Average Error: 0.4 → 0.4
Time: 2.4m
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right), \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right), 0.5\right)\]
\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({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right), 0.5\right)
double f(double u1, double u2) {
        double r12691824 = 1.0;
        double r12691825 = 6.0;
        double r12691826 = r12691824 / r12691825;
        double r12691827 = -2.0;
        double r12691828 = u1;
        double r12691829 = log(r12691828);
        double r12691830 = r12691827 * r12691829;
        double r12691831 = 0.5;
        double r12691832 = pow(r12691830, r12691831);
        double r12691833 = r12691826 * r12691832;
        double r12691834 = 2.0;
        double r12691835 = atan2(1.0, 0.0);
        double r12691836 = r12691834 * r12691835;
        double r12691837 = u2;
        double r12691838 = r12691836 * r12691837;
        double r12691839 = cos(r12691838);
        double r12691840 = r12691833 * r12691839;
        double r12691841 = r12691840 + r12691831;
        return r12691841;
}
↓
double f(double u1, double u2) {
        double r12691842 = atan2(1.0, 0.0);
        double r12691843 = 2.0;
        double r12691844 = r12691842 * r12691843;
        double r12691845 = u2;
        double r12691846 = r12691844 * r12691845;
        double r12691847 = cos(r12691846);
        double r12691848 = -2.0;
        double r12691849 = u1;
        double r12691850 = log(r12691849);
        double r12691851 = r12691848 * r12691850;
        double r12691852 = 0.5;
        double r12691853 = pow(r12691851, r12691852);
        double r12691854 = 0.16666666666666666;
        double r12691855 = r12691853 * r12691854;
        double r12691856 = fma(r12691847, r12691855, r12691852);
        return r12691856;
}

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. Using strategy rm

  4. Applied div-inv0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019128 +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))