Average Error: 0.4 → 0.3
Time: 30.7s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\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
double f(double u1, double u2) {
        double r86718 = 1.0;
        double r86719 = 6.0;
        double r86720 = r86718 / r86719;
        double r86721 = -2.0;
        double r86722 = u1;
        double r86723 = log(r86722);
        double r86724 = r86721 * r86723;
        double r86725 = 0.5;
        double r86726 = pow(r86724, r86725);
        double r86727 = r86720 * r86726;
        double r86728 = 2.0;
        double r86729 = atan2(1.0, 0.0);
        double r86730 = r86728 * r86729;
        double r86731 = u2;
        double r86732 = r86730 * r86731;
        double r86733 = cos(r86732);
        double r86734 = r86727 * r86733;
        double r86735 = r86734 + r86725;
        return r86735;
}
↓
double f(double u1, double u2) {
        double r86736 = 1.0;
        double r86737 = 6.0;
        double r86738 = r86736 / r86737;
        double r86739 = sqrt(r86738);
        double r86740 = -2.0;
        double r86741 = u1;
        double r86742 = log(r86741);
        double r86743 = r86740 * r86742;
        double r86744 = 0.5;
        double r86745 = pow(r86743, r86744);
        double r86746 = r86739 * r86745;
        double r86747 = r86739 * r86746;
        double r86748 = 2.0;
        double r86749 = atan2(1.0, 0.0);
        double r86750 = r86748 * r86749;
        double r86751 = u2;
        double r86752 = r86750 * r86751;
        double r86753 = cos(r86752);
        double r86754 = r86747 * r86753;
        double r86755 = r86754 + r86744;
        return r86755;
}

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 add-sqr-sqrt0.4

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

  4. Applied associate-*l*0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019209 +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))