Average Error: 0.4 → 0.3
Time: 27.6s
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 r80158 = 1.0;
        double r80159 = 6.0;
        double r80160 = r80158 / r80159;
        double r80161 = -2.0;
        double r80162 = u1;
        double r80163 = log(r80162);
        double r80164 = r80161 * r80163;
        double r80165 = 0.5;
        double r80166 = pow(r80164, r80165);
        double r80167 = r80160 * r80166;
        double r80168 = 2.0;
        double r80169 = atan2(1.0, 0.0);
        double r80170 = r80168 * r80169;
        double r80171 = u2;
        double r80172 = r80170 * r80171;
        double r80173 = cos(r80172);
        double r80174 = r80167 * r80173;
        double r80175 = r80174 + r80165;
        return r80175;
}
↓
double f(double u1, double u2) {
        double r80176 = 1.0;
        double r80177 = -2.0;
        double r80178 = u1;
        double r80179 = log(r80178);
        double r80180 = r80177 * r80179;
        double r80181 = 0.5;
        double r80182 = pow(r80180, r80181);
        double r80183 = 6.0;
        double r80184 = r80182 / r80183;
        double r80185 = r80176 * r80184;
        double r80186 = 2.0;
        double r80187 = atan2(1.0, 0.0);
        double r80188 = r80186 * r80187;
        double r80189 = u2;
        double r80190 = r80188 * r80189;
        double r80191 = cos(r80190);
        double r80192 = r80185 * r80191;
        double r80193 = r80192 + r80181;
        return r80193;
}

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 2019297 
(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))