Average Error: 0.4 → 0.3
Time: 32.6s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \sqrt{\frac{1}{6}}, 0.5\right)\]
\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(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \sqrt{\frac{1}{6}}, 0.5\right)
double f(double u1, double u2) {
        double r1120906 = 1.0;
        double r1120907 = 6.0;
        double r1120908 = r1120906 / r1120907;
        double r1120909 = -2.0;
        double r1120910 = u1;
        double r1120911 = log(r1120910);
        double r1120912 = r1120909 * r1120911;
        double r1120913 = 0.5;
        double r1120914 = pow(r1120912, r1120913);
        double r1120915 = r1120908 * r1120914;
        double r1120916 = 2.0;
        double r1120917 = atan2(1.0, 0.0);
        double r1120918 = r1120916 * r1120917;
        double r1120919 = u2;
        double r1120920 = r1120918 * r1120919;
        double r1120921 = cos(r1120920);
        double r1120922 = r1120915 * r1120921;
        double r1120923 = r1120922 + r1120913;
        return r1120923;
}
↓
double f(double u1, double u2) {
        double r1120924 = atan2(1.0, 0.0);
        double r1120925 = 2.0;
        double r1120926 = r1120924 * r1120925;
        double r1120927 = u2;
        double r1120928 = r1120926 * r1120927;
        double r1120929 = cos(r1120928);
        double r1120930 = 0.16666666666666666;
        double r1120931 = sqrt(r1120930);
        double r1120932 = -2.0;
        double r1120933 = u1;
        double r1120934 = log(r1120933);
        double r1120935 = r1120932 * r1120934;
        double r1120936 = 0.5;
        double r1120937 = pow(r1120935, r1120936);
        double r1120938 = r1120931 * r1120937;
        double r1120939 = r1120938 * r1120931;
        double r1120940 = fma(r1120929, r1120939, r1120936);
        return r1120940;
}

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

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019143 +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))