Average Error: 0.4 → 0.3
Time: 28.6s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\]
\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
↓
0.5 + \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(\pi \cdot 2\right) \cdot u2\right)
double f(double u1, double u2) {
        double r838423 = 1.0;
        double r838424 = 6.0;
        double r838425 = r838423 / r838424;
        double r838426 = -2.0;
        double r838427 = u1;
        double r838428 = log(r838427);
        double r838429 = r838426 * r838428;
        double r838430 = 0.5;
        double r838431 = pow(r838429, r838430);
        double r838432 = r838425 * r838431;
        double r838433 = 2.0;
        double r838434 = atan2(1.0, 0.0);
        double r838435 = r838433 * r838434;
        double r838436 = u2;
        double r838437 = r838435 * r838436;
        double r838438 = cos(r838437);
        double r838439 = r838432 * r838438;
        double r838440 = r838439 + r838430;
        return r838440;
}
↓
double f(double u1, double u2) {
        double r838441 = 0.5;
        double r838442 = -2.0;
        double r838443 = u1;
        double r838444 = log(r838443);
        double r838445 = r838442 * r838444;
        double r838446 = pow(r838445, r838441);
        double r838447 = 6.0;
        double r838448 = r838446 / r838447;
        double r838449 = atan2(1.0, 0.0);
        double r838450 = 2.0;
        double r838451 = r838449 * r838450;
        double r838452 = u2;
        double r838453 = r838451 * r838452;
        double r838454 = cos(r838453);
        double r838455 = r838448 * r838454;
        double r838456 = r838441 + r838455;
        return r838456;
}

Error

Bits error versus u1

Bits error versus u2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied associate-*l/0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019152 +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))