Average Error: 0.4 → 0.3
Time: 10.8s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r67949 = 1.0;
        double r67950 = 6.0;
        double r67951 = r67949 / r67950;
        double r67952 = -2.0;
        double r67953 = u1;
        double r67954 = log(r67953);
        double r67955 = r67952 * r67954;
        double r67956 = 0.5;
        double r67957 = pow(r67955, r67956);
        double r67958 = r67951 * r67957;
        double r67959 = 2.0;
        double r67960 = atan2(1.0, 0.0);
        double r67961 = r67959 * r67960;
        double r67962 = u2;
        double r67963 = r67961 * r67962;
        double r67964 = cos(r67963);
        double r67965 = r67958 * r67964;
        double r67966 = r67965 + r67956;
        return r67966;
}
↓
double f(double u1, double u2) {
        double r67967 = 1.0;
        double r67968 = -2.0;
        double r67969 = u1;
        double r67970 = log(r67969);
        double r67971 = r67968 * r67970;
        double r67972 = 0.5;
        double r67973 = pow(r67971, r67972);
        double r67974 = 6.0;
        double r67975 = r67973 / r67974;
        double r67976 = r67967 * r67975;
        double r67977 = 2.0;
        double r67978 = atan2(1.0, 0.0);
        double r67979 = r67977 * r67978;
        double r67980 = u2;
        double r67981 = r67979 * r67980;
        double r67982 = cos(r67981);
        double r67983 = r67976 * r67982;
        double r67984 = r67983 + r67972;
        return r67984;
}

Error

Bits error versus u1

Bits error versus u2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied div-inv0.4

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

  4. Applied associate-*l*0.4

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019362 
(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))