Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 17.6s

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 99.5%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

   5. div-sub 99.5%

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

   6. sub-neg 99.5%

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

   7. distribute-frac-neg 99.5%

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

   8. metadata-eval 99.5%

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

   9. neg-mul-1 99.5%

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

   10. associate-/l* 99.4%

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

   11. associate-/r/ 99.5%

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

   12. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.5%

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

   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 99.7%

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

   14. metadata-eval 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

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 99.5%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

   5. div-sub 99.5%

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

   6. sub-neg 99.5%

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

   7. distribute-frac-neg 99.5%

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

   8. metadata-eval 99.5%

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

   9. neg-mul-1 99.5%

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

   10. associate-/l* 99.4%

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

   11. associate-/r/ 99.5%

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

   12. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. +-commutative 99.5%

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

   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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {k}^{-0.5} \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{1 - k}{2}\right)} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (pow k -0.5) (pow (* n (* 2.0 PI)) (/ (- 1.0 k) 2.0))))

C

double code(double k, double n) {
	return pow(k, -0.5) * pow((n * (2.0 * ((double) M_PI))), ((1.0 - k) / 2.0));
}

Java

public static double code(double k, double n) {
	return Math.pow(k, -0.5) * Math.pow((n * (2.0 * Math.PI)), ((1.0 - k) / 2.0));
}

Python

def code(k, n):
	return math.pow(k, -0.5) * math.pow((n * (2.0 * math.pi)), ((1.0 - k) / 2.0))

Julia

function code(k, n)
	return Float64((k ^ -0.5) * (Float64(n * Float64(2.0 * pi)) ^ Float64(Float64(1.0 - k) / 2.0)))
end

MATLAB

function tmp = code(k, n)
	tmp = (k ^ -0.5) * ((n * (2.0 * pi)) ^ ((1.0 - k) / 2.0));
end

Wolfram

code[k_, n_] := N[(N[Power[k, -0.5], $MachinePrecision] * N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $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. expm1-log1p-u 96.1%

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

   2. expm1-udef 76.9%

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

   3. pow1/2 76.9%

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

   4. pow-flip 76.9%

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

   5. metadata-eval 76.9%

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

3. Applied egg-rr 76.9%

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

   1. expm1-def 96.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. expm1-log1p 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot n\right)\\ \mathbf{if}\;k \leq 1.85 \cdot 10^{-28}:\\ \;\;\;\;\frac{\sqrt{t_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* 2.0 n))))
   (if (<= k 1.85e-28)
     (/ (sqrt t_0) (sqrt k))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (2.0 * n);
	double tmp;
	if (k <= 1.85e-28) {
		tmp = sqrt(t_0) / sqrt(k);
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (2.0 * n);
	double tmp;
	if (k <= 1.85e-28) {
		tmp = Math.sqrt(t_0) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.pi * (2.0 * n)
	tmp = 0
	if k <= 1.85e-28:
		tmp = math.sqrt(t_0) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(2.0 * n))
	tmp = 0.0
	if (k <= 1.85e-28)
		tmp = Float64(sqrt(t_0) / sqrt(k));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = pi * (2.0 * n);
	tmp = 0.0;
	if (k <= 1.85e-28)
		tmp = sqrt(t_0) / sqrt(k);
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.85e-28], N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $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 < 1.8500000000000001e-28

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0 99.2%

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

      1. associate-*l/ 99.3%

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

      2. *-un-lft-identity 99.3%

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

      3. sqrt-unprod 99.4%

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

      4. *-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-*r* 99.4%

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

      9. *-commutative 99.4%

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

   4. Applied egg-rr 99.4%

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

   if 1.8500000000000001e-28 < k

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. 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 99.6%

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

      3. associate-*l/ 99.6%

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

      4. *-un-lft-identity 99.6%

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

      5. associate-*l/ 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. frac-times 99.6%

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

   3. Applied egg-rr 99.6%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.85 \cdot 10^{-28}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 5: 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 99.5%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

   5. div-sub 99.5%

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

   6. sub-neg 99.5%

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

   7. distribute-frac-neg 99.5%

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

   8. metadata-eval 99.5%

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

   9. neg-mul-1 99.5%

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

   10. associate-/l* 99.4%

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

   11. associate-/r/ 99.5%

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

   12. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 6: 54.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{+173}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\left(2 \cdot n\right) \cdot \frac{\pi}{k}\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.8e+173)
   (/ (sqrt (* PI (* 2.0 n))) (sqrt k))
   (pow (pow (* (* 2.0 n) (/ PI k)) 3.0) 0.16666666666666666)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.8e+173) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) / sqrt(k);
	} else {
		tmp = pow(pow(((2.0 * n) * (((double) M_PI) / k)), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.8e+173) {
		tmp = Math.sqrt((Math.PI * (2.0 * n))) / Math.sqrt(k);
	} else {
		tmp = Math.pow(Math.pow(((2.0 * n) * (Math.PI / k)), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 4.8e+173:
		tmp = math.sqrt((math.pi * (2.0 * n))) / math.sqrt(k)
	else:
		tmp = math.pow(math.pow(((2.0 * n) * (math.pi / k)), 3.0), 0.16666666666666666)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.8e+173)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) / sqrt(k));
	else
		tmp = (Float64(Float64(2.0 * n) * Float64(pi / k)) ^ 3.0) ^ 0.16666666666666666;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 4.8e+173)
		tmp = sqrt((pi * (2.0 * n))) / sqrt(k);
	else
		tmp = (((2.0 * n) * (pi / k)) ^ 3.0) ^ 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.8e+173], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(2.0 * n), $MachinePrecision] * N[(Pi / k), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.7999999999999998e173

   1. Initial program 99.4%

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

   2. Taylor expanded in k around 0 63.3%

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

      1. associate-*l/ 63.3%

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

      2. *-un-lft-identity 63.3%

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

      3. sqrt-unprod 63.4%

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

      4. *-commutative 63.4%

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

      5. *-commutative 63.4%

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

      6. associate-*r* 63.4%

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

      7. *-commutative 63.4%

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

      8. associate-*r* 63.4%

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

      9. *-commutative 63.4%

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

   4. Applied egg-rr 63.4%

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

   if 4.7999999999999998e173 < 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-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. associate-*l* 100.0%

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

      6. div-sub 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}}}\right)}^{3} \]

      8. div-inv 100.0%

         \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \color{blue}{k \cdot \frac{1}{2}}\right)}}{\sqrt{k}}}\right)}^{3} \]

      9. metadata-eval 100.0%

         \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot \color{blue}{0.5}\right)}}{\sqrt{k}}}\right)}^{3} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\sqrt{k}}}\right)}^{3}} \]

   4. Taylor expanded in k around 0 2.6%

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

      1. *-commutative 2.6%

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

      2. sqrt-unprod 2.6%

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

      3. associate-/l* 2.6%

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

   6. Applied egg-rr 2.6%

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

      1. pow1/2 2.6%

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

      2. metadata-eval 2.6%

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

      3. pow-pow 6.0%

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

      4. sqr-pow 6.0%

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

      5. pow-prod-down 24.8%

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

      6. pow-prod-up 24.8%

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

      7. associate-*r/ 24.8%

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

      8. *-commutative 24.8%

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

      9. div-inv 24.8%

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

      10. clear-num 24.8%

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

      11. metadata-eval 24.8%

          \[\leadsto {\left({\left(\left(n \cdot 2\right) \cdot \frac{\pi}{k}\right)}^{\color{blue}{3}}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      12. metadata-eval 24.8%

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

   8. Applied egg-rr 24.8%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{+173}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\left(2 \cdot n\right) \cdot \frac{\pi}{k}\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \]

Alternative 7: 49.8% 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 99.5%

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

2. Taylor expanded in k around 0 50.0%

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

   1. expm1-log1p-u 47.3%

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

   2. expm1-udef 47.5%

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

4. Applied egg-rr 38.1%

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

   1. expm1-def 37.8%

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

   2. expm1-log1p 39.6%

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

   3. associate-*r* 39.6%

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

   4. *-commutative 39.6%

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

   5. associate-/l* 39.6%

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

   6. associate-/r/ 39.6%

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

   7. *-commutative 39.6%

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

6. Simplified 39.6%

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

   1. sqrt-prod 50.1%

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

   2. *-commutative 50.1%

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

8. Applied egg-rr 50.1%

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

9. Final simplification 50.1%

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

Alternative 8: 49.8% 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 99.5%

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

2. Taylor expanded in k around 0 50.0%

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

   1. associate-*l/ 50.1%

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

   2. *-un-lft-identity 50.1%

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

   3. sqrt-unprod 50.1%

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

   4. *-commutative 50.1%

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

   5. *-commutative 50.1%

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

   6. associate-*r* 50.1%

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

   7. *-commutative 50.1%

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

   8. associate-*r* 50.1%

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

   9. *-commutative 50.1%

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

4. Applied egg-rr 50.1%

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

5. Final simplification 50.1%

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

Alternative 9: 38.3% 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 99.5%

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

   1. add-cube-cbrt 98.7%

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

   2. pow3 98.8%

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

   3. associate-*l/ 98.8%

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

   4. *-un-lft-identity 98.8%

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

   5. associate-*l* 98.8%

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

   6. div-sub 98.8%

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

   7. metadata-eval 98.8%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}}}\right)}^{3} \]

   8. div-inv 98.8%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \color{blue}{k \cdot \frac{1}{2}}\right)}}{\sqrt{k}}}\right)}^{3} \]

   9. metadata-eval 98.8%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot \color{blue}{0.5}\right)}}{\sqrt{k}}}\right)}^{3} \]

3. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\sqrt{k}}}\right)}^{3}} \]

4. Taylor expanded in k around 0 39.5%

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

   1. *-commutative 39.5%

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

   2. sqrt-unprod 39.6%

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

   3. associate-/l* 39.6%

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

6. Applied egg-rr 39.6%

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

   1. associate-/r/ 39.6%

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

8. Applied egg-rr 39.6%

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

9. Final simplification 39.6%

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

Alternative 10: 38.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(2.0 * N[(n / N[(k / 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. add-cube-cbrt 98.7%

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

   2. pow3 98.8%

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

   3. associate-*l/ 98.8%

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

   4. *-un-lft-identity 98.8%

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

   5. associate-*l* 98.8%

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

   6. div-sub 98.8%

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

   7. metadata-eval 98.8%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}}}\right)}^{3} \]

   8. div-inv 98.8%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - \color{blue}{k \cdot \frac{1}{2}}\right)}}{\sqrt{k}}}\right)}^{3} \]

   9. metadata-eval 98.8%

      \[\leadsto {\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot \color{blue}{0.5}\right)}}{\sqrt{k}}}\right)}^{3} \]

3. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\sqrt{k}}}\right)}^{3}} \]

4. Taylor expanded in k around 0 39.5%

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

   1. *-commutative 39.5%

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

   2. sqrt-unprod 39.6%

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

   3. associate-/l* 39.6%

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

6. Applied egg-rr 39.6%

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

7. Final simplification 39.6%

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

Reproduce

herbie shell --seed 2023325 
(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))))