Average Error: 0.4 → 0.3
Time: 10.4s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)
double f(double u1, double u2) {
        double r66459 = 1.0;
        double r66460 = 6.0;
        double r66461 = r66459 / r66460;
        double r66462 = -2.0;
        double r66463 = u1;
        double r66464 = log(r66463);
        double r66465 = r66462 * r66464;
        double r66466 = 0.5;
        double r66467 = pow(r66465, r66466);
        double r66468 = r66461 * r66467;
        double r66469 = 2.0;
        double r66470 = atan2(1.0, 0.0);
        double r66471 = r66469 * r66470;
        double r66472 = u2;
        double r66473 = r66471 * r66472;
        double r66474 = cos(r66473);
        double r66475 = r66468 * r66474;
        double r66476 = r66475 + r66466;
        return r66476;
}
↓
double f(double u1, double u2) {
        double r66477 = 1.0;
        double r66478 = -2.0;
        double r66479 = u1;
        double r66480 = log(r66479);
        double r66481 = r66478 * r66480;
        double r66482 = 0.5;
        double r66483 = pow(r66481, r66482);
        double r66484 = 6.0;
        double r66485 = r66483 / r66484;
        double r66486 = r66477 * r66485;
        double r66487 = 2.0;
        double r66488 = atan2(1.0, 0.0);
        double r66489 = r66487 * r66488;
        double r66490 = u2;
        double r66491 = r66489 * r66490;
        double r66492 = cos(r66491);
        double r66493 = fma(r66486, r66492, r66482);
        return r66493;
}

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. Simplified0.4

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

  4. Applied div-inv0.4

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

  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. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2020003 +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))