Average Error: 0.4 → 0.3
Time: 12.7s
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 r71721 = 1.0;
        double r71722 = 6.0;
        double r71723 = r71721 / r71722;
        double r71724 = -2.0;
        double r71725 = u1;
        double r71726 = log(r71725);
        double r71727 = r71724 * r71726;
        double r71728 = 0.5;
        double r71729 = pow(r71727, r71728);
        double r71730 = r71723 * r71729;
        double r71731 = 2.0;
        double r71732 = atan2(1.0, 0.0);
        double r71733 = r71731 * r71732;
        double r71734 = u2;
        double r71735 = r71733 * r71734;
        double r71736 = cos(r71735);
        double r71737 = r71730 * r71736;
        double r71738 = r71737 + r71728;
        return r71738;
}
↓
double f(double u1, double u2) {
        double r71739 = 1.0;
        double r71740 = 6.0;
        double r71741 = r71739 / r71740;
        double r71742 = sqrt(r71741);
        double r71743 = -2.0;
        double r71744 = u1;
        double r71745 = log(r71744);
        double r71746 = r71743 * r71745;
        double r71747 = 0.5;
        double r71748 = pow(r71746, r71747);
        double r71749 = r71742 * r71748;
        double r71750 = r71742 * r71749;
        double r71751 = 2.0;
        double r71752 = atan2(1.0, 0.0);
        double r71753 = r71751 * r71752;
        double r71754 = u2;
        double r71755 = r71753 * r71754;
        double r71756 = cos(r71755);
        double r71757 = r71750 * r71756;
        double r71758 = r71757 + r71747;
        return r71758;
}

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 2019318 
(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))