Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 14.5s

Alternatives: 12

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. expm1-log1p-u 95.5%

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

   2. expm1-udef 77.9%

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

   3. pow1/2 77.9%

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

   4. pow-flip 77.9%

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

   5. metadata-eval 77.9%

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

3. Applied egg-rr 77.9%

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

   1. expm1-def 95.5%

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

   2. expm1-log1p 99.2%

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

5. Simplified 99.2%

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

   1. div-sub 99.2%

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

   2. metadata-eval 99.2%

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

   3. pow-sub 99.3%

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

   4. pow1/2 99.3%

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

   5. associate-*l* 99.3%

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

   6. associate-*l* 99.3%

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

   7. div-inv 99.3%

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

   8. metadata-eval 99.3%

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

7. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. *-commutative 99.3%

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

   7. *-commutative 99.3%

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

9. Simplified 99.3%

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

10. Final simplification 99.3%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. *-commutative 99.1%

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

   2. associate-*r* 99.1%

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

   3. associate-/r/ 99.1%

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

   4. add-sqr-sqrt 99.0%

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

   5. sqrt-unprod 99.1%

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

   6. associate-*r* 99.1%

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

   7. *-commutative 99.1%

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

   8. associate-*r* 99.1%

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

   9. *-commutative 99.1%

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

   10. pow-prod-up 99.1%

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

3. Applied egg-rr 99.1%

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

   1. div-inv 99.1%

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

5. Applied egg-rr 99.1%

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

   1. *-commutative 99.1%

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

   2. associate-*r* 99.1%

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

   3. *-commutative 99.1%

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

   4. *-commutative 99.1%

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

   5. sqr-pow 99.1%

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

   6. sqr-pow 99.1%

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

7. Simplified 99.1%

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

   1. inv-pow 99.1%

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

   2. sqrt-pow1 99.1%

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

   3. pow-pow 99.2%

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

   4. *-commutative 99.2%

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

   5. associate-*l* 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-*l* 99.2%

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

   8. div-sub 99.2%

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

   9. metadata-eval 99.2%

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

   10. div-inv 99.2%

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

   11. metadata-eval 99.2%

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

9. Applied egg-rr 99.2%

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

    1. *-commutative 99.2%

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

    2. neg-mul-1 99.2%

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

    3. associate-*r* 99.2%

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

    4. *-commutative 99.2%

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

    5. associate-*l* 99.2%

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

    6. neg-sub0 99.2%

       \[\leadsto \frac{1}{\sqrt{k} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\color{blue}{\left(0 - \left(0.5 - k \cdot 0.5\right)\right)}}} \]

    7. metadata-eval 99.2%

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

    8. associate--r- 99.2%

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

    9. metadata-eval 99.2%

       \[\leadsto \frac{1}{\sqrt{k} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\left(\color{blue}{0} - 0.5\right) + k \cdot 0.5\right)}} \]

    10. metadata-eval 99.2%

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

11. Simplified 99.2%

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

    1. expm1-log1p-u 96.2%

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

    2. expm1-udef 83.9%

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

    3. associate-/r* 83.9%

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

    4. *-commutative 83.9%

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

    5. +-commutative 83.9%

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

    6. fma-def 83.9%

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

13. Applied egg-rr 83.9%

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

    1. expm1-def 96.2%

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

    2. expm1-log1p 99.2%

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

15. Simplified 99.2%

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

16. Final simplification 99.2%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(n \cdot 2\right) \cdot \pi\\ \mathbf{if}\;k \leq 3.05 \cdot 10^{-33}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{k} \cdot {t_0}^{\left(1 - k\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* n 2.0) PI)))
   (if (<= k 3.05e-33)
     (* (pow k -0.5) (sqrt t_0))
     (sqrt (* (/ 1.0 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 <= 3.05e-33) {
		tmp = pow(k, -0.5) * sqrt(t_0);
	} else {
		tmp = sqrt(((1.0 / 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 <= 3.05e-33) {
		tmp = Math.pow(k, -0.5) * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt(((1.0 / 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 <= 3.05e-33:
		tmp = math.pow(k, -0.5) * math.sqrt(t_0)
	else:
		tmp = math.sqrt(((1.0 / k) * math.pow(t_0, (1.0 - k))))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(Float64(n * 2.0) * pi)
	tmp = 0.0
	if (k <= 3.05e-33)
		tmp = Float64((k ^ -0.5) * sqrt(t_0));
	else
		tmp = sqrt(Float64(Float64(1.0 / 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 <= 3.05e-33)
		tmp = (k ^ -0.5) * sqrt(t_0);
	else
		tmp = sqrt(((1.0 / k) * (t_0 ^ (1.0 - k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(n * 2.0), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[k, 3.05e-33], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(1.0 / k), $MachinePrecision] * N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.0500000000000001e-33

   1. Initial program 98.5%

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

      1. expm1-log1p-u 91.3%

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

      2. expm1-udef 91.3%

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

      3. pow1/2 91.3%

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

      4. pow-flip 91.3%

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

      5. metadata-eval 91.3%

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

   3. Applied egg-rr 91.3%

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

      1. expm1-def 91.3%

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

      2. expm1-log1p 98.6%

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

   5. Simplified 98.6%

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

      1. div-sub 98.6%

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

      2. metadata-eval 98.6%

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

      3. pow-sub 98.6%

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

      4. pow1/2 98.6%

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

      5. associate-*l* 98.6%

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

      6. associate-*l* 98.6%

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

      7. div-inv 98.6%

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

      8. metadata-eval 98.6%

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

   7. Applied egg-rr 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-*r* 98.6%

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

      3. *-commutative 98.6%

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

      4. *-commutative 98.6%

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

      5. associate-*r* 98.6%

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

      6. *-commutative 98.6%

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

      7. *-commutative 98.6%

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

   9. Simplified 98.6%

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

   10. Taylor expanded in k around 0 98.6%

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

   if 3.0500000000000001e-33 < 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 99.5%

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

      6. expm1-udef 95.9%

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

   3. Applied egg-rr 95.9%

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

      1. expm1-def 99.5%

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

      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. *-commutative 99.7%

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

      4. associate-*r* 99.7%

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

      5. *-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. div-inv 99.7%

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

      2. *-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.05 \cdot 10^{-33}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\left(n \cdot 2\right) \cdot \pi}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{k} \cdot {\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(1 - k\right)}}\\ \end{array} \]

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(1.0 / k), $MachinePrecision]], $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 99.1%

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

   1. add-sqr-sqrt 99.0%

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

   2. sqrt-unprod 99.1%

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

   3. frac-times 99.1%

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

   4. metadata-eval 99.1%

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

   5. add-sqr-sqrt 99.2%

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

3. Applied egg-rr 99.2%

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

4. Final simplification 99.2%

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

Alternative 5: 99.5% 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 99.1%

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

   1. expm1-log1p-u 95.5%

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

   2. expm1-udef 77.9%

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

   3. pow1/2 77.9%

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

   4. pow-flip 77.9%

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

   5. metadata-eval 77.9%

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

3. Applied egg-rr 77.9%

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

   1. expm1-def 95.5%

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

   2. expm1-log1p 99.2%

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 6: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(n \cdot 2\right) \cdot \pi\\ \mathbf{if}\;k \leq 9 \cdot 10^{-34}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* n 2.0) PI)))
   (if (<= k 9e-34)
     (* (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 <= 9e-34) {
		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 <= 9e-34) {
		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 <= 9e-34:
		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(Float64(n * 2.0) * pi)
	tmp = 0.0
	if (k <= 9e-34)
		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 <= 9e-34)
		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), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[k, 9e-34], 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 < 9.00000000000000085e-34

   1. Initial program 98.5%

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

      1. expm1-log1p-u 91.3%

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

      2. expm1-udef 91.3%

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

      3. pow1/2 91.3%

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

      4. pow-flip 91.3%

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

      5. metadata-eval 91.3%

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

   3. Applied egg-rr 91.3%

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

      1. expm1-def 91.3%

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

      2. expm1-log1p 98.6%

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

   5. Simplified 98.6%

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

      1. div-sub 98.6%

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

      2. metadata-eval 98.6%

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

      3. pow-sub 98.6%

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

      4. pow1/2 98.6%

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

      5. associate-*l* 98.6%

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

      6. associate-*l* 98.6%

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

      7. div-inv 98.6%

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

      8. metadata-eval 98.6%

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

   7. Applied egg-rr 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-*r* 98.6%

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

      3. *-commutative 98.6%

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

      4. *-commutative 98.6%

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

      5. associate-*r* 98.6%

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

      6. *-commutative 98.6%

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

      7. *-commutative 98.6%

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

   9. Simplified 98.6%

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

   10. Taylor expanded in k around 0 98.6%

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

   if 9.00000000000000085e-34 < 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 99.5%

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

      6. expm1-udef 95.9%

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

   3. Applied egg-rr 95.9%

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

      1. expm1-def 99.5%

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

      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. *-commutative 99.7%

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

      4. associate-*r* 99.7%

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

      5. *-commutative 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-34}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\left(n \cdot 2\right) \cdot \pi}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 7: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. associate-*l/ 99.1%

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

   2. *-lft-identity 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-*l* 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 8: 87.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. *-commutative 99.1%

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

   2. *-commutative 99.1%

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

   3. associate-*r* 99.1%

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

   4. div-inv 99.1%

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

   5. expm1-log1p-u 96.2%

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

   6. expm1-udef 83.9%

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

3. Applied egg-rr 73.0%

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

   1. expm1-def 85.4%

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

   2. expm1-log1p 87.2%

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

   3. *-commutative 87.2%

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

   4. associate-*r* 87.2%

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

   5. *-commutative 87.2%

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

5. Simplified 87.2%

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

6. Final simplification 87.2%

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

Alternative 9: 37.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. *-commutative 99.1%

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

   2. *-commutative 99.1%

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

   3. associate-*r* 99.1%

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

   4. div-inv 99.1%

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

   5. expm1-log1p-u 96.2%

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

   6. expm1-udef 83.9%

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

3. Applied egg-rr 73.0%

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

   1. expm1-def 85.4%

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

   2. expm1-log1p 87.2%

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

   3. *-commutative 87.2%

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

   4. associate-*r* 87.2%

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

   5. *-commutative 87.2%

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

5. Simplified 87.2%

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

6. Taylor expanded in k around 0 40.7%

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

   1. associate-/l* 40.7%

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

   2. associate-/r/ 40.7%

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

8. Simplified 40.7%

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

   1. associate-*r* 40.7%

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

   2. sqrt-prod 40.9%

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

10. Applied egg-rr 40.9%

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

11. Final simplification 40.9%

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

Alternative 10: 37.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. *-commutative 99.1%

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

   2. *-commutative 99.1%

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

   3. associate-*r* 99.1%

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

   4. div-inv 99.1%

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

   5. expm1-log1p-u 96.2%

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

   6. expm1-udef 83.9%

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

3. Applied egg-rr 73.0%

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

   1. expm1-def 85.4%

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

   2. expm1-log1p 87.2%

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

   3. *-commutative 87.2%

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

   4. associate-*r* 87.2%

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

   5. *-commutative 87.2%

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

5. Simplified 87.2%

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

6. Taylor expanded in k around 0 40.7%

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

   1. associate-/l* 40.7%

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

   2. associate-/r/ 40.7%

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

8. Simplified 40.7%

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

   1. *-commutative 40.7%

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

   2. sqrt-prod 41.0%

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

   3. *-commutative 41.0%

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

10. Applied egg-rr 41.0%

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

11. Final simplification 41.0%

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

Alternative 11: 37.9% 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 99.1%

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

   1. *-commutative 99.1%

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

   2. *-commutative 99.1%

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

   3. associate-*r* 99.1%

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

   4. div-inv 99.1%

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

   5. expm1-log1p-u 96.2%

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

   6. expm1-udef 83.9%

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

3. Applied egg-rr 73.0%

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

   1. expm1-def 85.4%

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

   2. expm1-log1p 87.2%

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

   3. *-commutative 87.2%

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

   4. associate-*r* 87.2%

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

   5. *-commutative 87.2%

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

5. Simplified 87.2%

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

6. Taylor expanded in k around 0 40.7%

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

   1. associate-/l* 40.7%

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

   2. associate-/r/ 40.7%

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

8. Simplified 40.7%

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

9. Final simplification 40.7%

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

Alternative 12: 37.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. *-commutative 99.1%

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

   2. *-commutative 99.1%

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

   3. associate-*r* 99.1%

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

   4. div-inv 99.1%

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

   5. expm1-log1p-u 96.2%

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

   6. expm1-udef 83.9%

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

3. Applied egg-rr 73.0%

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

   1. expm1-def 85.4%

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

   2. expm1-log1p 87.2%

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

   3. *-commutative 87.2%

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

   4. associate-*r* 87.2%

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

   5. *-commutative 87.2%

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

5. Simplified 87.2%

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

6. Taylor expanded in k around 0 40.7%

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

   1. associate-/l* 40.7%

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

   2. associate-/r/ 40.7%

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

8. Simplified 40.7%

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

   1. *-commutative 40.7%

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

   2. clear-num 40.7%

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

   3. un-div-inv 40.8%

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

10. Applied egg-rr 40.8%

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

11. Final simplification 40.8%

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

Reproduce

herbie shell --seed 2023216 
(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))))