normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 13.9s

Alternatives: 10

Speedup: 1.5×

Specification

\[\left(0 \leq u1 \land u1 \leq 1\right) \land \left(0 \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2)))
  0.5))

C

double code(double u1, double u2) {
	return (((1.0 / 6.0) * pow((-2.0 * log(u1)), 0.5)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (((1.0 / 6.0) * Math.pow((-2.0 * Math.log(u1)), 0.5)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (((1.0 / 6.0) * math.pow((-2.0 * math.log(u1)), 0.5)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(Float64(Float64(1.0 / 6.0) * (Float64(-2.0 * log(u1)) ^ 0.5)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (((1.0 / 6.0) * ((-2.0 * log(u1)) ^ 0.5)) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[Power[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2)))
  0.5))

C

double code(double u1, double u2) {
	return (((1.0 / 6.0) * pow((-2.0 * log(u1)), 0.5)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (((1.0 / 6.0) * Math.pow((-2.0 * Math.log(u1)), 0.5)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (((1.0 / 6.0) * math.pow((-2.0 * math.log(u1)), 0.5)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(Float64(Float64(1.0 / 6.0) * (Float64(-2.0 * log(u1)) ^ 0.5)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (((1.0 / 6.0) * ((-2.0 * log(u1)) ^ 0.5)) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[Power[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \left(0.16666666666666666 \cdot \sqrt{2}\right), \sqrt{-\log u1}, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (cos (* PI (+ u2 u2))) (* 0.16666666666666666 (sqrt 2.0)))
  (sqrt (- (log u1)))
  0.5))

C

double code(double u1, double u2) {
	return fma((cos((((double) M_PI) * (u2 + u2))) * (0.16666666666666666 * sqrt(2.0))), sqrt(-log(u1)), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(cos(Float64(pi * Float64(u2 + u2))) * Float64(0.16666666666666666 * sqrt(2.0))), sqrt(Float64(-log(u1))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Cos[N[(Pi * N[(u2 + u2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.16666666666666666 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u1 around inf

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{\log \left(\frac{1}{u1}\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. log-rec N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. lower-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. lower-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. lower-sqrt.f64 99.5

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

5. Applied rewrites 99.5%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{-\log u1} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{1}{6}} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. lift-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. lift-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} \cdot \sqrt{2}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   9. pow1/2 N/A

      \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{{2}^{\frac{1}{2}}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot {2}^{\color{blue}{\left(\frac{1}{4} + \frac{1}{4}\right)}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   11. pow-prod-up N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\left({2}^{\frac{1}{4}} \cdot {2}^{\frac{1}{4}}\right)}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   12. pow-prod-down N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{{\left(2 \cdot 2\right)}^{\frac{1}{4}}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   13. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot {\color{blue}{4}}^{\frac{1}{4}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   14. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot {\color{blue}{\left(-2 \cdot -2\right)}}^{\frac{1}{4}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   15. pow-prod-down N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\left({-2}^{\frac{1}{4}} \cdot {-2}^{\frac{1}{4}}\right)}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   16. pow-prod-up N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{{-2}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   17. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot {-2}^{\color{blue}{\frac{1}{2}}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   18. pow1/2 N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\sqrt{-2}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   19. lift-sqrt.f64 N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\sqrt{-2}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   20. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left(\sqrt{-2} \cdot \frac{1}{6}\right)} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   21. lift-/.f64 N/A

       \[\leadsto \left(\left(\sqrt{-2} \cdot \color{blue}{\frac{1}{6}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   22. metadata-eval N/A

       \[\leadsto \left(\left(\sqrt{-2} \cdot \color{blue}{\frac{1}{6}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   23. lift-*.f64 N/A

       \[\leadsto \left(\color{blue}{\left(\sqrt{-2} \cdot \frac{1}{6}\right)} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   24. lower-*.f64 0.0

       \[\leadsto \color{blue}{\left(\left(\sqrt{-2} \cdot 0.16666666666666666\right) \cdot \sqrt{-\log u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

7. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\left(0.16666666666666666 \cdot \sqrt{2}\right) \cdot \sqrt{-\log u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
8. Step-by-step derivation

9. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \left(0.16666666666666666 \cdot \sqrt{2}\right), \sqrt{-\log u1}, 0.5\right)} \]
10. Add Preprocessing

Alternative 2: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{2}, 0.16666666666666666 \cdot \left(\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{-\log u1}\right), 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (sqrt 2.0)
  (* 0.16666666666666666 (* (cos (* PI (+ u2 u2))) (sqrt (- (log u1)))))
  0.5))

C

double code(double u1, double u2) {
	return fma(sqrt(2.0), (0.16666666666666666 * (cos((((double) M_PI) * (u2 + u2))) * sqrt(-log(u1)))), 0.5);
}

Julia

function code(u1, u2)
	return fma(sqrt(2.0), Float64(0.16666666666666666 * Float64(cos(Float64(pi * Float64(u2 + u2))) * sqrt(Float64(-log(u1))))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.16666666666666666 * N[(N[Cos[N[(Pi * N[(u2 + u2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u1 around inf

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{\log \left(\frac{1}{u1}\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. log-rec N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. lower-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. lower-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. lower-sqrt.f64 99.5

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

5. Applied rewrites 99.5%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{-\log u1} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{1}{6}} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. lift-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. lift-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} \cdot \sqrt{2}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   9. pow1/2 N/A

      \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{{2}^{\frac{1}{2}}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot {2}^{\color{blue}{\left(\frac{1}{4} + \frac{1}{4}\right)}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   11. pow-prod-up N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\left({2}^{\frac{1}{4}} \cdot {2}^{\frac{1}{4}}\right)}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   12. pow-prod-down N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{{\left(2 \cdot 2\right)}^{\frac{1}{4}}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   13. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot {\color{blue}{4}}^{\frac{1}{4}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   14. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot {\color{blue}{\left(-2 \cdot -2\right)}}^{\frac{1}{4}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   15. pow-prod-down N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\left({-2}^{\frac{1}{4}} \cdot {-2}^{\frac{1}{4}}\right)}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   16. pow-prod-up N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{{-2}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   17. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot {-2}^{\color{blue}{\frac{1}{2}}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   18. pow1/2 N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\sqrt{-2}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   19. lift-sqrt.f64 N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\sqrt{-2}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   20. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left(\sqrt{-2} \cdot \frac{1}{6}\right)} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   21. lift-/.f64 N/A

       \[\leadsto \left(\left(\sqrt{-2} \cdot \color{blue}{\frac{1}{6}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   22. metadata-eval N/A

       \[\leadsto \left(\left(\sqrt{-2} \cdot \color{blue}{\frac{1}{6}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   23. lift-*.f64 N/A

       \[\leadsto \left(\color{blue}{\left(\sqrt{-2} \cdot \frac{1}{6}\right)} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   24. lower-*.f64 0.0

       \[\leadsto \color{blue}{\left(\left(\sqrt{-2} \cdot 0.16666666666666666\right) \cdot \sqrt{-\log u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

7. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\left(0.16666666666666666 \cdot \sqrt{2}\right) \cdot \sqrt{-\log u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

9. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{2}, 0.16666666666666666 \cdot \left(\sqrt{-\log u1} \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right)\right), 0.5\right)} \]

10. Final simplification 99.5%

    \[\leadsto \mathsf{fma}\left(\sqrt{2}, 0.16666666666666666 \cdot \left(\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{-\log u1}\right), 0.5\right) \]
11. Add Preprocessing

Alternative 3: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (cos (* PI (+ u2 u2))) (sqrt (* (log u1) -2.0)))
  0.16666666666666666
  0.5))

C

double code(double u1, double u2) {
	return fma((cos((((double) M_PI) * (u2 + u2))) * sqrt((log(u1) * -2.0))), 0.16666666666666666, 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(cos(Float64(pi * Float64(u2 + u2))) * sqrt(Float64(log(u1) * -2.0))), 0.16666666666666666, 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Cos[N[(Pi * N[(u2 + u2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\left(-2 \cdot \color{blue}{\log u1}\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\color{blue}{\left(-2 \cdot \log u1\right)}}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. sqr-pow N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left({\left(-2 \cdot \log u1\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(-2 \cdot \log u1\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. pow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{\left({\left(-2 \cdot \log u1\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. lower-pow.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{\left({\left(-2 \cdot \log u1\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. metadata-eval N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(-2 \cdot \log u1\right)}^{\color{blue}{\frac{1}{4}}}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   7. metadata-eval N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(-2 \cdot \log u1\right)}^{\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{2}\right)}}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   8. lower-pow.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\color{blue}{\left({\left(-2 \cdot \log u1\right)}^{\left(\frac{1}{2} \cdot \frac{1}{2}\right)}\right)}}^{2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   9. metadata-eval 99.2

      \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(-2 \cdot \log u1\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Applied rewrites 99.2%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{\left({\left(-2 \cdot \log u1\right)}^{0.25}\right)}^{2}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
5. Step-by-step derivation

6. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\log u1 \cdot -2}, 0.5\right)} \]
7. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \cos \left(2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot u2\right)\right)\right) \cdot \sqrt{\log u1 \cdot -2} + \frac{1}{2} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \cos \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2\right)}\right)\right) \cdot \sqrt{\log u1 \cdot -2} + \frac{1}{2} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \cos \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)}\right) \cdot \sqrt{\log u1 \cdot -2} + \frac{1}{2} \]

   4. lift-cos.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)}\right) \cdot \sqrt{\log u1 \cdot -2} + \frac{1}{2} \]

   5. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\right)} \cdot \sqrt{\log u1 \cdot -2} + \frac{1}{2} \]

   6. lift-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\right) \cdot \sqrt{\color{blue}{\log u1} \cdot -2} + \frac{1}{2} \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\right) \cdot \sqrt{\color{blue}{\log u1 \cdot -2}} + \frac{1}{2} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\right) \cdot \color{blue}{\sqrt{\log u1 \cdot -2}} + \frac{1}{2} \]

   9. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\right)} \cdot \sqrt{\log u1 \cdot -2} + \frac{1}{2} \]

   10. associate-*l* N/A

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(\cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right) \cdot \sqrt{\log u1 \cdot -2}\right)} + \frac{1}{2} \]

   11. *-commutative N/A

       \[\leadsto \color{blue}{\left(\cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right) \cdot \sqrt{\log u1 \cdot -2}\right) \cdot \frac{1}{6}} + \frac{1}{2} \]

   12. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right) \cdot \sqrt{\log u1 \cdot -2}, \frac{1}{6}, \frac{1}{2}\right)} \]

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1 \cdot -2} \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right), 0.16666666666666666, 0.5\right)} \]

9. Final simplification 99.4%

   \[\leadsto \mathsf{fma}\left(\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right) \]
10. Add Preprocessing

Alternative 4: 98.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  0.5
  (*
   (* (/ 1.0 6.0) (* (sqrt 2.0) (sqrt (- (log u1)))))
   (fma u2 (* u2 (* -2.0 (* PI PI))) 1.0))))

C

double code(double u1, double u2) {
	return 0.5 + (((1.0 / 6.0) * (sqrt(2.0) * sqrt(-log(u1)))) * fma(u2, (u2 * (-2.0 * (((double) M_PI) * ((double) M_PI)))), 1.0));
}

Julia

function code(u1, u2)
	return Float64(0.5 + Float64(Float64(Float64(1.0 / 6.0) * Float64(sqrt(2.0) * sqrt(Float64(-log(u1))))) * fma(u2, Float64(u2 * Float64(-2.0 * Float64(pi * pi))), 1.0)))
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(u2 * N[(u2 * N[(-2.0 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u1 around inf

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{\log \left(\frac{1}{u1}\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. log-rec N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. lower-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. lower-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. lower-sqrt.f64 99.5

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

5. Applied rewrites 99.5%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{-\log u1} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

6. Taylor expanded in u2 around 0

   \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + \frac{1}{2} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} + \frac{1}{2} \]

   2. rem-square-sqrt N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{\left(\sqrt{-2} \cdot \sqrt{-2}\right)} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) + \frac{1}{2} \]

   3. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{{\left(\sqrt{-2}\right)}^{2}} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) + \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {\left(\sqrt{-2}\right)}^{2}} + 1\right) + \frac{1}{2} \]

   5. associate-*r* N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\left(\sqrt{-2}\right)}^{2}\right)} + 1\right) + \frac{1}{2} \]

   6. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{\left(\sqrt{-2} \cdot \sqrt{-2}\right)}\right) + 1\right) + \frac{1}{2} \]

   7. rem-square-sqrt N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{-2}\right) + 1\right) + \frac{1}{2} \]

   8. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left({u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) + \frac{1}{2} \]

   9. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) + \frac{1}{2} \]

   10. associate-*l* N/A

       \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + 1\right) + \frac{1}{2} \]

   11. lower-fma.f64 N/A

       \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right), 1\right)} + \frac{1}{2} \]

8. Applied rewrites 99.0%

   \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\left(\pi \cdot \pi\right) \cdot -2\right), 1\right)} + 0.5 \]

9. Final simplification 99.0%

   \[\leadsto 0.5 + \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right) \]
10. Add Preprocessing

Alternative 5: 98.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ 0.5 + \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right) \cdot \left(\sqrt{2} \cdot \left(0.16666666666666666 \cdot \sqrt{-\log u1}\right)\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  0.5
  (*
   (fma u2 (* u2 (* -2.0 (* PI PI))) 1.0)
   (* (sqrt 2.0) (* 0.16666666666666666 (sqrt (- (log u1))))))))

C

double code(double u1, double u2) {
	return 0.5 + (fma(u2, (u2 * (-2.0 * (((double) M_PI) * ((double) M_PI)))), 1.0) * (sqrt(2.0) * (0.16666666666666666 * sqrt(-log(u1)))));
}

Julia

function code(u1, u2)
	return Float64(0.5 + Float64(fma(u2, Float64(u2 * Float64(-2.0 * Float64(pi * pi))), 1.0) * Float64(sqrt(2.0) * Float64(0.16666666666666666 * sqrt(Float64(-log(u1)))))))
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[(N[(u2 * N[(u2 * N[(-2.0 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.16666666666666666 * N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u1 around inf

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{\log \left(\frac{1}{u1}\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. log-rec N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. lower-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. lower-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. lower-sqrt.f64 99.5

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

5. Applied rewrites 99.5%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{-\log u1} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

6. Taylor expanded in u2 around 0

   \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + \frac{1}{2} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} + \frac{1}{2} \]

   2. rem-square-sqrt N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{\left(\sqrt{-2} \cdot \sqrt{-2}\right)} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) + \frac{1}{2} \]

   3. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{{\left(\sqrt{-2}\right)}^{2}} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) + \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {\left(\sqrt{-2}\right)}^{2}} + 1\right) + \frac{1}{2} \]

   5. associate-*r* N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\left(\sqrt{-2}\right)}^{2}\right)} + 1\right) + \frac{1}{2} \]

   6. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{\left(\sqrt{-2} \cdot \sqrt{-2}\right)}\right) + 1\right) + \frac{1}{2} \]

   7. rem-square-sqrt N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{-2}\right) + 1\right) + \frac{1}{2} \]

   8. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left({u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) + \frac{1}{2} \]

   9. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) + \frac{1}{2} \]

   10. associate-*l* N/A

       \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + 1\right) + \frac{1}{2} \]

   11. lower-fma.f64 N/A

       \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right), 1\right)} + \frac{1}{2} \]

8. Applied rewrites 99.0%

   \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\left(\pi \cdot \pi\right) \cdot -2\right), 1\right)} + 0.5 \]
9. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left(\color{blue}{\frac{1}{6}} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -2\right), 1\right) + \frac{1}{2} \]

   2. lift-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -2\right), 1\right) + \frac{1}{2} \]

   3. lift-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -2\right), 1\right) + \frac{1}{2} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -2\right), 1\right) + \frac{1}{2} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -2\right), 1\right) + \frac{1}{2} \]

   6. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \sqrt{2}\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -2\right), 1\right) + \frac{1}{2} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \sqrt{2}\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -2\right), 1\right) + \frac{1}{2} \]

   8. lower-*.f64 99.0

      \[\leadsto \left(\color{blue}{\left(0.16666666666666666 \cdot \sqrt{-\log u1}\right)} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\left(\pi \cdot \pi\right) \cdot -2\right), 1\right) + 0.5 \]

10. Applied rewrites 99.0%

    \[\leadsto \color{blue}{\left(\left(0.16666666666666666 \cdot \sqrt{-\log u1}\right) \cdot \sqrt{2}\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\left(\pi \cdot \pi\right) \cdot -2\right), 1\right) + 0.5 \]

11. Final simplification 99.0%

    \[\leadsto 0.5 + \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right) \cdot \left(\sqrt{2} \cdot \left(0.16666666666666666 \cdot \sqrt{-\log u1}\right)\right) \]
12. Add Preprocessing

Alternative 6: 98.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(0.16666666666666666 \cdot \sqrt{-\log u1}\right) \cdot \mathsf{fma}\left(u2, -2 \cdot \left(u2 \cdot \left(\pi \cdot \pi\right)\right), 1\right), \sqrt{2}, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (*
   (* 0.16666666666666666 (sqrt (- (log u1))))
   (fma u2 (* -2.0 (* u2 (* PI PI))) 1.0))
  (sqrt 2.0)
  0.5))

C

double code(double u1, double u2) {
	return fma(((0.16666666666666666 * sqrt(-log(u1))) * fma(u2, (-2.0 * (u2 * (((double) M_PI) * ((double) M_PI)))), 1.0)), sqrt(2.0), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(Float64(0.16666666666666666 * sqrt(Float64(-log(u1)))) * fma(u2, Float64(-2.0 * Float64(u2 * Float64(pi * pi))), 1.0)), sqrt(2.0), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[(0.16666666666666666 * N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * N[(u2 * N[(-2.0 * N[(u2 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u1 around inf

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{\log \left(\frac{1}{u1}\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. log-rec N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. lower-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. lower-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. lower-sqrt.f64 99.5

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

5. Applied rewrites 99.5%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{-\log u1} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

6. Taylor expanded in u2 around 0

   \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + \frac{1}{2} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} + \frac{1}{2} \]

   2. rem-square-sqrt N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{\left(\sqrt{-2} \cdot \sqrt{-2}\right)} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) + \frac{1}{2} \]

   3. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{{\left(\sqrt{-2}\right)}^{2}} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) + \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {\left(\sqrt{-2}\right)}^{2}} + 1\right) + \frac{1}{2} \]

   5. associate-*r* N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\left(\sqrt{-2}\right)}^{2}\right)} + 1\right) + \frac{1}{2} \]

   6. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{\left(\sqrt{-2} \cdot \sqrt{-2}\right)}\right) + 1\right) + \frac{1}{2} \]

   7. rem-square-sqrt N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{-2}\right) + 1\right) + \frac{1}{2} \]

   8. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left({u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) + \frac{1}{2} \]

   9. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) + \frac{1}{2} \]

   10. associate-*l* N/A

       \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + 1\right) + \frac{1}{2} \]

   11. lower-fma.f64 N/A

       \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right), 1\right)} + \frac{1}{2} \]

8. Applied rewrites 99.0%

   \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\left(\pi \cdot \pi\right) \cdot -2\right), 1\right)} + 0.5 \]
9. Step-by-step derivation

10. Applied rewrites 99.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(u2, -2 \cdot \left(u2 \cdot \left(\pi \cdot \pi\right)\right), 1\right) \cdot \left(0.16666666666666666 \cdot \sqrt{-\log u1}\right), \sqrt{2}, 0.5\right)} \]

11. Final simplification 99.0%

    \[\leadsto \mathsf{fma}\left(\left(0.16666666666666666 \cdot \sqrt{-\log u1}\right) \cdot \mathsf{fma}\left(u2, -2 \cdot \left(u2 \cdot \left(\pi \cdot \pi\right)\right), 1\right), \sqrt{2}, 0.5\right) \]
12. Add Preprocessing

Alternative 7: 98.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(u2, -2 \cdot \left(u2 \cdot \left(\pi \cdot \pi\right)\right), 1\right) \cdot \sqrt{\log u1 \cdot \left(-2\right)}, 0.16666666666666666, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (fma u2 (* -2.0 (* u2 (* PI PI))) 1.0) (sqrt (* (log u1) (- 2.0))))
  0.16666666666666666
  0.5))

C

double code(double u1, double u2) {
	return fma((fma(u2, (-2.0 * (u2 * (((double) M_PI) * ((double) M_PI)))), 1.0) * sqrt((log(u1) * -2.0))), 0.16666666666666666, 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(fma(u2, Float64(-2.0 * Float64(u2 * Float64(pi * pi))), 1.0) * sqrt(Float64(log(u1) * Float64(-2.0)))), 0.16666666666666666, 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[(u2 * N[(-2.0 * N[(u2 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[Log[u1], $MachinePrecision] * (-2.0)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u1 around inf

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{\log \left(\frac{1}{u1}\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. log-rec N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. lower-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. lower-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. lower-sqrt.f64 99.5

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

5. Applied rewrites 99.5%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{-\log u1} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

6. Taylor expanded in u2 around 0

   \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + \frac{1}{2} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} + \frac{1}{2} \]

   2. rem-square-sqrt N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{\left(\sqrt{-2} \cdot \sqrt{-2}\right)} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) + \frac{1}{2} \]

   3. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{{\left(\sqrt{-2}\right)}^{2}} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) + \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {\left(\sqrt{-2}\right)}^{2}} + 1\right) + \frac{1}{2} \]

   5. associate-*r* N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\left(\sqrt{-2}\right)}^{2}\right)} + 1\right) + \frac{1}{2} \]

   6. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{\left(\sqrt{-2} \cdot \sqrt{-2}\right)}\right) + 1\right) + \frac{1}{2} \]

   7. rem-square-sqrt N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{-2}\right) + 1\right) + \frac{1}{2} \]

   8. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left({u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) + \frac{1}{2} \]

   9. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) + \frac{1}{2} \]

   10. associate-*l* N/A

       \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + 1\right) + \frac{1}{2} \]

   11. lower-fma.f64 N/A

       \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right), 1\right)} + \frac{1}{2} \]

8. Applied rewrites 99.0%

   \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\left(\pi \cdot \pi\right) \cdot -2\right), 1\right)} + 0.5 \]

10. Applied rewrites 98.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-\log u1 \cdot 2} \cdot \mathsf{fma}\left(u2, -2 \cdot \left(u2 \cdot \left(\pi \cdot \pi\right)\right), 1\right), 0.16666666666666666, 0.5\right)} \]

11. Final simplification 98.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(u2, -2 \cdot \left(u2 \cdot \left(\pi \cdot \pi\right)\right), 1\right) \cdot \sqrt{\log u1 \cdot \left(-2\right)}, 0.16666666666666666, 0.5\right) \]
12. Add Preprocessing

Alternative 8: 98.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, \pi \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 0.16666666666666666\right), \sqrt{\log u1 \cdot -2}, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (fma -0.3333333333333333 (* PI (* PI (* u2 u2))) 0.16666666666666666)
  (sqrt (* (log u1) -2.0))
  0.5))

C

double code(double u1, double u2) {
	return fma(fma(-0.3333333333333333, (((double) M_PI) * (((double) M_PI) * (u2 * u2))), 0.16666666666666666), sqrt((log(u1) * -2.0)), 0.5);
}

Julia

function code(u1, u2)
	return fma(fma(-0.3333333333333333, Float64(pi * Float64(pi * Float64(u2 * u2))), 0.16666666666666666), sqrt(Float64(log(u1) * -2.0)), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(-0.3333333333333333 * N[(Pi * N[(Pi * N[(u2 * u2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\left(-2 \cdot \color{blue}{\log u1}\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\color{blue}{\left(-2 \cdot \log u1\right)}}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. sqr-pow N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left({\left(-2 \cdot \log u1\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(-2 \cdot \log u1\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. pow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{\left({\left(-2 \cdot \log u1\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. lower-pow.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{\left({\left(-2 \cdot \log u1\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. metadata-eval N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(-2 \cdot \log u1\right)}^{\color{blue}{\frac{1}{4}}}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   7. metadata-eval N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(-2 \cdot \log u1\right)}^{\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{2}\right)}}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   8. lower-pow.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\color{blue}{\left({\left(-2 \cdot \log u1\right)}^{\left(\frac{1}{2} \cdot \frac{1}{2}\right)}\right)}}^{2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   9. metadata-eval 99.2

      \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(-2 \cdot \log u1\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Applied rewrites 99.2%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{\left({\left(-2 \cdot \log u1\right)}^{0.25}\right)}^{2}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
5. Step-by-step derivation

6. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\log u1 \cdot -2}, 0.5\right)} \]

7. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} + \frac{-1}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{6}}, \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   2. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{1}{6}\right)}, \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, {u2}^{2} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\left({u2}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\left({u2}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{\left({u2}^{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{PI}\left(\right), \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   9. lower-PI.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, \left(\left(u2 \cdot u2\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \mathsf{PI}\left(\right), \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   10. lower-PI.f64 98.9

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, \left(\left(u2 \cdot u2\right) \cdot \pi\right) \cdot \color{blue}{\pi}, 0.16666666666666666\right), \sqrt{\log u1 \cdot -2}, 0.5\right) \]

9. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.3333333333333333, \left(\left(u2 \cdot u2\right) \cdot \pi\right) \cdot \pi, 0.16666666666666666\right)}, \sqrt{\log u1 \cdot -2}, 0.5\right) \]

10. Final simplification 98.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, \pi \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 0.16666666666666666\right), \sqrt{\log u1 \cdot -2}, 0.5\right) \]
11. Add Preprocessing

Alternative 9: 98.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{2}, \sqrt{-\log u1}, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma (* 0.16666666666666666 (sqrt 2.0)) (sqrt (- (log u1))) 0.5))

C

double code(double u1, double u2) {
	return fma((0.16666666666666666 * sqrt(2.0)), sqrt(-log(u1)), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(0.16666666666666666 * sqrt(2.0)), sqrt(Float64(-log(u1))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(0.16666666666666666 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u1 around inf

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{\log \left(\frac{1}{u1}\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. log-rec N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. lower-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. lower-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. lower-sqrt.f64 99.5

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

5. Applied rewrites 99.5%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{-\log u1} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{1}{6}} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. lift-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. lift-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} \cdot \sqrt{2}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   9. pow1/2 N/A

      \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{{2}^{\frac{1}{2}}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot {2}^{\color{blue}{\left(\frac{1}{4} + \frac{1}{4}\right)}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   11. pow-prod-up N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\left({2}^{\frac{1}{4}} \cdot {2}^{\frac{1}{4}}\right)}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   12. pow-prod-down N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{{\left(2 \cdot 2\right)}^{\frac{1}{4}}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   13. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot {\color{blue}{4}}^{\frac{1}{4}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   14. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot {\color{blue}{\left(-2 \cdot -2\right)}}^{\frac{1}{4}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   15. pow-prod-down N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\left({-2}^{\frac{1}{4}} \cdot {-2}^{\frac{1}{4}}\right)}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   16. pow-prod-up N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{{-2}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   17. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot {-2}^{\color{blue}{\frac{1}{2}}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   18. pow1/2 N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\sqrt{-2}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   19. lift-sqrt.f64 N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\sqrt{-2}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   20. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left(\sqrt{-2} \cdot \frac{1}{6}\right)} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   21. lift-/.f64 N/A

       \[\leadsto \left(\left(\sqrt{-2} \cdot \color{blue}{\frac{1}{6}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   22. metadata-eval N/A

       \[\leadsto \left(\left(\sqrt{-2} \cdot \color{blue}{\frac{1}{6}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   23. lift-*.f64 N/A

       \[\leadsto \left(\color{blue}{\left(\sqrt{-2} \cdot \frac{1}{6}\right)} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   24. lower-*.f64 0.0

       \[\leadsto \color{blue}{\left(\left(\sqrt{-2} \cdot 0.16666666666666666\right) \cdot \sqrt{-\log u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

7. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\left(0.16666666666666666 \cdot \sqrt{2}\right) \cdot \sqrt{-\log u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
8. Step-by-step derivation

9. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \left(0.16666666666666666 \cdot \sqrt{2}\right), \sqrt{-\log u1}, 0.5\right)} \]

10. Taylor expanded in u2 around 0

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot \sqrt{2}}, \sqrt{\mathsf{neg}\left(\log u1\right)}, \frac{1}{2}\right) \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot \sqrt{2}}, \sqrt{\mathsf{neg}\left(\log u1\right)}, \frac{1}{2}\right) \]

    2. lower-sqrt.f64 98.6

       \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot \color{blue}{\sqrt{2}}, \sqrt{-\log u1}, 0.5\right) \]

12. Applied rewrites 98.6%

    \[\leadsto \mathsf{fma}\left(\color{blue}{0.16666666666666666 \cdot \sqrt{2}}, \sqrt{-\log u1}, 0.5\right) \]
13. Add Preprocessing

Alternative 10: 98.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma (sqrt (* (log u1) -2.0)) 0.16666666666666666 0.5))

C

double code(double u1, double u2) {
	return fma(sqrt((log(u1) * -2.0)), 0.16666666666666666, 0.5);
}

Julia

function code(u1, u2)
	return fma(sqrt(Float64(log(u1) * -2.0)), 0.16666666666666666, 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{6} \cdot \left(\sqrt{\log u1} \cdot \sqrt{-2}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(\sqrt{\log u1} \cdot \sqrt{-2}\right) + \frac{1}{2}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{\log u1} \cdot \sqrt{-2}\right) \cdot \frac{1}{6}} + \frac{1}{2} \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\sqrt{\log u1} \cdot \left(\sqrt{-2} \cdot \frac{1}{6}\right)} + \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\log u1} \cdot \color{blue}{\left(\frac{1}{6} \cdot \sqrt{-2}\right)} + \frac{1}{2} \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1}, \frac{1}{6} \cdot \sqrt{-2}, \frac{1}{2}\right)} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\log u1}}, \frac{1}{6} \cdot \sqrt{-2}, \frac{1}{2}\right) \]

   7. lower-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log u1}}, \frac{1}{6} \cdot \sqrt{-2}, \frac{1}{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1}, \color{blue}{\sqrt{-2} \cdot \frac{1}{6}}, \frac{1}{2}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1}, \color{blue}{\sqrt{-2} \cdot \frac{1}{6}}, \frac{1}{2}\right) \]

   10. lower-sqrt.f64 0.0

       \[\leadsto \mathsf{fma}\left(\sqrt{\log u1}, \color{blue}{\sqrt{-2}} \cdot 0.16666666666666666, 0.5\right) \]

5. Applied rewrites 0.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1}, \sqrt{-2} \cdot 0.16666666666666666, 0.5\right)} \]
6. Step-by-step derivation

   1. lift-log.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\log u1}} \cdot \left(\sqrt{-2} \cdot \frac{1}{6}\right) + \frac{1}{2} \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\log u1}} \cdot \left(\sqrt{-2} \cdot \frac{1}{6}\right) + \frac{1}{2} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{\log u1} \cdot \left(\color{blue}{\sqrt{-2}} \cdot \frac{1}{6}\right) + \frac{1}{2} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\log u1} \cdot \color{blue}{\left(\sqrt{-2} \cdot \frac{1}{6}\right)} + \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{-2} \cdot \frac{1}{6}\right) \cdot \sqrt{\log u1}} + \frac{1}{2} \]

   6. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{-2} \cdot \frac{1}{6}\right)} \cdot \sqrt{\log u1} + \frac{1}{2} \]

   7. metadata-eval N/A

      \[\leadsto \left(\sqrt{-2} \cdot \color{blue}{\frac{1}{6}}\right) \cdot \sqrt{\log u1} + \frac{1}{2} \]

   8. lift-/.f64 N/A

      \[\leadsto \left(\sqrt{-2} \cdot \color{blue}{\frac{1}{6}}\right) \cdot \sqrt{\log u1} + \frac{1}{2} \]

   9. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \sqrt{-2}\right)} \cdot \sqrt{\log u1} + \frac{1}{2} \]

   10. associate-*r* N/A

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(\sqrt{-2} \cdot \sqrt{\log u1}\right)} + \frac{1}{2} \]

   11. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{6} \cdot \left(\color{blue}{\sqrt{-2}} \cdot \sqrt{\log u1}\right) + \frac{1}{2} \]

   12. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{6} \cdot \left(\sqrt{-2} \cdot \color{blue}{\sqrt{\log u1}}\right) + \frac{1}{2} \]

   13. sqrt-prod N/A

       \[\leadsto \frac{1}{6} \cdot \color{blue}{\sqrt{-2 \cdot \log u1}} + \frac{1}{2} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{1}{6} \cdot \sqrt{\color{blue}{-2 \cdot \log u1}} + \frac{1}{2} \]

   15. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{6} \cdot \color{blue}{\sqrt{-2 \cdot \log u1}} + \frac{1}{2} \]

   16. *-commutative N/A

       \[\leadsto \color{blue}{\sqrt{-2 \cdot \log u1} \cdot \frac{1}{6}} + \frac{1}{2} \]

   17. lower-fma.f64 98.4

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, \frac{1}{6}, 0.5\right)} \]

7. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (and (<= 0.0 u1) (<= u1 1.0)) (and (<= 0.0 u2) (<= u2 1.0)))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))