Average Error: 0.4 → 0.3
Time: 11.3s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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(\frac{1 \cdot {\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 r67884 = 1.0;
        double r67885 = 6.0;
        double r67886 = r67884 / r67885;
        double r67887 = -2.0;
        double r67888 = u1;
        double r67889 = log(r67888);
        double r67890 = r67887 * r67889;
        double r67891 = 0.5;
        double r67892 = pow(r67890, r67891);
        double r67893 = r67886 * r67892;
        double r67894 = 2.0;
        double r67895 = atan2(1.0, 0.0);
        double r67896 = r67894 * r67895;
        double r67897 = u2;
        double r67898 = r67896 * r67897;
        double r67899 = cos(r67898);
        double r67900 = r67893 * r67899;
        double r67901 = r67900 + r67891;
        return r67901;
}
↓
double f(double u1, double u2) {
        double r67902 = 1.0;
        double r67903 = -2.0;
        double r67904 = u1;
        double r67905 = log(r67904);
        double r67906 = r67903 * r67905;
        double r67907 = 0.5;
        double r67908 = pow(r67906, r67907);
        double r67909 = r67902 * r67908;
        double r67910 = 6.0;
        double r67911 = r67909 / r67910;
        double r67912 = 2.0;
        double r67913 = atan2(1.0, 0.0);
        double r67914 = r67912 * r67913;
        double r67915 = u2;
        double r67916 = r67914 * r67915;
        double r67917 = cos(r67916);
        double r67918 = fma(r67911, r67917, r67907);
        return r67918;
}

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 associate-*l/0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2020046 +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))