Average Error: 0.4 → 0.3
Time: 18.5s
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 r70581 = 1.0;
        double r70582 = 6.0;
        double r70583 = r70581 / r70582;
        double r70584 = -2.0;
        double r70585 = u1;
        double r70586 = log(r70585);
        double r70587 = r70584 * r70586;
        double r70588 = 0.5;
        double r70589 = pow(r70587, r70588);
        double r70590 = r70583 * r70589;
        double r70591 = 2.0;
        double r70592 = atan2(1.0, 0.0);
        double r70593 = r70591 * r70592;
        double r70594 = u2;
        double r70595 = r70593 * r70594;
        double r70596 = cos(r70595);
        double r70597 = r70590 * r70596;
        double r70598 = r70597 + r70588;
        return r70598;
}
↓
double f(double u1, double u2) {
        double r70599 = 1.0;
        double r70600 = -2.0;
        double r70601 = u1;
        double r70602 = log(r70601);
        double r70603 = r70600 * r70602;
        double r70604 = 0.5;
        double r70605 = pow(r70603, r70604);
        double r70606 = r70599 * r70605;
        double r70607 = 6.0;
        double r70608 = r70606 / r70607;
        double r70609 = 2.0;
        double r70610 = atan2(1.0, 0.0);
        double r70611 = r70609 * r70610;
        double r70612 = u2;
        double r70613 = r70611 * r70612;
        double r70614 = cos(r70613);
        double r70615 = fma(r70608, r70614, r70604);
        return r70615;
}

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 2020047 +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))