Average Error: 0.4 → 0.4
Time: 10.6s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\frac{1}{6 \cdot {\left(\frac{1}{{-1}^{1} \cdot \left({-2}^{1} \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1}\right)}\right)}^{0.5}}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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}{{-1}^{1} \cdot \left({-2}^{1} \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1}\right)}\right)}^{0.5}}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)
double f(double u1, double u2) {
        double r63623 = 1.0;
        double r63624 = 6.0;
        double r63625 = r63623 / r63624;
        double r63626 = -2.0;
        double r63627 = u1;
        double r63628 = log(r63627);
        double r63629 = r63626 * r63628;
        double r63630 = 0.5;
        double r63631 = pow(r63629, r63630);
        double r63632 = r63625 * r63631;
        double r63633 = 2.0;
        double r63634 = atan2(1.0, 0.0);
        double r63635 = r63633 * r63634;
        double r63636 = u2;
        double r63637 = r63635 * r63636;
        double r63638 = cos(r63637);
        double r63639 = r63632 * r63638;
        double r63640 = r63639 + r63630;
        return r63640;
}
↓
double f(double u1, double u2) {
        double r63641 = 1.0;
        double r63642 = 6.0;
        double r63643 = -1.0;
        double r63644 = 1.0;
        double r63645 = pow(r63643, r63644);
        double r63646 = -2.0;
        double r63647 = pow(r63646, r63644);
        double r63648 = u1;
        double r63649 = r63641 / r63648;
        double r63650 = log(r63649);
        double r63651 = pow(r63650, r63644);
        double r63652 = r63647 * r63651;
        double r63653 = r63645 * r63652;
        double r63654 = r63641 / r63653;
        double r63655 = 0.5;
        double r63656 = pow(r63654, r63655);
        double r63657 = r63642 * r63656;
        double r63658 = r63641 / r63657;
        double r63659 = 2.0;
        double r63660 = atan2(1.0, 0.0);
        double r63661 = r63659 * r63660;
        double r63662 = u2;
        double r63663 = r63661 * r63662;
        double r63664 = cos(r63663);
        double r63665 = fma(r63658, r63664, r63655);
        return r63665;
}

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 inf 0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2020001 +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))