Average Error: 0.4 → 0.4
Time: 4.6m
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right), \left(\frac{1}{{\left(\frac{1}{{-1}^{1.0} \cdot \left({-2}^{1.0} \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1.0}\right)}\right)}^{0.5} \cdot 6}\right), 0.5\right)\]
\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(\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right), \left(\frac{1}{{\left(\frac{1}{{-1}^{1.0} \cdot \left({-2}^{1.0} \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1.0}\right)}\right)}^{0.5} \cdot 6}\right), 0.5\right)
double f(double u1, double u2) {
        double r32788467 = 1.0;
        double r32788468 = 6.0;
        double r32788469 = r32788467 / r32788468;
        double r32788470 = -2.0;
        double r32788471 = u1;
        double r32788472 = log(r32788471);
        double r32788473 = r32788470 * r32788472;
        double r32788474 = 0.5;
        double r32788475 = pow(r32788473, r32788474);
        double r32788476 = r32788469 * r32788475;
        double r32788477 = 2.0;
        double r32788478 = atan2(1.0, 0.0);
        double r32788479 = r32788477 * r32788478;
        double r32788480 = u2;
        double r32788481 = r32788479 * r32788480;
        double r32788482 = cos(r32788481);
        double r32788483 = r32788476 * r32788482;
        double r32788484 = r32788483 + r32788474;
        return r32788484;
}
↓
double f(double u1, double u2) {
        double r32788485 = atan2(1.0, 0.0);
        double r32788486 = 2.0;
        double r32788487 = r32788485 * r32788486;
        double r32788488 = u2;
        double r32788489 = r32788487 * r32788488;
        double r32788490 = cos(r32788489);
        double r32788491 = 1.0;
        double r32788492 = -1.0;
        double r32788493 = 1.0;
        double r32788494 = pow(r32788492, r32788493);
        double r32788495 = -2.0;
        double r32788496 = pow(r32788495, r32788493);
        double r32788497 = u1;
        double r32788498 = r32788491 / r32788497;
        double r32788499 = log(r32788498);
        double r32788500 = pow(r32788499, r32788493);
        double r32788501 = r32788496 * r32788500;
        double r32788502 = r32788494 * r32788501;
        double r32788503 = r32788491 / r32788502;
        double r32788504 = 0.5;
        double r32788505 = pow(r32788503, r32788504);
        double r32788506 = 6.0;
        double r32788507 = r32788505 * r32788506;
        double r32788508 = r32788491 / r32788507;
        double r32788509 = fma(r32788490, r32788508, r32788504);
        return r32788509;
}

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

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

  4. Applied *-un-lft-identity0.3

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

  5. Applied associate-/l*0.3

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

  6. Taylor expanded around inf 0.4

    \[\leadsto \mathsf{fma}\left(\left(\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right), \left(\frac{1}{\color{blue}{6 \cdot {\left(\frac{1}{{-1}^{1.0} \cdot \left({-2}^{1.0} \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1.0}\right)}\right)}^{0.5}}}\right), 0.5\right)\]

  7. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right), \left(\frac{1}{{\left(\frac{1}{{-1}^{1.0} \cdot \left({-2}^{1.0} \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1.0}\right)}\right)}^{0.5} \cdot 6}\right), 0.5\right)\]

Reproduce

herbie shell --seed 2019120 +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))