Average Error: 0.4 → 0.3
Time: 10.7s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)
double f(double u1, double u2) {
        double r62624 = 1.0;
        double r62625 = 6.0;
        double r62626 = r62624 / r62625;
        double r62627 = -2.0;
        double r62628 = u1;
        double r62629 = log(r62628);
        double r62630 = r62627 * r62629;
        double r62631 = 0.5;
        double r62632 = pow(r62630, r62631);
        double r62633 = r62626 * r62632;
        double r62634 = 2.0;
        double r62635 = atan2(1.0, 0.0);
        double r62636 = r62634 * r62635;
        double r62637 = u2;
        double r62638 = r62636 * r62637;
        double r62639 = cos(r62638);
        double r62640 = r62633 * r62639;
        double r62641 = r62640 + r62631;
        return r62641;
}
↓
double f(double u1, double u2) {
        double r62642 = 1.0;
        double r62643 = -2.0;
        double r62644 = u1;
        double r62645 = log(r62644);
        double r62646 = r62643 * r62645;
        double r62647 = 0.5;
        double r62648 = pow(r62646, r62647);
        double r62649 = r62642 * r62648;
        double r62650 = 6.0;
        double r62651 = r62649 / r62650;
        double r62652 = 2.0;
        double r62653 = atan2(1.0, 0.0);
        double r62654 = r62652 * r62653;
        double r62655 = u2;
        double r62656 = r62654 * r62655;
        double r62657 = cos(r62656);
        double r62658 = fma(r62651, r62657, r62647);
        return r62658;
}

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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019353 +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))