Average Error: 0.4 → 0.3
Time: 30.6s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, 0.5\right)\]
\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), \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, 0.5\right)
double f(double u1, double u2) {
        double r1192474 = 1.0;
        double r1192475 = 6.0;
        double r1192476 = r1192474 / r1192475;
        double r1192477 = -2.0;
        double r1192478 = u1;
        double r1192479 = log(r1192478);
        double r1192480 = r1192477 * r1192479;
        double r1192481 = 0.5;
        double r1192482 = pow(r1192480, r1192481);
        double r1192483 = r1192476 * r1192482;
        double r1192484 = 2.0;
        double r1192485 = atan2(1.0, 0.0);
        double r1192486 = r1192484 * r1192485;
        double r1192487 = u2;
        double r1192488 = r1192486 * r1192487;
        double r1192489 = cos(r1192488);
        double r1192490 = r1192483 * r1192489;
        double r1192491 = r1192490 + r1192481;
        return r1192491;
}
↓
double f(double u1, double u2) {
        double r1192492 = atan2(1.0, 0.0);
        double r1192493 = 2.0;
        double r1192494 = r1192492 * r1192493;
        double r1192495 = u2;
        double r1192496 = r1192494 * r1192495;
        double r1192497 = cos(r1192496);
        double r1192498 = -2.0;
        double r1192499 = u1;
        double r1192500 = log(r1192499);
        double r1192501 = r1192498 * r1192500;
        double r1192502 = 0.5;
        double r1192503 = pow(r1192501, r1192502);
        double r1192504 = 6.0;
        double r1192505 = r1192503 / r1192504;
        double r1192506 = fma(r1192497, r1192505, r1192502);
        return r1192506;
}

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. Simplified0.3

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

  3. Final simplification0.3

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

Reproduce

herbie shell --seed 2019135 +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))