Average Error: 0.4 → 0.4
Time: 2.4m
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot {\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{0.5}\right), \frac{1}{6}, 0.5\right)\]
\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(\left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot {\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{0.5}\right), \frac{1}{6}, 0.5\right)
double f(double u1, double u2) {
        double r12321982 = 1.0;
        double r12321983 = 6.0;
        double r12321984 = r12321982 / r12321983;
        double r12321985 = -2.0;
        double r12321986 = u1;
        double r12321987 = log(r12321986);
        double r12321988 = r12321985 * r12321987;
        double r12321989 = 0.5;
        double r12321990 = pow(r12321988, r12321989);
        double r12321991 = r12321984 * r12321990;
        double r12321992 = 2.0;
        double r12321993 = atan2(1.0, 0.0);
        double r12321994 = r12321992 * r12321993;
        double r12321995 = u2;
        double r12321996 = r12321994 * r12321995;
        double r12321997 = cos(r12321996);
        double r12321998 = r12321991 * r12321997;
        double r12321999 = r12321998 + r12321989;
        return r12321999;
}
↓
double f(double u1, double u2) {
        double r12322000 = 2.0;
        double r12322001 = u2;
        double r12322002 = atan2(1.0, 0.0);
        double r12322003 = r12322001 * r12322002;
        double r12322004 = r12322000 * r12322003;
        double r12322005 = cos(r12322004);
        double r12322006 = u1;
        double r12322007 = log(r12322006);
        double r12322008 = 1.0;
        double r12322009 = pow(r12322007, r12322008);
        double r12322010 = -2.0;
        double r12322011 = pow(r12322010, r12322008);
        double r12322012 = r12322009 * r12322011;
        double r12322013 = 0.5;
        double r12322014 = pow(r12322012, r12322013);
        double r12322015 = r12322005 * r12322014;
        double r12322016 = 0.16666666666666666;
        double r12322017 = fma(r12322015, r12322016, r12322013);
        return r12322017;
}

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. Taylor expanded around -inf 62.0

    \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({\left({\left(\log -1 - \log \left(\frac{-1}{u1}\right)\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{0.5} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) + 0.5}\]

  4. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{0.5} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right), \frac{1}{6}, 0.5\right)}\]

  5. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot {\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{0.5}\right), \frac{1}{6}, 0.5\right)\]

Reproduce

herbie shell --seed 2019121 +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))