Average Error: 0.4 → 0.3
Time: 10.7s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)
double f(double u1, double u2) {
        double r64914 = 1.0;
        double r64915 = 6.0;
        double r64916 = r64914 / r64915;
        double r64917 = -2.0;
        double r64918 = u1;
        double r64919 = log(r64918);
        double r64920 = r64917 * r64919;
        double r64921 = 0.5;
        double r64922 = pow(r64920, r64921);
        double r64923 = r64916 * r64922;
        double r64924 = 2.0;
        double r64925 = atan2(1.0, 0.0);
        double r64926 = r64924 * r64925;
        double r64927 = u2;
        double r64928 = r64926 * r64927;
        double r64929 = cos(r64928);
        double r64930 = r64923 * r64929;
        double r64931 = r64930 + r64921;
        return r64931;
}
↓
double f(double u1, double u2) {
        double r64932 = 1.0;
        double r64933 = -2.0;
        double r64934 = u1;
        double r64935 = log(r64934);
        double r64936 = r64933 * r64935;
        double r64937 = 0.5;
        double r64938 = pow(r64936, r64937);
        double r64939 = 6.0;
        double r64940 = r64938 / r64939;
        double r64941 = r64932 * r64940;
        double r64942 = 2.0;
        double r64943 = atan2(1.0, 0.0);
        double r64944 = r64942 * r64943;
        double r64945 = u2;
        double r64946 = r64944 * r64945;
        double r64947 = cos(r64946);
        double r64948 = fma(r64941, r64947, r64937);
        return r64948;
}

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

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

  4. Applied div-inv0.4

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

  5. Applied associate-*l*0.4

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (<= 0.0 u1 1) (<= 0.0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))