Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 11.4s

Alternatives: 6

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 2.4513265290210897e+253
  2. n = -128495097033.07954

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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around inf

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. exp-prod N/A

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

   7. *-commutative N/A

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

   8. pow-lowering-pow.f64 N/A

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

   9. rem-exp-log N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. rem-exp-log N/A

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

   14. *-lowering-*.f64 N/A

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

   15. rem-exp-log N/A

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

   16. *-commutative N/A

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

   17. *-lowering-*.f64 N/A

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

   18. PI-lowering-PI.f64 N/A

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

   19. sub-neg N/A

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

   20. mul-1-neg N/A

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

5. Simplified 99.6%

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

7. Applied egg-rr 99.6%

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

Alternative 2: 49.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. PI-lowering-PI.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 39.7

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

5. Simplified 39.7%

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

   1. sqrt-unprod N/A

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

   2. associate-*l/ N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. sqrt-undiv N/A

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

   6. /-lowering-/.f64 N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. PI-lowering-PI.f64 N/A

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

   14. sqrt-lowering-sqrt.f64 51.5

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

7. Applied egg-rr 51.5%

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

Alternative 3: 49.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. PI-lowering-PI.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 39.7

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

5. Simplified 39.7%

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

   1. sqrt-unprod N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. /-lowering-/.f64 N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. PI-lowering-PI.f64 39.8

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

7. Applied egg-rr 39.8%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-/l* N/A

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

   4. sqrt-unprod N/A

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

   5. sqrt-unprod N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. PI-lowering-PI.f64 N/A

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

   11. sqrt-unprod N/A

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

   12. sqrt-lowering-sqrt.f64 N/A

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

   13. *-lowering-*.f64 51.5

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

9. Applied egg-rr 51.5%

   \[\leadsto \color{blue}{\sqrt{\frac{\pi}{k}} \cdot \sqrt{n \cdot 2}} \]
10. Add Preprocessing

Alternative 4: 49.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. PI-lowering-PI.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 39.7

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

5. Simplified 39.7%

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

   1. sqrt-unprod N/A

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

   2. associate-/l* N/A

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

   3. associate-*l* N/A

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

   4. sqrt-prod N/A

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

   5. pow1/2 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. pow1/2 N/A

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

   8. sqrt-lowering-sqrt.f64 N/A

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

   9. sqrt-lowering-sqrt.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. PI-lowering-PI.f64 51.5

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

7. Applied egg-rr 51.5%

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

8. Final simplification 51.5%

   \[\leadsto \sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}} \]
9. Add Preprocessing

Alternative 5: 38.3% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. PI-lowering-PI.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 39.7

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

5. Simplified 39.7%

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

   1. sqrt-unprod N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. /-lowering-/.f64 N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. PI-lowering-PI.f64 39.8

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

7. Applied egg-rr 39.8%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. PI-lowering-PI.f64 39.9

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

9. Applied egg-rr 39.9%

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

10. Final simplification 39.9%

    \[\leadsto \sqrt{n \cdot \frac{\pi \cdot 2}{k}} \]
11. Add Preprocessing

Alternative 6: 38.3% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. PI-lowering-PI.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 39.7

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

5. Simplified 39.7%

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

   1. sqrt-unprod N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. /-lowering-/.f64 N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. PI-lowering-PI.f64 39.8

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

7. Applied egg-rr 39.8%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f64 N/A

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

   6. PI-lowering-PI.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. /-lowering-/.f64 39.8

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

9. Applied egg-rr 39.8%

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

    1. *-commutative N/A

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

    2. associate-*l/ N/A

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

    3. associate-/l* N/A

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

    4. associate-*l* N/A

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

    5. *-lowering-*.f64 N/A

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

    6. *-lowering-*.f64 N/A

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

    7. /-lowering-/.f64 N/A

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

    8. PI-lowering-PI.f64 39.8

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

11. Applied egg-rr 39.8%

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

12. Final simplification 39.8%

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

Reproduce

herbie shell --seed 2024199 
(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))))