Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.5%

Time: 18.8s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. *-commutative 99.5%

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

   3. add-sqr-sqrt 99.3%

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

   4. inv-pow 99.3%

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

   5. sqrt-pow2 99.3%

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

   6. metadata-eval 99.3%

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

   7. sqrt-unprod 99.5%

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

   8. associate-*r* 99.5%

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

   9. *-commutative 99.5%

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

   10. associate-*r* 99.5%

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

   11. *-commutative 99.5%

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

   12. pow-prod-up 99.5%

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

5. Applied egg-rr 99.5%

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

   1. pow-sub 99.7%

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

   2. pow1 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;k \leq 7.8 \cdot 10^{-47}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{t_0}^{\left(1 - k\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* n (* 2.0 PI))))
   (if (<= k 7.8e-47)
     (* (pow k -0.5) (sqrt t_0))
     (/ 1.0 (sqrt (/ k (pow t_0 (- 1.0 k))))))))

C

double code(double k, double n) {
	double t_0 = n * (2.0 * ((double) M_PI));
	double tmp;
	if (k <= 7.8e-47) {
		tmp = pow(k, -0.5) * sqrt(t_0);
	} else {
		tmp = 1.0 / sqrt((k / pow(t_0, (1.0 - k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = n * (2.0 * Math.PI);
	double tmp;
	if (k <= 7.8e-47) {
		tmp = Math.pow(k, -0.5) * Math.sqrt(t_0);
	} else {
		tmp = 1.0 / Math.sqrt((k / Math.pow(t_0, (1.0 - k))));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = n * (2.0 * math.pi)
	tmp = 0
	if k <= 7.8e-47:
		tmp = math.pow(k, -0.5) * math.sqrt(t_0)
	else:
		tmp = 1.0 / math.sqrt((k / math.pow(t_0, (1.0 - k))))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(n * Float64(2.0 * pi))
	tmp = 0.0
	if (k <= 7.8e-47)
		tmp = Float64((k ^ -0.5) * sqrt(t_0));
	else
		tmp = Float64(1.0 / sqrt(Float64(k / (t_0 ^ Float64(1.0 - k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = n * (2.0 * pi);
	tmp = 0.0;
	if (k <= 7.8e-47)
		tmp = (k ^ -0.5) * sqrt(t_0);
	else
		tmp = 1.0 / sqrt((k / (t_0 ^ (1.0 - k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 7.8e-47], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(k / N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 7.79999999999999956e-47

   1. Initial program 99.2%

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

      1. associate-*l/ 99.4%

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

      2. *-lft-identity 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-*l* 99.4%

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

   3. Simplified 99.4%

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

      1. div-inv 99.2%

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

      2. *-commutative 99.2%

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

      3. add-sqr-sqrt 98.9%

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

      4. inv-pow 98.9%

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

      5. sqrt-pow2 98.9%

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

      6. metadata-eval 98.9%

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

      7. sqrt-unprod 99.3%

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

      8. associate-*r* 99.3%

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

      9. *-commutative 99.3%

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

      10. associate-*r* 99.3%

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

      11. *-commutative 99.3%

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

      12. pow-prod-up 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in k around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*r* 99.3%

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

   8. Simplified 99.3%

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

   if 7.79999999999999956e-47 < 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. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

      1. div-inv 99.7%

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

      2. *-commutative 99.7%

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

      3. add-sqr-sqrt 99.6%

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

      4. inv-pow 99.6%

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

      5. sqrt-pow2 99.6%

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

      6. metadata-eval 99.6%

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

      7. sqrt-unprod 99.7%

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

      8. associate-*r* 99.7%

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

      9. *-commutative 99.7%

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

      10. associate-*r* 99.7%

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

      11. *-commutative 99.7%

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

      12. pow-prod-up 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. sqrt-pow1 99.7%

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

      3. associate-*r* 99.7%

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

      4. sqrt-pow1 99.7%

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

      5. add-sqr-sqrt 99.6%

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

      6. sqrt-unprod 99.7%

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

      7. sqrt-prod 99.6%

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

      8. pow-prod-up 99.7%

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

      9. metadata-eval 99.7%

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

      10. inv-pow 99.7%

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

      11. div-inv 99.7%

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

      12. clear-num 99.7%

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

      13. sqrt-div 99.7%

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

      14. metadata-eval 99.7%

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

      15. associate-*r* 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-*r* 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.8 \cdot 10^{-47}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(2 \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;k \leq 3.5 \cdot 10^{-47}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* n (* 2.0 PI))))
   (if (<= k 3.5e-47)
     (* (pow k -0.5) (sqrt t_0))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = n * (2.0 * ((double) M_PI));
	double tmp;
	if (k <= 3.5e-47) {
		tmp = pow(k, -0.5) * sqrt(t_0);
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = n * (2.0 * Math.PI);
	double tmp;
	if (k <= 3.5e-47) {
		tmp = Math.pow(k, -0.5) * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = n * (2.0 * math.pi)
	tmp = 0
	if k <= 3.5e-47:
		tmp = math.pow(k, -0.5) * math.sqrt(t_0)
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(n * Float64(2.0 * pi))
	tmp = 0.0
	if (k <= 3.5e-47)
		tmp = Float64((k ^ -0.5) * sqrt(t_0));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = n * (2.0 * pi);
	tmp = 0.0;
	if (k <= 3.5e-47)
		tmp = (k ^ -0.5) * sqrt(t_0);
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 3.5e-47], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.4999999999999998e-47

   1. Initial program 99.2%

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

      1. associate-*l/ 99.4%

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

      2. *-lft-identity 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-*l* 99.4%

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

   3. Simplified 99.4%

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

      1. div-inv 99.2%

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

      2. *-commutative 99.2%

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

      3. add-sqr-sqrt 98.9%

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

      4. inv-pow 98.9%

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

      5. sqrt-pow2 98.9%

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

      6. metadata-eval 98.9%

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

      7. sqrt-unprod 99.3%

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

      8. associate-*r* 99.3%

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

      9. *-commutative 99.3%

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

      10. associate-*r* 99.3%

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

      11. *-commutative 99.3%

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

      12. pow-prod-up 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in k around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*r* 99.3%

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

   8. Simplified 99.3%

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

   if 3.4999999999999998e-47 < 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. *-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*r* 99.7%

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

      4. div-inv 99.7%

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

      5. expm1-log1p-u 99.3%

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

      6. expm1-udef 93.6%

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

   3. Applied egg-rr 93.6%

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

      1. expm1-def 99.3%

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

      2. expm1-log1p 99.7%

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

      3. associate-*r* 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-47}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(2 \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 5: 49.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. *-commutative 99.5%

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

   3. add-sqr-sqrt 99.3%

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

   4. inv-pow 99.3%

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

   5. sqrt-pow2 99.3%

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

   6. metadata-eval 99.3%

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

   7. sqrt-unprod 99.5%

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

   8. associate-*r* 99.5%

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

   9. *-commutative 99.5%

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

   10. associate-*r* 99.5%

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

   11. *-commutative 99.5%

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

   12. pow-prod-up 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in k around 0 47.1%

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

   1. *-commutative 47.1%

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

   2. associate-*r* 47.1%

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

8. Simplified 47.1%

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

9. Final simplification 47.1%

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

Alternative 6: 38.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(Pi * N[(2.0 / N[(k / 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. 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%

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. *-commutative 99.5%

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

   3. add-sqr-sqrt 99.3%

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

   4. inv-pow 99.3%

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

   5. sqrt-pow2 99.3%

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

   6. metadata-eval 99.3%

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

   7. sqrt-unprod 99.5%

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

   8. associate-*r* 99.5%

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

   9. *-commutative 99.5%

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

   10. associate-*r* 99.5%

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

   11. *-commutative 99.5%

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

   12. pow-prod-up 99.5%

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

5. Applied egg-rr 99.5%

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

   1. *-commutative 99.5%

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

   2. sqrt-pow1 99.5%

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

   3. associate-*r* 99.5%

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

   4. sqrt-pow1 99.5%

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

   5. add-sqr-sqrt 99.3%

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

   6. sqrt-unprod 99.5%

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

   7. sqrt-prod 88.9%

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

   8. pow-prod-up 89.0%

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

   9. metadata-eval 89.0%

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

   10. inv-pow 89.0%

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

   11. div-inv 89.0%

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

   12. add-cbrt-cube 80.7%

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

   13. pow1/3 78.8%

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

7. Applied egg-rr 78.8%

   \[\leadsto \color{blue}{{\left({\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
8. Step-by-step derivation

   1. unpow1/3 80.7%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5}}} \]

   2. associate-*r* 80.7%

      \[\leadsto \sqrt[3]{{\left(\frac{{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot n\right)}}^{\left(1 - k\right)}}{k}\right)}^{1.5}} \]

9. Simplified 80.7%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5}}} \]

10. Taylor expanded in k around 0 29.7%

    \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{2 \cdot \left(n \cdot \pi\right)}}{k}\right)}^{1.5}} \]
11. Step-by-step derivation

    1. associate-*r* 29.7%

       \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(2 \cdot n\right) \cdot \pi}}{k}\right)}^{1.5}} \]

12. Simplified 29.7%

    \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(2 \cdot n\right) \cdot \pi}}{k}\right)}^{1.5}} \]
13. Step-by-step derivation

    1. expm1-log1p-u 28.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{\left(2 \cdot n\right) \cdot \pi}{k}\right)}^{1.5}}\right)\right)} \]

    2. expm1-udef 28.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{\left(2 \cdot n\right) \cdot \pi}{k}\right)}^{1.5}}\right)} - 1} \]

    3. pow1/3 28.0%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{\left(2 \cdot n\right) \cdot \pi}{k}\right)}^{1.5}\right)}^{0.3333333333333333}}\right)} - 1 \]

    4. pow-pow 34.4%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{\left(2 \cdot n\right) \cdot \pi}{k}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} - 1 \]

    5. metadata-eval 34.4%

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

    6. pow1/2 34.4%

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

    7. associate-/l* 34.4%

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

14. Applied egg-rr 34.4%

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

    1. expm1-def 34.9%

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

    2. expm1-log1p 36.6%

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

    3. associate-/r/ 36.6%

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

    4. *-commutative 36.6%

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

    5. associate-/l* 36.6%

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

16. Simplified 36.6%

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

17. Final simplification 36.6%

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

Alternative 7: 38.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. *-commutative 99.5%

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

   3. add-sqr-sqrt 99.3%

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

   4. inv-pow 99.3%

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

   5. sqrt-pow2 99.3%

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

   6. metadata-eval 99.3%

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

   7. sqrt-unprod 99.5%

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

   8. associate-*r* 99.5%

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

   9. *-commutative 99.5%

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

   10. associate-*r* 99.5%

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

   11. *-commutative 99.5%

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

   12. pow-prod-up 99.5%

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

5. Applied egg-rr 99.5%

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

   1. *-commutative 99.5%

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

   2. sqrt-pow1 99.5%

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

   3. associate-*r* 99.5%

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

   4. sqrt-pow1 99.5%

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

   5. add-sqr-sqrt 99.3%

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

   6. sqrt-unprod 99.5%

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

   7. sqrt-prod 88.9%

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

   8. pow-prod-up 89.0%

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

   9. metadata-eval 89.0%

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

   10. inv-pow 89.0%

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

   11. div-inv 89.0%

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

   12. add-cbrt-cube 80.7%

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

   13. pow1/3 78.8%

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

7. Applied egg-rr 78.8%

   \[\leadsto \color{blue}{{\left({\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
8. Step-by-step derivation

   1. unpow1/3 80.7%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5}}} \]

   2. associate-*r* 80.7%

      \[\leadsto \sqrt[3]{{\left(\frac{{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot n\right)}}^{\left(1 - k\right)}}{k}\right)}^{1.5}} \]

9. Simplified 80.7%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5}}} \]

10. Taylor expanded in k around 0 29.7%

    \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{2 \cdot \left(n \cdot \pi\right)}}{k}\right)}^{1.5}} \]
11. Step-by-step derivation

    1. associate-*r* 29.7%

       \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(2 \cdot n\right) \cdot \pi}}{k}\right)}^{1.5}} \]

12. Simplified 29.7%

    \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\left(2 \cdot n\right) \cdot \pi}}{k}\right)}^{1.5}} \]
13. Step-by-step derivation

    1. expm1-log1p-u 28.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{\left(2 \cdot n\right) \cdot \pi}{k}\right)}^{1.5}}\right)\right)} \]

    2. expm1-udef 28.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{\left(2 \cdot n\right) \cdot \pi}{k}\right)}^{1.5}}\right)} - 1} \]

    3. pow1/3 28.0%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{\left(2 \cdot n\right) \cdot \pi}{k}\right)}^{1.5}\right)}^{0.3333333333333333}}\right)} - 1 \]

    4. pow-pow 34.4%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{\left(2 \cdot n\right) \cdot \pi}{k}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} - 1 \]

    5. metadata-eval 34.4%

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

    6. pow1/2 34.4%

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

    7. associate-/l* 34.4%

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

14. Applied egg-rr 34.4%

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

    1. expm1-def 34.9%

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

    2. expm1-log1p 36.6%

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

    3. associate-/r/ 36.6%

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

16. Simplified 36.6%

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

17. Final simplification 36.6%

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

Reproduce

herbie shell --seed 2023227 
(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))))