Average Error: 0.4 → 0.3
Time: 11.9s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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 r69715 = 1.0;
        double r69716 = 6.0;
        double r69717 = r69715 / r69716;
        double r69718 = -2.0;
        double r69719 = u1;
        double r69720 = log(r69719);
        double r69721 = r69718 * r69720;
        double r69722 = 0.5;
        double r69723 = pow(r69721, r69722);
        double r69724 = r69717 * r69723;
        double r69725 = 2.0;
        double r69726 = atan2(1.0, 0.0);
        double r69727 = r69725 * r69726;
        double r69728 = u2;
        double r69729 = r69727 * r69728;
        double r69730 = cos(r69729);
        double r69731 = r69724 * r69730;
        double r69732 = r69731 + r69722;
        return r69732;
}
↓
double f(double u1, double u2) {
        double r69733 = 1.0;
        double r69734 = -2.0;
        double r69735 = u1;
        double r69736 = log(r69735);
        double r69737 = r69734 * r69736;
        double r69738 = 0.5;
        double r69739 = pow(r69737, r69738);
        double r69740 = r69733 * r69739;
        double r69741 = 6.0;
        double r69742 = r69740 / r69741;
        double r69743 = 2.0;
        double r69744 = atan2(1.0, 0.0);
        double r69745 = r69743 * r69744;
        double r69746 = u2;
        double r69747 = r69745 * r69746;
        double r69748 = cos(r69747);
        double r69749 = r69742 * r69748;
        double r69750 = r69749 + r69738;
        return r69750;
}

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 2020035 
(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))