Average Error: 0.4 → 0.3
Time: 32.8s
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 r1372949 = 1.0;
        double r1372950 = 6.0;
        double r1372951 = r1372949 / r1372950;
        double r1372952 = -2.0;
        double r1372953 = u1;
        double r1372954 = log(r1372953);
        double r1372955 = r1372952 * r1372954;
        double r1372956 = 0.5;
        double r1372957 = pow(r1372955, r1372956);
        double r1372958 = r1372951 * r1372957;
        double r1372959 = 2.0;
        double r1372960 = atan2(1.0, 0.0);
        double r1372961 = r1372959 * r1372960;
        double r1372962 = u2;
        double r1372963 = r1372961 * r1372962;
        double r1372964 = cos(r1372963);
        double r1372965 = r1372958 * r1372964;
        double r1372966 = r1372965 + r1372956;
        return r1372966;
}
↓
double f(double u1, double u2) {
        double r1372967 = atan2(1.0, 0.0);
        double r1372968 = 2.0;
        double r1372969 = r1372967 * r1372968;
        double r1372970 = u2;
        double r1372971 = r1372969 * r1372970;
        double r1372972 = cos(r1372971);
        double r1372973 = 0.16666666666666666;
        double r1372974 = sqrt(r1372973);
        double r1372975 = -2.0;
        double r1372976 = u1;
        double r1372977 = log(r1372976);
        double r1372978 = r1372975 * r1372977;
        double r1372979 = 0.5;
        double r1372980 = pow(r1372978, r1372979);
        double r1372981 = r1372974 * r1372980;
        double r1372982 = r1372981 * r1372974;
        double r1372983 = fma(r1372972, r1372982, r1372979);
        return r1372983;
}

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 2019144 +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))