Average Error: 0.4 → 0.4
Time: 27.3s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\frac{1}{\frac{6}{{\left(-2 \cdot \log u1\right)}^{0.5}}}, \cos \left(\left(\left(2 \cdot \pi\right) \cdot \sqrt{u2}\right) \cdot \sqrt{u2}\right), 0.5\right)\]
\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}{\frac{6}{{\left(-2 \cdot \log u1\right)}^{0.5}}}, \cos \left(\left(\left(2 \cdot \pi\right) \cdot \sqrt{u2}\right) \cdot \sqrt{u2}\right), 0.5\right)
double f(double u1, double u2) {
        double r67816 = 1.0;
        double r67817 = 6.0;
        double r67818 = r67816 / r67817;
        double r67819 = -2.0;
        double r67820 = u1;
        double r67821 = log(r67820);
        double r67822 = r67819 * r67821;
        double r67823 = 0.5;
        double r67824 = pow(r67822, r67823);
        double r67825 = r67818 * r67824;
        double r67826 = 2.0;
        double r67827 = atan2(1.0, 0.0);
        double r67828 = r67826 * r67827;
        double r67829 = u2;
        double r67830 = r67828 * r67829;
        double r67831 = cos(r67830);
        double r67832 = r67825 * r67831;
        double r67833 = r67832 + r67823;
        return r67833;
}
↓
double f(double u1, double u2) {
        double r67834 = 1.0;
        double r67835 = 6.0;
        double r67836 = -2.0;
        double r67837 = u1;
        double r67838 = log(r67837);
        double r67839 = r67836 * r67838;
        double r67840 = 0.5;
        double r67841 = pow(r67839, r67840);
        double r67842 = r67835 / r67841;
        double r67843 = r67834 / r67842;
        double r67844 = 2.0;
        double r67845 = atan2(1.0, 0.0);
        double r67846 = r67844 * r67845;
        double r67847 = u2;
        double r67848 = sqrt(r67847);
        double r67849 = r67846 * r67848;
        double r67850 = r67849 * r67848;
        double r67851 = cos(r67850);
        double r67852 = fma(r67843, r67851, r67840);
        return r67852;
}

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. Using strategy rm

  6. Applied add-sqr-sqrt0.3

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

  7. Applied associate-*r*0.3

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

  9. Applied associate-/l*0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019325 +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))