Average Error: 0.4 → 0.3
Time: 14.3s
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 r72588 = 1.0;
        double r72589 = 6.0;
        double r72590 = r72588 / r72589;
        double r72591 = -2.0;
        double r72592 = u1;
        double r72593 = log(r72592);
        double r72594 = r72591 * r72593;
        double r72595 = 0.5;
        double r72596 = pow(r72594, r72595);
        double r72597 = r72590 * r72596;
        double r72598 = 2.0;
        double r72599 = atan2(1.0, 0.0);
        double r72600 = r72598 * r72599;
        double r72601 = u2;
        double r72602 = r72600 * r72601;
        double r72603 = cos(r72602);
        double r72604 = r72597 * r72603;
        double r72605 = r72604 + r72595;
        return r72605;
}
↓
double f(double u1, double u2) {
        double r72606 = 1.0;
        double r72607 = -2.0;
        double r72608 = u1;
        double r72609 = log(r72608);
        double r72610 = r72607 * r72609;
        double r72611 = 0.5;
        double r72612 = pow(r72610, r72611);
        double r72613 = r72606 * r72612;
        double r72614 = 6.0;
        double r72615 = r72613 / r72614;
        double r72616 = 2.0;
        double r72617 = atan2(1.0, 0.0);
        double r72618 = r72616 * r72617;
        double r72619 = u2;
        double r72620 = r72618 * r72619;
        double r72621 = cos(r72620);
        double r72622 = fma(r72615, r72621, r72611);
        return r72622;
}

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 associate-*l/0.3

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

  5. 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 2020046 +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))