Average Error: 0.4 → 0.4
Time: 30.6s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(1 \cdot \frac{1}{\frac{6}{{\left(-2 \cdot \log u1\right)}^{0.5}}}, \cos \left(\left(\left(2 \cdot \pi\right) \cdot \sqrt{u2}\right) \cdot \sqrt{u2}\right), 0.5\right)\]
\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(1 \cdot \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 r68754 = 1.0;
        double r68755 = 6.0;
        double r68756 = r68754 / r68755;
        double r68757 = -2.0;
        double r68758 = u1;
        double r68759 = log(r68758);
        double r68760 = r68757 * r68759;
        double r68761 = 0.5;
        double r68762 = pow(r68760, r68761);
        double r68763 = r68756 * r68762;
        double r68764 = 2.0;
        double r68765 = atan2(1.0, 0.0);
        double r68766 = r68764 * r68765;
        double r68767 = u2;
        double r68768 = r68766 * r68767;
        double r68769 = cos(r68768);
        double r68770 = r68763 * r68769;
        double r68771 = r68770 + r68761;
        return r68771;
}
↓
double f(double u1, double u2) {
        double r68772 = 1.0;
        double r68773 = 1.0;
        double r68774 = 6.0;
        double r68775 = -2.0;
        double r68776 = u1;
        double r68777 = log(r68776);
        double r68778 = r68775 * r68777;
        double r68779 = 0.5;
        double r68780 = pow(r68778, r68779);
        double r68781 = r68774 / r68780;
        double r68782 = r68773 / r68781;
        double r68783 = r68772 * r68782;
        double r68784 = 2.0;
        double r68785 = atan2(1.0, 0.0);
        double r68786 = r68784 * r68785;
        double r68787 = u2;
        double r68788 = sqrt(r68787);
        double r68789 = r68786 * r68788;
        double r68790 = r68789 * r68788;
        double r68791 = cos(r68790);
        double r68792 = fma(r68783, r68791, r68779);
        return r68792;
}

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 div-inv0.4

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

  5. Applied associate-*l*0.4

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

  6. Simplified0.3

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

  8. Applied add-sqr-sqrt0.3

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

  9. Applied associate-*r*0.3

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

  11. Applied clear-num0.4

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

  12. Final simplification0.4

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

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))