

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

    1. add-sqr-sqrt 41.0%

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

    2. add-sqr-sqrt 41.0%

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

    3. frac-2neg 41.0%

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

    4. div-inv 41.0%

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

    5. associate-*r* 40.9%

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

    6. associate-/r/ 41.0%

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

    7. associate-*r* 41.0%

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

    8. sqrt-prod 0.0%

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

    9. add-sqr-sqrt 0.0%

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

    10. sqrt-unprod 0.0%

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

    11. sqr-neg 0.0%

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

    12. sqrt-unprod 0.0%

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

    13. add-sqr-sqrt 0.0%

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

    14. *-commutative 0.0%

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

    15. clear-num 0.0%

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

12. Applied egg-rr 46.4%

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

13. Final simplification 46.4%

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

Alternative 7: 49.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-unprod 94.1%

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

   3. associate-*l/ 94.1%

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

   4. *-un-lft-identity 94.1%

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

   5. associate-*l/ 94.1%

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

   6. *-un-lft-identity 94.1%

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

   7. frac-times 94.1%

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

3. Applied egg-rr 94.1%

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

5. Simplified 94.2%

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

6. Taylor expanded in k around 0 41.0%

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

   1. associate-/l* 41.0%

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

8. Simplified 41.0%

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

   1. associate-/r/ 41.0%

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

10. Applied egg-rr 41.0%

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

    1. frac-2neg 41.0%

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

    2. div-inv 41.0%

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

    3. associate-*r* 40.9%

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

    4. associate-/r/ 41.0%

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

    5. sqrt-prod 40.9%

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

    6. associate-/r/ 40.9%

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

    7. associate-*r* 40.9%

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

    8. div-inv 41.0%

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

    9. frac-2neg 41.0%

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

    10. associate-*l/ 41.0%

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

    11. clear-num 40.9%

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

    12. sqrt-prod 41.0%

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

    13. div-inv 41.0%

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

    14. associate-/r/ 41.0%

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

    15. sqrt-prod 46.5%

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

12. Applied egg-rr 46.5%

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

13. Final simplification 46.5%

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

Alternative 8: 49.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in k around 0 46.3%

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

   1. associate-*l/ 46.3%

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

   2. *-un-lft-identity 46.3%

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

   3. sqrt-unprod 46.5%

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

   4. *-commutative 46.5%

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

   5. *-commutative 46.5%

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

   6. associate-*r* 46.5%

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

   7. *-commutative 46.5%

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

   8. associate-*r* 46.5%

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

   9. *-commutative 46.5%

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

4. Applied egg-rr 46.5%

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

5. Final simplification 46.5%

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

Alternative 9: 38.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-unprod 94.1%

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

   3. associate-*l/ 94.1%

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

   4. *-un-lft-identity 94.1%

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

   5. associate-*l/ 94.1%

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

   6. *-un-lft-identity 94.1%

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

   7. frac-times 94.1%

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

3. Applied egg-rr 94.1%

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

5. Simplified 94.2%

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

6. Taylor expanded in k around 0 41.0%

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

   1. associate-/l* 41.0%

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

8. Simplified 41.0%

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

   1. associate-*r/ 41.0%

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

   2. *-commutative 41.0%

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

   3. associate-/l* 41.0%

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

   4. *-commutative 41.0%

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

   5. sqrt-undiv 46.5%

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

   6. clear-num 46.5%

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

   7. sqrt-undiv 41.4%

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

   8. *-commutative 41.4%

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

   9. associate-*r* 41.4%

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

   10. *-commutative 41.4%

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

   11. associate-*l* 41.4%

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

10. Applied egg-rr 41.4%

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

    1. *-commutative 41.4%

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

    2. *-commutative 41.4%

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

    3. associate-*l* 41.4%

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

    4. *-commutative 41.4%

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

12. Simplified 41.4%

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

13. Final simplification 41.4%

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

Alternative 10: 38.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-unprod 94.1%

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

   3. associate-*l/ 94.1%

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

   4. *-un-lft-identity 94.1%

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

   5. associate-*l/ 94.1%

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

   6. *-un-lft-identity 94.1%

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

   7. frac-times 94.1%

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

3. Applied egg-rr 94.1%

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

5. Simplified 94.2%

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

6. Taylor expanded in k around 0 41.0%

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

   1. associate-/l* 41.0%

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

8. Simplified 41.0%

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

   1. associate-/r/ 41.0%

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

10. Applied egg-rr 41.0%

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

11. Final simplification 41.0%

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

Reproduce

herbie shell --seed 2023334 
(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))))