Average Error: 0.4 → 0.3
Time: 17.6s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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(\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)
double f(double u1, double u2) {
        double r63730 = 1.0;
        double r63731 = 6.0;
        double r63732 = r63730 / r63731;
        double r63733 = -2.0;
        double r63734 = u1;
        double r63735 = log(r63734);
        double r63736 = r63733 * r63735;
        double r63737 = 0.5;
        double r63738 = pow(r63736, r63737);
        double r63739 = r63732 * r63738;
        double r63740 = 2.0;
        double r63741 = atan2(1.0, 0.0);
        double r63742 = r63740 * r63741;
        double r63743 = u2;
        double r63744 = r63742 * r63743;
        double r63745 = cos(r63744);
        double r63746 = r63739 * r63745;
        double r63747 = r63746 + r63737;
        return r63747;
}
↓
double f(double u1, double u2) {
        double r63748 = 1.0;
        double r63749 = -2.0;
        double r63750 = u1;
        double r63751 = log(r63750);
        double r63752 = r63749 * r63751;
        double r63753 = 0.5;
        double r63754 = pow(r63752, r63753);
        double r63755 = r63748 * r63754;
        double r63756 = 6.0;
        double r63757 = r63755 / r63756;
        double r63758 = 2.0;
        double r63759 = atan2(1.0, 0.0);
        double r63760 = r63758 * r63759;
        double r63761 = u2;
        double r63762 = r63760 * r63761;
        double r63763 = cos(r63762);
        double r63764 = fma(r63757, r63763, r63753);
        return r63764;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)}\]
  3. Using strategy rm

  4. Applied associate-*l/0.3

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

  5. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (<= 0.0 u1 1) (<= 0.0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))