Average Error: 0.4 → 0.4
Time: 31.9s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}, 0.5\right)\]
\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), \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}, 0.5\right)
double f(double u1, double u2) {
        double r732782 = 1.0;
        double r732783 = 6.0;
        double r732784 = r732782 / r732783;
        double r732785 = -2.0;
        double r732786 = u1;
        double r732787 = log(r732786);
        double r732788 = r732785 * r732787;
        double r732789 = 0.5;
        double r732790 = pow(r732788, r732789);
        double r732791 = r732784 * r732790;
        double r732792 = 2.0;
        double r732793 = atan2(1.0, 0.0);
        double r732794 = r732792 * r732793;
        double r732795 = u2;
        double r732796 = r732794 * r732795;
        double r732797 = cos(r732796);
        double r732798 = r732791 * r732797;
        double r732799 = r732798 + r732789;
        return r732799;
}
↓
double f(double u1, double u2) {
        double r732800 = atan2(1.0, 0.0);
        double r732801 = 2.0;
        double r732802 = r732800 * r732801;
        double r732803 = u2;
        double r732804 = r732802 * r732803;
        double r732805 = cos(r732804);
        double r732806 = -2.0;
        double r732807 = u1;
        double r732808 = log(r732807);
        double r732809 = r732806 * r732808;
        double r732810 = 0.5;
        double r732811 = pow(r732809, r732810);
        double r732812 = 0.16666666666666666;
        double r732813 = sqrt(r732812);
        double r732814 = r732811 * r732813;
        double r732815 = r732814 * r732813;
        double r732816 = fma(r732805, r732815, r732810);
        return r732816;
}

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

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*r*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019151 +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))