Average Error: 0.4 → 0.3
Time: 27.9s
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 r686566 = 1.0;
        double r686567 = 6.0;
        double r686568 = r686566 / r686567;
        double r686569 = -2.0;
        double r686570 = u1;
        double r686571 = log(r686570);
        double r686572 = r686569 * r686571;
        double r686573 = 0.5;
        double r686574 = pow(r686572, r686573);
        double r686575 = r686568 * r686574;
        double r686576 = 2.0;
        double r686577 = atan2(1.0, 0.0);
        double r686578 = r686576 * r686577;
        double r686579 = u2;
        double r686580 = r686578 * r686579;
        double r686581 = cos(r686580);
        double r686582 = r686575 * r686581;
        double r686583 = r686582 + r686573;
        return r686583;
}
↓
double f(double u1, double u2) {
        double r686584 = atan2(1.0, 0.0);
        double r686585 = 2.0;
        double r686586 = r686584 * r686585;
        double r686587 = u2;
        double r686588 = r686586 * r686587;
        double r686589 = cos(r686588);
        double r686590 = -2.0;
        double r686591 = u1;
        double r686592 = log(r686591);
        double r686593 = r686590 * r686592;
        double r686594 = 0.5;
        double r686595 = pow(r686593, r686594);
        double r686596 = 0.16666666666666666;
        double r686597 = sqrt(r686596);
        double r686598 = r686595 * r686597;
        double r686599 = r686598 * r686597;
        double r686600 = fma(r686589, r686599, r686594);
        return r686600;
}

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))