Average Error: 0.4 → 0.3
Time: 12.0s
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 r71015 = 1.0;
        double r71016 = 6.0;
        double r71017 = r71015 / r71016;
        double r71018 = -2.0;
        double r71019 = u1;
        double r71020 = log(r71019);
        double r71021 = r71018 * r71020;
        double r71022 = 0.5;
        double r71023 = pow(r71021, r71022);
        double r71024 = r71017 * r71023;
        double r71025 = 2.0;
        double r71026 = atan2(1.0, 0.0);
        double r71027 = r71025 * r71026;
        double r71028 = u2;
        double r71029 = r71027 * r71028;
        double r71030 = cos(r71029);
        double r71031 = r71024 * r71030;
        double r71032 = r71031 + r71022;
        return r71032;
}
↓
double f(double u1, double u2) {
        double r71033 = 1.0;
        double r71034 = -2.0;
        double r71035 = u1;
        double r71036 = log(r71035);
        double r71037 = r71034 * r71036;
        double r71038 = 0.5;
        double r71039 = pow(r71037, r71038);
        double r71040 = 6.0;
        double r71041 = r71039 / r71040;
        double r71042 = r71033 * r71041;
        double r71043 = 2.0;
        double r71044 = atan2(1.0, 0.0);
        double r71045 = r71043 * r71044;
        double r71046 = u2;
        double r71047 = r71045 * r71046;
        double r71048 = cos(r71047);
        double r71049 = r71042 * r71048;
        double r71050 = r71049 + r71038;
        return r71050;
}

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 2020001 +o rules:numerics
(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))