normal distribution

Percentage Accurate: 99.4% → 99.4%

Time: 4.6s

Alternatives: 3

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: 99.4% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (cos (* u2 (+ (PI) (PI)))) (sqrt (* (log u1) -2.0)))
  0.16666666666666666
  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-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 2: 98.7% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (sqrt (* -2.0 (log u1)))
  (fma (* (PI) (PI)) (* -0.3333333333333333 (* u2 u2)) 0.16666666666666666)
  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-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 99.4

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.4

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.4

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

   13. lift-/.f64 N/A

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

   14. metadata-eval 99.4

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

   15. lift-pow.f64 N/A

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

   16. unpow1/2 N/A

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

   17. lower-sqrt.f64 99.4

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-PI.f64 N/A

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

   11. lower-PI.f64 98.7

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

7. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -0.3333333333333333, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 0.16666666666666666\right)} \cdot \sqrt{\log u1 \cdot -2} + 0.5 \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 98.7

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

9. Applied rewrites 98.7%

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

Alternative 3: 98.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-log.f64 0.0

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

5. Applied rewrites 0.0%

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

7. Applied rewrites 98.4%

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

Reproduce

herbie shell --seed 2025017 
(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))