Average Error: 0.4 → 0.3
Time: 1.9m
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 r75678 = 1.0;
        double r75679 = 6.0;
        double r75680 = r75678 / r75679;
        double r75681 = -2.0;
        double r75682 = u1;
        double r75683 = log(r75682);
        double r75684 = r75681 * r75683;
        double r75685 = 0.5;
        double r75686 = pow(r75684, r75685);
        double r75687 = r75680 * r75686;
        double r75688 = 2.0;
        double r75689 = atan2(1.0, 0.0);
        double r75690 = r75688 * r75689;
        double r75691 = u2;
        double r75692 = r75690 * r75691;
        double r75693 = cos(r75692);
        double r75694 = r75687 * r75693;
        double r75695 = r75694 + r75685;
        return r75695;
}
↓
double f(double u1, double u2) {
        double r75696 = 1.0;
        double r75697 = 6.0;
        double r75698 = r75696 / r75697;
        double r75699 = sqrt(r75698);
        double r75700 = -2.0;
        double r75701 = u1;
        double r75702 = log(r75701);
        double r75703 = r75700 * r75702;
        double r75704 = 0.5;
        double r75705 = pow(r75703, r75704);
        double r75706 = r75699 * r75705;
        double r75707 = r75699 * r75706;
        double r75708 = 2.0;
        double r75709 = atan2(1.0, 0.0);
        double r75710 = r75708 * r75709;
        double r75711 = u2;
        double r75712 = r75710 * r75711;
        double r75713 = cos(r75712);
        double r75714 = r75707 * r75713;
        double r75715 = r75714 + r75704;
        return r75715;
}

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 2019202 
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0.0 u1 1) (<= 0.0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))