Average Error: 0.4 → 0.4
Time: 2.2m
Precision: 64
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[(\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right) + 0.5)_*\]
double f(double u1, double u2) {
        double r11137629 = 1.0;
        double r11137630 = 6.0;
        double r11137631 = r11137629 / r11137630;
        double r11137632 = -2.0;
        double r11137633 = u1;
        double r11137634 = log(r11137633);
        double r11137635 = r11137632 * r11137634;
        double r11137636 = 0.5;
        double r11137637 = pow(r11137635, r11137636);
        double r11137638 = r11137631 * r11137637;
        double r11137639 = 2.0;
        double r11137640 = atan2(1.0, 0.0);
        double r11137641 = r11137639 * r11137640;
        double r11137642 = u2;
        double r11137643 = r11137641 * r11137642;
        double r11137644 = cos(r11137643);
        double r11137645 = r11137638 * r11137644;
        double r11137646 = r11137645 + r11137636;
        return r11137646;
}
↓
double f(double u1, double u2) {
        double r11137647 = atan2(1.0, 0.0);
        double r11137648 = 2.0;
        double r11137649 = r11137647 * r11137648;
        double r11137650 = u2;
        double r11137651 = r11137649 * r11137650;
        double r11137652 = cos(r11137651);
        double r11137653 = -2.0;
        double r11137654 = u1;
        double r11137655 = log(r11137654);
        double r11137656 = r11137653 * r11137655;
        double r11137657 = 0.5;
        double r11137658 = pow(r11137656, r11137657);
        double r11137659 = 0.16666666666666666;
        double r11137660 = r11137658 * r11137659;
        double r11137661 = fma(r11137652, r11137660, r11137657);
        return r11137661;
}


\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(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right) + 0.5)_*

Error

Bits error versus u1

Bits error versus u2

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. Simplified0.3

    \[\leadsto \color{blue}{(\left(\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right) \cdot \left(\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) + 0.5)_*}\]
  3. Using strategy rm

  4. Applied clear-num0.3

    \[\leadsto (\left(\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{\frac{6}{{\left(-2 \cdot \log u1\right)}^{0.5}}}\right)} + 0.5)_*\]
  5. Using strategy rm

  6. Applied div-inv0.4

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

  7. Applied *-un-lft-identity0.4

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

  8. Applied times-frac0.4

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

  9. Simplified0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0 u1 1) (<= 0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))