Average Error: 0.4 → 0.3
Time: 27.5s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\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
double f(double u1, double u2) {
        double r67952 = 1.0;
        double r67953 = 6.0;
        double r67954 = r67952 / r67953;
        double r67955 = -2.0;
        double r67956 = u1;
        double r67957 = log(r67956);
        double r67958 = r67955 * r67957;
        double r67959 = 0.5;
        double r67960 = pow(r67958, r67959);
        double r67961 = r67954 * r67960;
        double r67962 = 2.0;
        double r67963 = atan2(1.0, 0.0);
        double r67964 = r67962 * r67963;
        double r67965 = u2;
        double r67966 = r67964 * r67965;
        double r67967 = cos(r67966);
        double r67968 = r67961 * r67967;
        double r67969 = r67968 + r67959;
        return r67969;
}
↓
double f(double u1, double u2) {
        double r67970 = 1.0;
        double r67971 = 6.0;
        double r67972 = r67970 / r67971;
        double r67973 = sqrt(r67972);
        double r67974 = -2.0;
        double r67975 = u1;
        double r67976 = log(r67975);
        double r67977 = r67974 * r67976;
        double r67978 = 0.5;
        double r67979 = pow(r67977, r67978);
        double r67980 = r67973 * r67979;
        double r67981 = r67973 * r67980;
        double r67982 = 2.0;
        double r67983 = atan2(1.0, 0.0);
        double r67984 = r67982 * r67983;
        double r67985 = u2;
        double r67986 = r67984 * r67985;
        double r67987 = cos(r67986);
        double r67988 = r67981 * r67987;
        double r67989 = r67988 + r67978;
        return r67989;
}

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 add-sqr-sqrt0.4

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

  4. Applied associate-*l*0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019325 
(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))