Average Error: 0.4 → 0.3
Time: 32.2s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}, 0.5\right)\]
\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(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}, 0.5\right)
double f(double u1, double u2) {
        double r700591 = 1.0;
        double r700592 = 6.0;
        double r700593 = r700591 / r700592;
        double r700594 = -2.0;
        double r700595 = u1;
        double r700596 = log(r700595);
        double r700597 = r700594 * r700596;
        double r700598 = 0.5;
        double r700599 = pow(r700597, r700598);
        double r700600 = r700593 * r700599;
        double r700601 = 2.0;
        double r700602 = atan2(1.0, 0.0);
        double r700603 = r700601 * r700602;
        double r700604 = u2;
        double r700605 = r700603 * r700604;
        double r700606 = cos(r700605);
        double r700607 = r700600 * r700606;
        double r700608 = r700607 + r700598;
        return r700608;
}
↓
double f(double u1, double u2) {
        double r700609 = atan2(1.0, 0.0);
        double r700610 = 2.0;
        double r700611 = r700609 * r700610;
        double r700612 = u2;
        double r700613 = r700611 * r700612;
        double r700614 = cos(r700613);
        double r700615 = -2.0;
        double r700616 = u1;
        double r700617 = log(r700616);
        double r700618 = r700615 * r700617;
        double r700619 = 0.5;
        double r700620 = pow(r700618, r700619);
        double r700621 = 0.16666666666666666;
        double r700622 = sqrt(r700621);
        double r700623 = r700620 * r700622;
        double r700624 = r700623 * r700622;
        double r700625 = fma(r700614, r700624, r700619);
        return r700625;
}

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

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*r*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019153 +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))