Average Error: 0.4 → 0.3
Time: 33.3s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\cos \left(\left(u2 \cdot \pi\right) \cdot 2\right), 1 \cdot \frac{1}{\frac{6}{{\left(\log u1 \cdot -2\right)}^{0.5}}}, 0.5\right)\]
\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(u2 \cdot \pi\right) \cdot 2\right), 1 \cdot \frac{1}{\frac{6}{{\left(\log u1 \cdot -2\right)}^{0.5}}}, 0.5\right)
double f(double u1, double u2) {
        double r1817365 = 1.0;
        double r1817366 = 6.0;
        double r1817367 = r1817365 / r1817366;
        double r1817368 = -2.0;
        double r1817369 = u1;
        double r1817370 = log(r1817369);
        double r1817371 = r1817368 * r1817370;
        double r1817372 = 0.5;
        double r1817373 = pow(r1817371, r1817372);
        double r1817374 = r1817367 * r1817373;
        double r1817375 = 2.0;
        double r1817376 = atan2(1.0, 0.0);
        double r1817377 = r1817375 * r1817376;
        double r1817378 = u2;
        double r1817379 = r1817377 * r1817378;
        double r1817380 = cos(r1817379);
        double r1817381 = r1817374 * r1817380;
        double r1817382 = r1817381 + r1817372;
        return r1817382;
}
↓
double f(double u1, double u2) {
        double r1817383 = u2;
        double r1817384 = atan2(1.0, 0.0);
        double r1817385 = r1817383 * r1817384;
        double r1817386 = 2.0;
        double r1817387 = r1817385 * r1817386;
        double r1817388 = cos(r1817387);
        double r1817389 = 1.0;
        double r1817390 = 1.0;
        double r1817391 = 6.0;
        double r1817392 = u1;
        double r1817393 = log(r1817392);
        double r1817394 = -2.0;
        double r1817395 = r1817393 * r1817394;
        double r1817396 = 0.5;
        double r1817397 = pow(r1817395, r1817396);
        double r1817398 = r1817391 / r1817397;
        double r1817399 = r1817390 / r1817398;
        double r1817400 = r1817389 * r1817399;
        double r1817401 = fma(r1817388, r1817400, r1817396);
        return r1817401;
}

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

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

  4. Applied div-inv0.4

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

  5. Applied associate-*l*0.4

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

  6. Simplified0.3

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

  8. Applied clear-num0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0.0 u1 1.0) (<= 0.0 u2 1.0))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))