Average Error: 0.4 → 0.3
Time: 34.5s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right), \left(\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right), 0.5\right)\]
\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(\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right), \left(\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right), 0.5\right)
double f(double u1, double u2) {
        double r1474653 = 1.0;
        double r1474654 = 6.0;
        double r1474655 = r1474653 / r1474654;
        double r1474656 = -2.0;
        double r1474657 = u1;
        double r1474658 = log(r1474657);
        double r1474659 = r1474656 * r1474658;
        double r1474660 = 0.5;
        double r1474661 = pow(r1474659, r1474660);
        double r1474662 = r1474655 * r1474661;
        double r1474663 = 2.0;
        double r1474664 = atan2(1.0, 0.0);
        double r1474665 = r1474663 * r1474664;
        double r1474666 = u2;
        double r1474667 = r1474665 * r1474666;
        double r1474668 = cos(r1474667);
        double r1474669 = r1474662 * r1474668;
        double r1474670 = r1474669 + r1474660;
        return r1474670;
}
↓
double f(double u1, double u2) {
        double r1474671 = atan2(1.0, 0.0);
        double r1474672 = 2.0;
        double r1474673 = r1474671 * r1474672;
        double r1474674 = u2;
        double r1474675 = r1474673 * r1474674;
        double r1474676 = cos(r1474675);
        double r1474677 = -2.0;
        double r1474678 = u1;
        double r1474679 = log(r1474678);
        double r1474680 = r1474677 * r1474679;
        double r1474681 = 0.5;
        double r1474682 = pow(r1474680, r1474681);
        double r1474683 = 6.0;
        double r1474684 = r1474682 / r1474683;
        double r1474685 = fma(r1474676, r1474684, r1474681);
        return r1474685;
}

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

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

  3. Final simplification0.3

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

Reproduce

herbie shell --seed 2019130 +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))