Average Error: 0.4 → 0.3
Time: 10.6s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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(1 \cdot \frac{{\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 r63400 = 1.0;
        double r63401 = 6.0;
        double r63402 = r63400 / r63401;
        double r63403 = -2.0;
        double r63404 = u1;
        double r63405 = log(r63404);
        double r63406 = r63403 * r63405;
        double r63407 = 0.5;
        double r63408 = pow(r63406, r63407);
        double r63409 = r63402 * r63408;
        double r63410 = 2.0;
        double r63411 = atan2(1.0, 0.0);
        double r63412 = r63410 * r63411;
        double r63413 = u2;
        double r63414 = r63412 * r63413;
        double r63415 = cos(r63414);
        double r63416 = r63409 * r63415;
        double r63417 = r63416 + r63407;
        return r63417;
}
↓
double f(double u1, double u2) {
        double r63418 = 1.0;
        double r63419 = -2.0;
        double r63420 = u1;
        double r63421 = log(r63420);
        double r63422 = r63419 * r63421;
        double r63423 = 0.5;
        double r63424 = pow(r63422, r63423);
        double r63425 = 6.0;
        double r63426 = r63424 / r63425;
        double r63427 = r63418 * r63426;
        double r63428 = 2.0;
        double r63429 = atan2(1.0, 0.0);
        double r63430 = r63428 * r63429;
        double r63431 = u2;
        double r63432 = r63430 * r63431;
        double r63433 = cos(r63432);
        double r63434 = fma(r63427, r63433, r63423);
        return r63434;
}

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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019346 +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))