Average Error: 0.4 → 0.3
Time: 15.8s
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 r75749 = 1.0;
        double r75750 = 6.0;
        double r75751 = r75749 / r75750;
        double r75752 = -2.0;
        double r75753 = u1;
        double r75754 = log(r75753);
        double r75755 = r75752 * r75754;
        double r75756 = 0.5;
        double r75757 = pow(r75755, r75756);
        double r75758 = r75751 * r75757;
        double r75759 = 2.0;
        double r75760 = atan2(1.0, 0.0);
        double r75761 = r75759 * r75760;
        double r75762 = u2;
        double r75763 = r75761 * r75762;
        double r75764 = cos(r75763);
        double r75765 = r75758 * r75764;
        double r75766 = r75765 + r75756;
        return r75766;
}
↓
double f(double u1, double u2) {
        double r75767 = 1.0;
        double r75768 = -2.0;
        double r75769 = u1;
        double r75770 = log(r75769);
        double r75771 = r75768 * r75770;
        double r75772 = 0.5;
        double r75773 = pow(r75771, r75772);
        double r75774 = 6.0;
        double r75775 = r75773 / r75774;
        double r75776 = r75767 * r75775;
        double r75777 = 2.0;
        double r75778 = atan2(1.0, 0.0);
        double r75779 = r75777 * r75778;
        double r75780 = u2;
        double r75781 = r75779 * r75780;
        double r75782 = cos(r75781);
        double r75783 = r75776 * r75782;
        double r75784 = r75783 + r75772;
        return r75784;
}

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 2020046 
(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))