Average Error: 0.4 → 0.4
Time: 30.2s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(1 \cdot \frac{1}{\frac{6}{{\left(-2 \cdot \log u1\right)}^{0.5}}}, \cos \left(\left(\left(2 \cdot \pi\right) \cdot \sqrt{u2}\right) \cdot \sqrt{u2}\right), 0.5\right)\]
\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(1 \cdot \frac{1}{\frac{6}{{\left(-2 \cdot \log u1\right)}^{0.5}}}, \cos \left(\left(\left(2 \cdot \pi\right) \cdot \sqrt{u2}\right) \cdot \sqrt{u2}\right), 0.5\right)
double f(double u1, double u2) {
        double r68611 = 1.0;
        double r68612 = 6.0;
        double r68613 = r68611 / r68612;
        double r68614 = -2.0;
        double r68615 = u1;
        double r68616 = log(r68615);
        double r68617 = r68614 * r68616;
        double r68618 = 0.5;
        double r68619 = pow(r68617, r68618);
        double r68620 = r68613 * r68619;
        double r68621 = 2.0;
        double r68622 = atan2(1.0, 0.0);
        double r68623 = r68621 * r68622;
        double r68624 = u2;
        double r68625 = r68623 * r68624;
        double r68626 = cos(r68625);
        double r68627 = r68620 * r68626;
        double r68628 = r68627 + r68618;
        return r68628;
}
↓
double f(double u1, double u2) {
        double r68629 = 1.0;
        double r68630 = 1.0;
        double r68631 = 6.0;
        double r68632 = -2.0;
        double r68633 = u1;
        double r68634 = log(r68633);
        double r68635 = r68632 * r68634;
        double r68636 = 0.5;
        double r68637 = pow(r68635, r68636);
        double r68638 = r68631 / r68637;
        double r68639 = r68630 / r68638;
        double r68640 = r68629 * r68639;
        double r68641 = 2.0;
        double r68642 = atan2(1.0, 0.0);
        double r68643 = r68641 * r68642;
        double r68644 = u2;
        double r68645 = sqrt(r68644);
        double r68646 = r68643 * r68645;
        double r68647 = r68646 * r68645;
        double r68648 = cos(r68647);
        double r68649 = fma(r68640, r68648, r68636);
        return r68649;
}

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 div-inv0.4

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

  5. Applied associate-*l*0.4

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

  6. Simplified0.3

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

  8. Applied add-sqr-sqrt0.3

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

  9. Applied associate-*r*0.3

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

  11. Applied clear-num0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019325 +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))