Average Error: 0.4 → 0.4
Time: 32.4s
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 r1552632 = 1.0;
        double r1552633 = 6.0;
        double r1552634 = r1552632 / r1552633;
        double r1552635 = -2.0;
        double r1552636 = u1;
        double r1552637 = log(r1552636);
        double r1552638 = r1552635 * r1552637;
        double r1552639 = 0.5;
        double r1552640 = pow(r1552638, r1552639);
        double r1552641 = r1552634 * r1552640;
        double r1552642 = 2.0;
        double r1552643 = atan2(1.0, 0.0);
        double r1552644 = r1552642 * r1552643;
        double r1552645 = u2;
        double r1552646 = r1552644 * r1552645;
        double r1552647 = cos(r1552646);
        double r1552648 = r1552641 * r1552647;
        double r1552649 = r1552648 + r1552639;
        return r1552649;
}
↓
double f(double u1, double u2) {
        double r1552650 = atan2(1.0, 0.0);
        double r1552651 = 2.0;
        double r1552652 = r1552650 * r1552651;
        double r1552653 = u2;
        double r1552654 = r1552652 * r1552653;
        double r1552655 = cos(r1552654);
        double r1552656 = -2.0;
        double r1552657 = u1;
        double r1552658 = log(r1552657);
        double r1552659 = r1552656 * r1552658;
        double r1552660 = 0.5;
        double r1552661 = pow(r1552659, r1552660);
        double r1552662 = 0.16666666666666666;
        double r1552663 = sqrt(r1552662);
        double r1552664 = r1552661 * r1552663;
        double r1552665 = r1552664 * r1552663;
        double r1552666 = fma(r1552655, r1552665, r1552660);
        return r1552666;
}

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. Simplified0.3

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

  4. Applied div-inv0.4

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

  5. Simplified0.4

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

  7. Applied add-sqr-sqrt0.4

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

  8. Applied associate-*r*0.4

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

  9. 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 2019164 +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))