Average Error: 0.4 → 0.3
Time: 3.3m
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right), \left(\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right), 0.5\right)\]
\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(\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right), \left(\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right), 0.5\right)
double f(double u1, double u2) {
        double r17723572 = 1.0;
        double r17723573 = 6.0;
        double r17723574 = r17723572 / r17723573;
        double r17723575 = -2.0;
        double r17723576 = u1;
        double r17723577 = log(r17723576);
        double r17723578 = r17723575 * r17723577;
        double r17723579 = 0.5;
        double r17723580 = pow(r17723578, r17723579);
        double r17723581 = r17723574 * r17723580;
        double r17723582 = 2.0;
        double r17723583 = atan2(1.0, 0.0);
        double r17723584 = r17723582 * r17723583;
        double r17723585 = u2;
        double r17723586 = r17723584 * r17723585;
        double r17723587 = cos(r17723586);
        double r17723588 = r17723581 * r17723587;
        double r17723589 = r17723588 + r17723579;
        return r17723589;
}
↓
double f(double u1, double u2) {
        double r17723590 = atan2(1.0, 0.0);
        double r17723591 = 2.0;
        double r17723592 = r17723590 * r17723591;
        double r17723593 = u2;
        double r17723594 = r17723592 * r17723593;
        double r17723595 = cos(r17723594);
        double r17723596 = -2.0;
        double r17723597 = u1;
        double r17723598 = log(r17723597);
        double r17723599 = r17723596 * r17723598;
        double r17723600 = 0.5;
        double r17723601 = pow(r17723599, r17723600);
        double r17723602 = 6.0;
        double r17723603 = r17723601 / r17723602;
        double r17723604 = fma(r17723595, r17723603, r17723600);
        return r17723604;
}

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. Simplified0.3

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

  3. Final simplification0.3

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0 u1 1) (<= 0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))