Average Error: 0.4 → 0.4
Time: 29.5s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right), \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\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)
double f(double u1, double u2) {
        double r78582 = 1.0;
        double r78583 = 6.0;
        double r78584 = r78582 / r78583;
        double r78585 = -2.0;
        double r78586 = u1;
        double r78587 = log(r78586);
        double r78588 = r78585 * r78587;
        double r78589 = 0.5;
        double r78590 = pow(r78588, r78589);
        double r78591 = r78584 * r78590;
        double r78592 = 2.0;
        double r78593 = atan2(1.0, 0.0);
        double r78594 = r78592 * r78593;
        double r78595 = u2;
        double r78596 = r78594 * r78595;
        double r78597 = cos(r78596);
        double r78598 = r78591 * r78597;
        double r78599 = r78598 + r78589;
        return r78599;
}
↓
double f(double u1, double u2) {
        double r78600 = 1.0;
        double r78601 = 6.0;
        double r78602 = r78600 / r78601;
        double r78603 = sqrt(r78602);
        double r78604 = -2.0;
        double r78605 = u1;
        double r78606 = log(r78605);
        double r78607 = r78604 * r78606;
        double r78608 = 0.5;
        double r78609 = pow(r78607, r78608);
        double r78610 = r78603 * r78609;
        double r78611 = r78603 * r78610;
        double r78612 = 2.0;
        double r78613 = atan2(1.0, 0.0);
        double r78614 = r78612 * r78613;
        double r78615 = u2;
        double r78616 = r78614 * r78615;
        double r78617 = cos(r78616);
        double r78618 = fma(r78611, r78617, r78608);
        return r78618;
}

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 add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019303 +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))