Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 12.7s

Alternatives: 11

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 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*l* 99.4%

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

   5. div-sub 99.4%

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

   6. sub-neg 99.4%

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

   7. distribute-frac-neg 99.4%

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

   8. metadata-eval 99.4%

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

   9. neg-mul-1 99.4%

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

   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/ 99.4%

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

   12. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. div-inv 99.4%

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

   2. unpow-prod-up 99.6%

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

   3. pow1/2 99.6%

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

   4. associate-*l* 99.6%

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

   5. associate-*r* 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*l* 99.6%

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

   8. associate-*r* 99.6%

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

   9. *-commutative 99.6%

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

   10. associate-*l* 99.6%

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

   11. *-commutative 99.6%

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

   12. inv-pow 99.6%

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

   13. sqrt-pow2 99.6%

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

   14. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

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 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*l* 99.4%

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

   5. div-sub 99.4%

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

   6. sub-neg 99.4%

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

   7. distribute-frac-neg 99.4%

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

   8. metadata-eval 99.4%

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

   9. neg-mul-1 99.4%

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

   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/ 99.4%

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

   12. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. +-commutative 99.4%

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

   2. unpow-prod-up 99.6%

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

   3. associate-*r* 99.6%

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

   4. *-commutative 99.6%

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

   5. associate-*l* 99.6%

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

   6. *-commutative 99.6%

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

   7. pow1/2 99.6%

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

   8. associate-*r* 99.6%

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

   9. *-commutative 99.6%

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

   10. associate-*l* 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

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 99.4%

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

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 99.4%

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

   3. frac-times 99.4%

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

   4. metadata-eval 99.4%

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

   5. add-sqr-sqrt 99.5%

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

3. Applied egg-rr 99.5%

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

4. Final simplification 99.5%

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

Alternative 4: 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 99.4%

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

   1. expm1-log1p-u 49.6%

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

   2. expm1-udef 68.0%

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

   3. pow1/2 68.0%

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

   4. pow-flip 68.0%

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

   5. metadata-eval 68.0%

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

3. Applied egg-rr 72.1%

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

   1. expm1-def 49.6%

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

   2. expm1-log1p 53.1%

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-48}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(2 \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.5e-48)
   (* (pow k -0.5) (sqrt (* n (* 2.0 PI))))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.5e-48) {
		tmp = pow(k, -0.5) * sqrt((n * (2.0 * ((double) M_PI))));
	} else {
		tmp = sqrt((pow((((double) M_PI) * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.5e-48) {
		tmp = Math.pow(k, -0.5) * Math.sqrt((n * (2.0 * Math.PI)));
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.5e-48:
		tmp = math.pow(k, -0.5) * math.sqrt((n * (2.0 * math.pi)))
	else:
		tmp = math.sqrt((math.pow((math.pi * (2.0 * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.5e-48)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(n * Float64(2.0 * pi))));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(2.0 * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.5e-48)
		tmp = (k ^ -0.5) * sqrt((n * (2.0 * pi)));
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.5e-48], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.49999999999999991e-48

   1. Initial program 99.3%

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

   2. Taylor expanded in k around 0 99.3%

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

      1. expm1-log1p-u 92.2%

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

      2. expm1-udef 92.2%

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

      3. pow1/2 92.2%

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

      4. pow-flip 92.2%

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

      5. metadata-eval 92.2%

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

   4. Applied egg-rr 92.2%

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

      1. expm1-def 92.2%

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

      2. expm1-log1p 99.3%

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

   6. Simplified 99.3%

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

      1. sqrt-unprod 99.4%

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

      2. *-commutative 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-*r* 99.4%

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

   8. Applied egg-rr 99.4%

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

   if 3.49999999999999991e-48 < 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 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-48}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(2 \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 6: 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 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*l* 99.4%

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

   5. div-sub 99.4%

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

   6. sub-neg 99.4%

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

   7. distribute-frac-neg 99.4%

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

   8. metadata-eval 99.4%

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

   9. neg-mul-1 99.4%

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

   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/ 99.4%

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

   12. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 7: 55.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{+148}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(2 \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{n}{\frac{\frac{k}{2}}{\pi}}\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.2e+148)
   (* (pow k -0.5) (sqrt (* n (* 2.0 PI))))
   (pow (pow (/ n (/ (/ k 2.0) PI)) 3.0) 0.16666666666666666)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.2e+148) {
		tmp = pow(k, -0.5) * sqrt((n * (2.0 * ((double) M_PI))));
	} else {
		tmp = pow(pow((n / ((k / 2.0) / ((double) M_PI))), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.2e+148) {
		tmp = Math.pow(k, -0.5) * Math.sqrt((n * (2.0 * Math.PI)));
	} else {
		tmp = Math.pow(Math.pow((n / ((k / 2.0) / Math.PI)), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 4.2e+148:
		tmp = math.pow(k, -0.5) * math.sqrt((n * (2.0 * math.pi)))
	else:
		tmp = math.pow(math.pow((n / ((k / 2.0) / math.pi)), 3.0), 0.16666666666666666)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.2e+148)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(n * Float64(2.0 * pi))));
	else
		tmp = (Float64(n / Float64(Float64(k / 2.0) / pi)) ^ 3.0) ^ 0.16666666666666666;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 4.2e+148)
		tmp = (k ^ -0.5) * sqrt((n * (2.0 * pi)));
	else
		tmp = ((n / ((k / 2.0) / pi)) ^ 3.0) ^ 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.2e+148], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(n / N[(N[(k / 2.0), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.19999999999999998e148

   1. Initial program 99.3%

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

   2. Taylor expanded in k around 0 68.8%

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

      1. expm1-log1p-u 64.2%

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

      2. expm1-udef 75.1%

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

      3. pow1/2 75.1%

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

      4. pow-flip 75.1%

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

      5. metadata-eval 75.1%

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

   4. Applied egg-rr 75.1%

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

      1. expm1-def 64.2%

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

      2. expm1-log1p 68.9%

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

   6. Simplified 68.9%

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

      1. sqrt-unprod 68.9%

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

      2. *-commutative 68.9%

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

      3. *-commutative 68.9%

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

      4. associate-*r* 68.9%

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

   8. Applied egg-rr 68.9%

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

   if 4.19999999999999998e148 < 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. Taylor expanded in k around 0 2.7%

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

      1. expm1-log1p-u 2.7%

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

      2. expm1-udef 38.8%

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

      3. *-commutative 38.8%

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

      4. sqrt-unprod 38.8%

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

      5. *-commutative 38.8%

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

      6. *-commutative 38.8%

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

      7. div-inv 38.8%

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

      8. sqrt-undiv 38.8%

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

      9. associate-*r* 38.8%

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

   4. Applied egg-rr 38.8%

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

      1. expm1-def 2.6%

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

      2. expm1-log1p 2.6%

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

      3. associate-*l* 2.6%

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

      4. associate-/l* 2.6%

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

      5. associate-/r/ 2.6%

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

      6. *-commutative 2.6%

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

   6. Simplified 2.6%

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

   7. Taylor expanded in k around 0 2.6%

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

      1. pow1/2 2.6%

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

      2. *-commutative 2.6%

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

      3. associate-*l/ 2.6%

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

      4. associate-*r/ 2.6%

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

      5. *-commutative 2.6%

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

      6. associate-*r* 2.6%

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

      7. metadata-eval 2.6%

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

      8. pow-pow 4.0%

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

      9. sqr-pow 4.0%

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

      10. pow-prod-down 13.4%

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

      11. pow-prod-up 13.4%

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

      12. clear-num 13.4%

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

      13. un-div-inv 13.4%

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

      14. div-inv 13.4%

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

      15. metadata-eval 13.4%

          \[\leadsto {\left({\left(\pi \cdot \frac{n}{k \cdot \color{blue}{0.5}}\right)}^{\left(1.5 + 1.5\right)}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      16. metadata-eval 13.4%

          \[\leadsto {\left({\left(\pi \cdot \frac{n}{k \cdot 0.5}\right)}^{\color{blue}{3}}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

      17. metadata-eval 13.4%

          \[\leadsto {\left({\left(\pi \cdot \frac{n}{k \cdot 0.5}\right)}^{3}\right)}^{\color{blue}{0.16666666666666666}} \]

   9. Applied egg-rr 13.4%

      \[\leadsto \color{blue}{{\left({\left(\pi \cdot \frac{n}{k \cdot 0.5}\right)}^{3}\right)}^{0.16666666666666666}} \]
   10. Step-by-step derivation

       1. *-commutative 13.4%

          \[\leadsto {\left({\color{blue}{\left(\frac{n}{k \cdot 0.5} \cdot \pi\right)}}^{3}\right)}^{0.16666666666666666} \]

       2. associate-/l/ 13.4%

          \[\leadsto {\left({\left(\color{blue}{\frac{\frac{n}{0.5}}{k}} \cdot \pi\right)}^{3}\right)}^{0.16666666666666666} \]

       3. metadata-eval 13.4%

          \[\leadsto {\left({\left(\frac{\frac{n}{\color{blue}{\frac{1}{2}}}}{k} \cdot \pi\right)}^{3}\right)}^{0.16666666666666666} \]

       4. associate-/l* 13.4%

          \[\leadsto {\left({\left(\frac{\color{blue}{\frac{n \cdot 2}{1}}}{k} \cdot \pi\right)}^{3}\right)}^{0.16666666666666666} \]

       5. /-rgt-identity 13.4%

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

       6. associate-/r/ 13.4%

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

       7. associate-/l* 13.4%

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

       8. associate-/l/ 13.4%

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

       9. associate-/r* 13.4%

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

   11. Simplified 13.4%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{+148}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(2 \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{n}{\frac{\frac{k}{2}}{\pi}}\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \]

Alternative 8: 50.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in k around 0 53.1%

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

   1. expm1-log1p-u 49.6%

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

   2. expm1-udef 68.0%

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

   3. pow1/2 68.0%

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

   4. pow-flip 68.0%

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

   5. metadata-eval 68.0%

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

4. Applied egg-rr 68.0%

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

   1. expm1-def 49.6%

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

   2. expm1-log1p 53.1%

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

6. Simplified 53.1%

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

   1. sqrt-unprod 53.1%

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

   2. *-commutative 53.1%

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

   3. *-commutative 53.1%

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

   4. associate-*r* 53.1%

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

8. Applied egg-rr 53.1%

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

9. Final simplification 53.1%

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

Alternative 9: 50.6% 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 99.4%

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

2. Taylor expanded in k around 0 53.1%

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

   1. expm1-log1p-u 49.8%

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

   2. expm1-udef 50.8%

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

   3. *-commutative 50.8%

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

   4. sqrt-unprod 50.8%

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

   5. *-commutative 50.8%

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

   6. *-commutative 50.8%

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

   7. div-inv 50.8%

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

   8. sqrt-undiv 37.4%

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

   9. associate-*r* 37.4%

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

4. Applied egg-rr 37.4%

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

   1. expm1-def 36.4%

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

   2. expm1-log1p 38.2%

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

   3. associate-*l* 38.2%

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

   4. associate-/l* 38.1%

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

   5. associate-/r/ 38.2%

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

   6. *-commutative 38.2%

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

6. Simplified 38.2%

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

   1. sqrt-prod 53.1%

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

   2. *-commutative 53.1%

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

8. Applied egg-rr 53.1%

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

9. Final simplification 53.1%

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

Alternative 10: 39.0% 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 99.4%

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

2. Taylor expanded in k around 0 53.1%

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

   1. expm1-log1p-u 49.8%

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

   2. expm1-udef 50.8%

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

   3. *-commutative 50.8%

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

   4. sqrt-unprod 50.8%

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

   5. *-commutative 50.8%

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

   6. *-commutative 50.8%

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

   7. div-inv 50.8%

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

   8. sqrt-undiv 37.4%

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

   9. associate-*r* 37.4%

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

4. Applied egg-rr 37.4%

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

   1. expm1-def 36.4%

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

   2. expm1-log1p 38.2%

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

   3. associate-*l* 38.2%

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

   4. associate-/l* 38.1%

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

   5. associate-/r/ 38.2%

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

   6. *-commutative 38.2%

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

6. Simplified 38.2%

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

7. Taylor expanded in k around 0 38.2%

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

   1. associate-*l/ 38.2%

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

9. Applied egg-rr 38.2%

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

10. Final simplification 38.2%

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

Alternative 11: 39.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. 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 38.1%

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

   1. *-commutative 38.1%

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

   2. associate-/l* 38.1%

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

6. Simplified 38.1%

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

   1. sqrt-unprod 38.2%

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

   2. associate-/l* 38.2%

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

   3. associate-*r/ 38.2%

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

   4. associate-*l/ 38.2%

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

   5. *-commutative 38.2%

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

   6. *-commutative 38.2%

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

   7. associate-*l* 38.2%

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

8. Applied egg-rr 38.2%

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

9. Final simplification 38.2%

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

Reproduce

herbie shell --seed 2023339 
(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))))