normal distribution

Average Error: 0.4 → 0.2

Time: 32.0s

Precision: 64

\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]

Math

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

↓

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

C

double f(double u1, double u2) {
        double r82486 = 1.0;
        double r82487 = 6.0;
        double r82488 = r82486 / r82487;
        double r82489 = -2.0;
        double r82490 = u1;
        double r82491 = log(r82490);
        double r82492 = r82489 * r82491;
        double r82493 = 0.5;
        double r82494 = pow(r82492, r82493);
        double r82495 = r82488 * r82494;
        double r82496 = 2.0;
        double r82497 = atan2(1.0, 0.0);
        double r82498 = r82496 * r82497;
        double r82499 = u2;
        double r82500 = r82498 * r82499;
        double r82501 = cos(r82500);
        double r82502 = r82495 * r82501;
        double r82503 = r82502 + r82493;
        return r82503;
}

↓

double f(double u1, double u2) {
        double r82504 = -2.0;
        double r82505 = u1;
        double r82506 = log(r82505);
        double r82507 = r82504 * r82506;
        double r82508 = 2.0;
        double r82509 = 0.5;
        double r82510 = r82508 * r82509;
        double r82511 = pow(r82507, r82510);
        double r82512 = 6.0;
        double r82513 = r82512 * r82512;
        double r82514 = r82511 / r82513;
        double r82515 = 0.5;
        double r82516 = pow(r82514, r82515);
        double r82517 = 1.0;
        double r82518 = r82516 * r82517;
        double r82519 = 2.0;
        double r82520 = atan2(1.0, 0.0);
        double r82521 = r82519 * r82520;
        double r82522 = u2;
        double r82523 = r82521 * r82522;
        double r82524 = cos(r82523);
        double r82525 = fma(r82518, r82524, r82509);
        return r82525;
}

Error

Bits error versus u1

Bits error versus u2

Derivation

1. Initial program 0.4

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

2. Simplified 0.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)}\]
3. Using strategy rm

4. Applied div-inv 0.4

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

5. Applied associate-*l* 0.4

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

6. Simplified 0.3

   \[\leadsto \mathsf{fma}\left(1 \cdot \color{blue}{\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
7. Using strategy rm

8. Applied add-sqr-sqrt 0.6

   \[\leadsto \mathsf{fma}\left(1 \cdot \color{blue}{\left(\sqrt{\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}} \cdot \sqrt{\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}}\right)}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
9. Using strategy rm

10. Applied pow1 0.6

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

11. Applied sqrt-pow1 0.6

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

12. Applied pow1 0.6

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

13. Applied sqrt-pow1 0.6

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

14. Applied pow-prod-down 0.3

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

15. Simplified 0.2

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

16. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (<= 0.0 u1 1) (<= 0.0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))