Average Error: 0.4 → 0.3
Time: 1.5m
Precision: 64
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\]
double f(double u1, double u2) {
        double r7539436 = 1.0;
        double r7539437 = 6.0;
        double r7539438 = r7539436 / r7539437;
        double r7539439 = -2.0;
        double r7539440 = u1;
        double r7539441 = log(r7539440);
        double r7539442 = r7539439 * r7539441;
        double r7539443 = 0.5;
        double r7539444 = pow(r7539442, r7539443);
        double r7539445 = r7539438 * r7539444;
        double r7539446 = 2.0;
        double r7539447 = atan2(1.0, 0.0);
        double r7539448 = r7539446 * r7539447;
        double r7539449 = u2;
        double r7539450 = r7539448 * r7539449;
        double r7539451 = cos(r7539450);
        double r7539452 = r7539445 * r7539451;
        double r7539453 = r7539452 + r7539443;
        return r7539453;
}
↓
double f(double u1, double u2) {
        double r7539454 = 0.5;
        double r7539455 = -2.0;
        double r7539456 = u1;
        double r7539457 = log(r7539456);
        double r7539458 = r7539455 * r7539457;
        double r7539459 = pow(r7539458, r7539454);
        double r7539460 = 6.0;
        double r7539461 = r7539459 / r7539460;
        double r7539462 = atan2(1.0, 0.0);
        double r7539463 = 2.0;
        double r7539464 = r7539462 * r7539463;
        double r7539465 = u2;
        double r7539466 = r7539464 * r7539465;
        double r7539467 = cos(r7539466);
        double r7539468 = r7539461 * r7539467;
        double r7539469 = r7539454 + r7539468;
        return r7539469;
}


\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)

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. 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

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

  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 2019101 
(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))