Average Error: 0.4 → 0.3
Time: 10.9s
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 r68715 = 1.0;
        double r68716 = 6.0;
        double r68717 = r68715 / r68716;
        double r68718 = -2.0;
        double r68719 = u1;
        double r68720 = log(r68719);
        double r68721 = r68718 * r68720;
        double r68722 = 0.5;
        double r68723 = pow(r68721, r68722);
        double r68724 = r68717 * r68723;
        double r68725 = 2.0;
        double r68726 = atan2(1.0, 0.0);
        double r68727 = r68725 * r68726;
        double r68728 = u2;
        double r68729 = r68727 * r68728;
        double r68730 = cos(r68729);
        double r68731 = r68724 * r68730;
        double r68732 = r68731 + r68722;
        return r68732;
}
↓
double f(double u1, double u2) {
        double r68733 = 1.0;
        double r68734 = -2.0;
        double r68735 = u1;
        double r68736 = log(r68735);
        double r68737 = r68734 * r68736;
        double r68738 = 0.5;
        double r68739 = pow(r68737, r68738);
        double r68740 = 6.0;
        double r68741 = r68739 / r68740;
        double r68742 = r68733 * r68741;
        double r68743 = 2.0;
        double r68744 = atan2(1.0, 0.0);
        double r68745 = r68743 * r68744;
        double r68746 = u2;
        double r68747 = r68745 * r68746;
        double r68748 = cos(r68747);
        double r68749 = r68742 * r68748;
        double r68750 = r68749 + r68738;
        return r68750;
}

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 2020057 +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))