Average Error: 0.4 → 0.3
Time: 35.3s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) + 0.5\]
\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
↓
\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) + 0.5
double f(double u1, double u2) {
        double r1573936 = 1.0;
        double r1573937 = 6.0;
        double r1573938 = r1573936 / r1573937;
        double r1573939 = -2.0;
        double r1573940 = u1;
        double r1573941 = log(r1573940);
        double r1573942 = r1573939 * r1573941;
        double r1573943 = 0.5;
        double r1573944 = pow(r1573942, r1573943);
        double r1573945 = r1573938 * r1573944;
        double r1573946 = 2.0;
        double r1573947 = atan2(1.0, 0.0);
        double r1573948 = r1573946 * r1573947;
        double r1573949 = u2;
        double r1573950 = r1573948 * r1573949;
        double r1573951 = cos(r1573950);
        double r1573952 = r1573945 * r1573951;
        double r1573953 = r1573952 + r1573943;
        return r1573953;
}
↓
double f(double u1, double u2) {
        double r1573954 = -2.0;
        double r1573955 = u1;
        double r1573956 = log(r1573955);
        double r1573957 = r1573954 * r1573956;
        double r1573958 = 0.5;
        double r1573959 = pow(r1573957, r1573958);
        double r1573960 = 6.0;
        double r1573961 = r1573959 / r1573960;
        double r1573962 = u2;
        double r1573963 = 2.0;
        double r1573964 = atan2(1.0, 0.0);
        double r1573965 = r1573963 * r1573964;
        double r1573966 = r1573962 * r1573965;
        double r1573967 = cos(r1573966);
        double r1573968 = r1573961 * r1573967;
        double r1573969 = r1573968 + r1573958;
        return r1573969;
}

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. Simplified0.4

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

  4. Applied div-inv0.4

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

  5. Applied associate-*l*0.4

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019143 
(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))