Average Error: 0.4 → 0.4
Time: 31.2s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \log \left(e^{{\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}}\right) \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\]
\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
↓
0.5 + \log \left(e^{{\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}}\right) \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)
double f(double u1, double u2) {
        double r1610938 = 1.0;
        double r1610939 = 6.0;
        double r1610940 = r1610938 / r1610939;
        double r1610941 = -2.0;
        double r1610942 = u1;
        double r1610943 = log(r1610942);
        double r1610944 = r1610941 * r1610943;
        double r1610945 = 0.5;
        double r1610946 = pow(r1610944, r1610945);
        double r1610947 = r1610940 * r1610946;
        double r1610948 = 2.0;
        double r1610949 = atan2(1.0, 0.0);
        double r1610950 = r1610948 * r1610949;
        double r1610951 = u2;
        double r1610952 = r1610950 * r1610951;
        double r1610953 = cos(r1610952);
        double r1610954 = r1610947 * r1610953;
        double r1610955 = r1610954 + r1610945;
        return r1610955;
}
↓
double f(double u1, double u2) {
        double r1610956 = 0.5;
        double r1610957 = -2.0;
        double r1610958 = u1;
        double r1610959 = log(r1610958);
        double r1610960 = r1610957 * r1610959;
        double r1610961 = pow(r1610960, r1610956);
        double r1610962 = 0.16666666666666666;
        double r1610963 = r1610961 * r1610962;
        double r1610964 = exp(r1610963);
        double r1610965 = log(r1610964);
        double r1610966 = u2;
        double r1610967 = 2.0;
        double r1610968 = atan2(1.0, 0.0);
        double r1610969 = r1610967 * r1610968;
        double r1610970 = r1610966 * r1610969;
        double r1610971 = cos(r1610970);
        double r1610972 = r1610965 * r1610971;
        double r1610973 = r1610956 + r1610972;
        return r1610973;
}

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. Using strategy rm

  6. Applied add-log-exp0.4

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

  7. Simplified0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019163 +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))