normal distribution

Percentage Accurate: 99.4% → 99.7%

Time: 7.2s

Alternatives: 2

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{-0.05555555555555555 \cdot \log u1}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right), 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma (sqrt (* -0.05555555555555555 (log u1))) (cos (* (* (PI) 2.0) u2)) 0.5))

Derivation

1. Initial program 99.4%

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

   1. lift-pow.f64 N/A

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

   2. unpow1/2 N/A

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

   3. lower-sqrt.f64 99.4

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.4

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

4. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lift-/.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. pow1/2 N/A

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

   9. metadata-eval N/A

      \[\leadsto \left(\left|\frac{1}{6}\right| \cdot {\left(\log u1 \cdot -2\right)}^{\color{blue}{\left(\frac{1}{8} \cdot 4\right)}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   10. pow-pow N/A

       \[\leadsto \left(\left|\frac{1}{6}\right| \cdot \color{blue}{{\left({\left(\log u1 \cdot -2\right)}^{\frac{1}{8}}\right)}^{4}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   11. lift-pow.f64 N/A

       \[\leadsto \left(\left|\frac{1}{6}\right| \cdot {\color{blue}{\left({\left(\log u1 \cdot -2\right)}^{\frac{1}{8}}\right)}}^{4}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   12. sqr-pow N/A

       \[\leadsto \left(\left|\frac{1}{6}\right| \cdot \color{blue}{\left({\left({\left(\log u1 \cdot -2\right)}^{\frac{1}{8}}\right)}^{\left(\frac{4}{2}\right)} \cdot {\left({\left(\log u1 \cdot -2\right)}^{\frac{1}{8}}\right)}^{\left(\frac{4}{2}\right)}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   13. sqr-abs-rev N/A

       \[\leadsto \left(\left|\frac{1}{6}\right| \cdot \color{blue}{\left(\left|{\left({\left(\log u1 \cdot -2\right)}^{\frac{1}{8}}\right)}^{\left(\frac{4}{2}\right)}\right| \cdot \left|{\left({\left(\log u1 \cdot -2\right)}^{\frac{1}{8}}\right)}^{\left(\frac{4}{2}\right)}\right|\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   14. mul-fabs N/A

       \[\leadsto \left(\left|\frac{1}{6}\right| \cdot \color{blue}{\left|{\left({\left(\log u1 \cdot -2\right)}^{\frac{1}{8}}\right)}^{\left(\frac{4}{2}\right)} \cdot {\left({\left(\log u1 \cdot -2\right)}^{\frac{1}{8}}\right)}^{\left(\frac{4}{2}\right)}\right|}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   15. sqr-pow N/A

       \[\leadsto \left(\left|\frac{1}{6}\right| \cdot \left|\color{blue}{{\left({\left(\log u1 \cdot -2\right)}^{\frac{1}{8}}\right)}^{4}}\right|\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   16. lift-pow.f64 N/A

       \[\leadsto \left(\left|\frac{1}{6}\right| \cdot \left|{\color{blue}{\left({\left(\log u1 \cdot -2\right)}^{\frac{1}{8}}\right)}}^{4}\right|\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   17. pow-pow N/A

       \[\leadsto \left(\left|\frac{1}{6}\right| \cdot \left|\color{blue}{{\left(\log u1 \cdot -2\right)}^{\left(\frac{1}{8} \cdot 4\right)}}\right|\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   18. metadata-eval N/A

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

   19. pow1/2 N/A

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

   20. lift-sqrt.f64 N/A

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

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\sqrt{0.027777777777777776 \cdot \left(\log u1 \cdot -2\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2}} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} + \frac{1}{2} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. sqrt-prod N/A

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{36}} \cdot \sqrt{\log u1 \cdot -2}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. metadata-eval N/A

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

   7. lift-sqrt.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-0.05555555555555555 \cdot \log u1}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right), 0.5\right)} \]
9. Add Preprocessing

Alternative 2: 98.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{-0.05555555555555555 \cdot \log u1} + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+ (sqrt (* -0.05555555555555555 (log u1))) 0.5))

C

double code(double u1, double u2) {
	return sqrt((-0.05555555555555555 * log(u1))) + 0.5;
}

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = sqrt(((-0.05555555555555555d0) * log(u1))) + 0.5d0
end function

Java

public static double code(double u1, double u2) {
	return Math.sqrt((-0.05555555555555555 * Math.log(u1))) + 0.5;
}

Python

def code(u1, u2):
	return math.sqrt((-0.05555555555555555 * math.log(u1))) + 0.5

Julia

function code(u1, u2)
	return Float64(sqrt(Float64(-0.05555555555555555 * log(u1))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = sqrt((-0.05555555555555555 * log(u1))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(-0.05555555555555555 * N[Log[u1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

4. Applied rewrites 98.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-neg-in N/A

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

   6. distribute-rgt-neg-out N/A

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

   7. lift-neg.f64 N/A

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

   8. *-commutative N/A

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

   9. sqrt-unprod N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. sqr-pow N/A

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

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{-\log u1} \cdot \left({2}^{\color{blue}{\frac{1}{4}}} \cdot {2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), \frac{1}{6}, \frac{1}{2}\right) \]

   14. metadata-eval N/A

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

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{-\log u1} \cdot \left({2}^{\left(\frac{1}{2} \cdot \frac{1}{2}\right)} \cdot {2}^{\color{blue}{\frac{1}{4}}}\right), \frac{1}{6}, \frac{1}{2}\right) \]

   16. metadata-eval N/A

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

   17. pow-prod-down N/A

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

   18. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{-\log u1} \cdot {\color{blue}{4}}^{\left(\frac{1}{2} \cdot \frac{1}{2}\right)}, \frac{1}{6}, \frac{1}{2}\right) \]

   19. metadata-eval N/A

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

   20. pow-prod-down N/A

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

   21. pow-sqr N/A

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

   22. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{-\log u1} \cdot {-2}^{\left(2 \cdot \color{blue}{\frac{1}{4}}\right)}, \frac{1}{6}, \frac{1}{2}\right) \]

   23. metadata-eval N/A

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

   24. lower-*.f64 N/A

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

6. Applied rewrites 98.5%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-unprod N/A

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

   5. lift-neg.f64 N/A

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

   6. distribute-lft-neg-out N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lift-*.f64 N/A

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

   10. lift-sqrt.f64 98.4

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

   11. lower-fma.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-+.f64 98.4

       \[\leadsto \color{blue}{\sqrt{\log u1 \cdot -2} \cdot 0.16666666666666666 + 0.5} \]

8. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\sqrt{-0.05555555555555555 \cdot \log u1} + 0.5} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024340 
(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))