Average Error: 0.4 → 0.3
Time: 34.3s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, 0.5\right)\]
\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(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, 0.5\right)
double f(double u1, double u2) {
        double r1706803 = 1.0;
        double r1706804 = 6.0;
        double r1706805 = r1706803 / r1706804;
        double r1706806 = -2.0;
        double r1706807 = u1;
        double r1706808 = log(r1706807);
        double r1706809 = r1706806 * r1706808;
        double r1706810 = 0.5;
        double r1706811 = pow(r1706809, r1706810);
        double r1706812 = r1706805 * r1706811;
        double r1706813 = 2.0;
        double r1706814 = atan2(1.0, 0.0);
        double r1706815 = r1706813 * r1706814;
        double r1706816 = u2;
        double r1706817 = r1706815 * r1706816;
        double r1706818 = cos(r1706817);
        double r1706819 = r1706812 * r1706818;
        double r1706820 = r1706819 + r1706810;
        return r1706820;
}
↓
double f(double u1, double u2) {
        double r1706821 = atan2(1.0, 0.0);
        double r1706822 = 2.0;
        double r1706823 = r1706821 * r1706822;
        double r1706824 = u2;
        double r1706825 = r1706823 * r1706824;
        double r1706826 = cos(r1706825);
        double r1706827 = -2.0;
        double r1706828 = u1;
        double r1706829 = log(r1706828);
        double r1706830 = r1706827 * r1706829;
        double r1706831 = 0.5;
        double r1706832 = pow(r1706830, r1706831);
        double r1706833 = 6.0;
        double r1706834 = r1706832 / r1706833;
        double r1706835 = fma(r1706826, r1706834, r1706831);
        return r1706835;
}

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

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

  3. Final simplification0.3

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

Reproduce

herbie shell --seed 2019158 +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))