Average Error: 0.4 → 0.4
Time: 11.9s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\frac{1}{6 \cdot {\left(\frac{1}{{\left(\log u1\right)}^{1} \cdot {-2}^{1}}\right)}^{0.5}}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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
↓
\mathsf{fma}\left(\frac{1}{6 \cdot {\left(\frac{1}{{\left(\log u1\right)}^{1} \cdot {-2}^{1}}\right)}^{0.5}}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)
double f(double u1, double u2) {
        double r66637 = 1.0;
        double r66638 = 6.0;
        double r66639 = r66637 / r66638;
        double r66640 = -2.0;
        double r66641 = u1;
        double r66642 = log(r66641);
        double r66643 = r66640 * r66642;
        double r66644 = 0.5;
        double r66645 = pow(r66643, r66644);
        double r66646 = r66639 * r66645;
        double r66647 = 2.0;
        double r66648 = atan2(1.0, 0.0);
        double r66649 = r66647 * r66648;
        double r66650 = u2;
        double r66651 = r66649 * r66650;
        double r66652 = cos(r66651);
        double r66653 = r66646 * r66652;
        double r66654 = r66653 + r66644;
        return r66654;
}
↓
double f(double u1, double u2) {
        double r66655 = 1.0;
        double r66656 = 6.0;
        double r66657 = u1;
        double r66658 = log(r66657);
        double r66659 = 1.0;
        double r66660 = pow(r66658, r66659);
        double r66661 = -2.0;
        double r66662 = pow(r66661, r66659);
        double r66663 = r66660 * r66662;
        double r66664 = r66655 / r66663;
        double r66665 = 0.5;
        double r66666 = pow(r66664, r66665);
        double r66667 = r66656 * r66666;
        double r66668 = r66655 / r66667;
        double r66669 = 2.0;
        double r66670 = atan2(1.0, 0.0);
        double r66671 = r66669 * r66670;
        double r66672 = u2;
        double r66673 = r66671 * r66672;
        double r66674 = cos(r66673);
        double r66675 = fma(r66668, r66674, r66665);
        return r66675;
}

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. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)}\]
  3. Using strategy rm

  4. Applied associate-*l/0.3

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

  6. Applied clear-num0.3

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

  7. Taylor expanded around 0 0.4

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

  8. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\frac{1}{6 \cdot {\left(\frac{1}{{\left(\log u1\right)}^{1} \cdot {-2}^{1}}\right)}^{0.5}}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(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))