Average Error: 0.4 → 0.3
Time: 11.3s
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 r65666 = 1.0;
        double r65667 = 6.0;
        double r65668 = r65666 / r65667;
        double r65669 = -2.0;
        double r65670 = u1;
        double r65671 = log(r65670);
        double r65672 = r65669 * r65671;
        double r65673 = 0.5;
        double r65674 = pow(r65672, r65673);
        double r65675 = r65668 * r65674;
        double r65676 = 2.0;
        double r65677 = atan2(1.0, 0.0);
        double r65678 = r65676 * r65677;
        double r65679 = u2;
        double r65680 = r65678 * r65679;
        double r65681 = cos(r65680);
        double r65682 = r65675 * r65681;
        double r65683 = r65682 + r65673;
        return r65683;
}
↓
double f(double u1, double u2) {
        double r65684 = 1.0;
        double r65685 = 6.0;
        double r65686 = r65684 / r65685;
        double r65687 = sqrt(r65686);
        double r65688 = -2.0;
        double r65689 = u1;
        double r65690 = log(r65689);
        double r65691 = r65688 * r65690;
        double r65692 = 0.5;
        double r65693 = pow(r65691, r65692);
        double r65694 = r65687 * r65693;
        double r65695 = r65687 * r65694;
        double r65696 = 2.0;
        double r65697 = atan2(1.0, 0.0);
        double r65698 = r65696 * r65697;
        double r65699 = u2;
        double r65700 = r65698 * r65699;
        double r65701 = cos(r65700);
        double r65702 = r65695 * r65701;
        double r65703 = r65702 + r65692;
        return r65703;
}

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 2019347 
(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))