Average Error: 0.4 → 0.3
Time: 28.3s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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(\frac{1 \cdot {\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 r81495 = 1.0;
        double r81496 = 6.0;
        double r81497 = r81495 / r81496;
        double r81498 = -2.0;
        double r81499 = u1;
        double r81500 = log(r81499);
        double r81501 = r81498 * r81500;
        double r81502 = 0.5;
        double r81503 = pow(r81501, r81502);
        double r81504 = r81497 * r81503;
        double r81505 = 2.0;
        double r81506 = atan2(1.0, 0.0);
        double r81507 = r81505 * r81506;
        double r81508 = u2;
        double r81509 = r81507 * r81508;
        double r81510 = cos(r81509);
        double r81511 = r81504 * r81510;
        double r81512 = r81511 + r81502;
        return r81512;
}
↓
double f(double u1, double u2) {
        double r81513 = 1.0;
        double r81514 = -2.0;
        double r81515 = u1;
        double r81516 = log(r81515);
        double r81517 = r81514 * r81516;
        double r81518 = 0.5;
        double r81519 = pow(r81517, r81518);
        double r81520 = r81513 * r81519;
        double r81521 = 6.0;
        double r81522 = r81520 / r81521;
        double r81523 = 2.0;
        double r81524 = atan2(1.0, 0.0);
        double r81525 = r81523 * r81524;
        double r81526 = u2;
        double r81527 = r81525 * r81526;
        double r81528 = cos(r81527);
        double r81529 = fma(r81522, r81528, r81518);
        return r81529;
}

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 *-un-lft-identity0.4

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

  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

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019208 +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))