Average Error: 0.4 → 0.3
Time: 11.3s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)
double f(double u1, double u2) {
        double r67502 = 1.0;
        double r67503 = 6.0;
        double r67504 = r67502 / r67503;
        double r67505 = -2.0;
        double r67506 = u1;
        double r67507 = log(r67506);
        double r67508 = r67505 * r67507;
        double r67509 = 0.5;
        double r67510 = pow(r67508, r67509);
        double r67511 = r67504 * r67510;
        double r67512 = 2.0;
        double r67513 = atan2(1.0, 0.0);
        double r67514 = r67512 * r67513;
        double r67515 = u2;
        double r67516 = r67514 * r67515;
        double r67517 = cos(r67516);
        double r67518 = r67511 * r67517;
        double r67519 = r67518 + r67509;
        return r67519;
}
↓
double f(double u1, double u2) {
        double r67520 = 1.0;
        double r67521 = -2.0;
        double r67522 = u1;
        double r67523 = log(r67522);
        double r67524 = r67521 * r67523;
        double r67525 = 0.5;
        double r67526 = pow(r67524, r67525);
        double r67527 = r67520 * r67526;
        double r67528 = 6.0;
        double r67529 = r67527 / r67528;
        double r67530 = 2.0;
        double r67531 = atan2(1.0, 0.0);
        double r67532 = r67530 * r67531;
        double r67533 = u2;
        double r67534 = r67532 * r67533;
        double r67535 = cos(r67534);
        double r67536 = fma(r67529, r67535, r67525);
        return r67536;
}

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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019352 +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))