Average Error: 0.4 → 0.4
Time: 10.6s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)
double f(double u1, double u2) {
        double r62745 = 1.0;
        double r62746 = 6.0;
        double r62747 = r62745 / r62746;
        double r62748 = -2.0;
        double r62749 = u1;
        double r62750 = log(r62749);
        double r62751 = r62748 * r62750;
        double r62752 = 0.5;
        double r62753 = pow(r62751, r62752);
        double r62754 = r62747 * r62753;
        double r62755 = 2.0;
        double r62756 = atan2(1.0, 0.0);
        double r62757 = r62755 * r62756;
        double r62758 = u2;
        double r62759 = r62757 * r62758;
        double r62760 = cos(r62759);
        double r62761 = r62754 * r62760;
        double r62762 = r62761 + r62752;
        return r62762;
}
↓
double f(double u1, double u2) {
        double r62763 = 1.0;
        double r62764 = 6.0;
        double r62765 = r62763 / r62764;
        double r62766 = -2.0;
        double r62767 = u1;
        double r62768 = log(r62767);
        double r62769 = r62766 * r62768;
        double r62770 = 0.5;
        double r62771 = pow(r62769, r62770);
        double r62772 = r62765 * r62771;
        double r62773 = 2.0;
        double r62774 = atan2(1.0, 0.0);
        double r62775 = r62773 * r62774;
        double r62776 = u2;
        double r62777 = r62775 * r62776;
        double r62778 = cos(r62777);
        double r62779 = fma(r62772, r62778, r62770);
        return r62779;
}

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 sqr-pow0.6

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

  5. Applied associate-*r*0.5

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

  7. Applied associate-*l*0.6

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020100 +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))