Average Error: 0.4 → 0.3
Time: 24.9s
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 r80095 = 1.0;
        double r80096 = 6.0;
        double r80097 = r80095 / r80096;
        double r80098 = -2.0;
        double r80099 = u1;
        double r80100 = log(r80099);
        double r80101 = r80098 * r80100;
        double r80102 = 0.5;
        double r80103 = pow(r80101, r80102);
        double r80104 = r80097 * r80103;
        double r80105 = 2.0;
        double r80106 = atan2(1.0, 0.0);
        double r80107 = r80105 * r80106;
        double r80108 = u2;
        double r80109 = r80107 * r80108;
        double r80110 = cos(r80109);
        double r80111 = r80104 * r80110;
        double r80112 = r80111 + r80102;
        return r80112;
}
↓
double f(double u1, double u2) {
        double r80113 = 1.0;
        double r80114 = -2.0;
        double r80115 = u1;
        double r80116 = log(r80115);
        double r80117 = r80114 * r80116;
        double r80118 = 0.5;
        double r80119 = pow(r80117, r80118);
        double r80120 = 6.0;
        double r80121 = r80119 / r80120;
        double r80122 = r80113 * r80121;
        double r80123 = 2.0;
        double r80124 = atan2(1.0, 0.0);
        double r80125 = r80123 * r80124;
        double r80126 = u2;
        double r80127 = r80125 * r80126;
        double r80128 = cos(r80127);
        double r80129 = r80122 * r80128;
        double r80130 = r80129 + r80118;
        return r80130;
}

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 2019298 
(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))