Average Error: 0.4 → 0.3
Time: 26.3s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r75719 = 1.0;
        double r75720 = 6.0;
        double r75721 = r75719 / r75720;
        double r75722 = -2.0;
        double r75723 = u1;
        double r75724 = log(r75723);
        double r75725 = r75722 * r75724;
        double r75726 = 0.5;
        double r75727 = pow(r75725, r75726);
        double r75728 = r75721 * r75727;
        double r75729 = 2.0;
        double r75730 = atan2(1.0, 0.0);
        double r75731 = r75729 * r75730;
        double r75732 = u2;
        double r75733 = r75731 * r75732;
        double r75734 = cos(r75733);
        double r75735 = r75728 * r75734;
        double r75736 = r75735 + r75726;
        return r75736;
}
↓
double f(double u1, double u2) {
        double r75737 = 1.0;
        double r75738 = -2.0;
        double r75739 = u1;
        double r75740 = log(r75739);
        double r75741 = r75738 * r75740;
        double r75742 = 0.5;
        double r75743 = pow(r75741, r75742);
        double r75744 = r75737 * r75743;
        double r75745 = 6.0;
        double r75746 = r75744 / r75745;
        double r75747 = 2.0;
        double r75748 = atan2(1.0, 0.0);
        double r75749 = r75747 * r75748;
        double r75750 = u2;
        double r75751 = r75749 * r75750;
        double r75752 = cos(r75751);
        double r75753 = r75746 * r75752;
        double r75754 = r75753 + r75742;
        return r75754;
}

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 associate-*l/0.3

    \[\leadsto \color{blue}{\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  4. Final simplification0.3

    \[\leadsto \frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

Reproduce

herbie shell --seed 2019235 
(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))