Average Error: 0.4 → 0.3
Time: 13.8s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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 r61933 = 1.0;
        double r61934 = 6.0;
        double r61935 = r61933 / r61934;
        double r61936 = -2.0;
        double r61937 = u1;
        double r61938 = log(r61937);
        double r61939 = r61936 * r61938;
        double r61940 = 0.5;
        double r61941 = pow(r61939, r61940);
        double r61942 = r61935 * r61941;
        double r61943 = 2.0;
        double r61944 = atan2(1.0, 0.0);
        double r61945 = r61943 * r61944;
        double r61946 = u2;
        double r61947 = r61945 * r61946;
        double r61948 = cos(r61947);
        double r61949 = r61942 * r61948;
        double r61950 = r61949 + r61940;
        return r61950;
}
↓
double f(double u1, double u2) {
        double r61951 = 1.0;
        double r61952 = 6.0;
        double r61953 = r61951 / r61952;
        double r61954 = sqrt(r61953);
        double r61955 = -2.0;
        double r61956 = u1;
        double r61957 = log(r61956);
        double r61958 = r61955 * r61957;
        double r61959 = 0.5;
        double r61960 = pow(r61958, r61959);
        double r61961 = r61954 * r61960;
        double r61962 = r61954 * r61961;
        double r61963 = 2.0;
        double r61964 = atan2(1.0, 0.0);
        double r61965 = r61963 * r61964;
        double r61966 = u2;
        double r61967 = r61965 * r61966;
        double r61968 = cos(r61967);
        double r61969 = r61962 * r61968;
        double r61970 = r61969 + r61959;
        return r61970;
}

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 2020047 
(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))