Average Error: 0.4 → 0.4
Time: 32.1s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right), \left(\frac{1}{{\left(\frac{1}{{-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}}\right)}^{0.5} \cdot 6}\right), 0.5\right)\]
\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}{{-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}}\right)}^{0.5} \cdot 6}\right), 0.5\right)
double f(double u1, double u2) {
        double r1060377 = 1.0;
        double r1060378 = 6.0;
        double r1060379 = r1060377 / r1060378;
        double r1060380 = -2.0;
        double r1060381 = u1;
        double r1060382 = log(r1060381);
        double r1060383 = r1060380 * r1060382;
        double r1060384 = 0.5;
        double r1060385 = pow(r1060383, r1060384);
        double r1060386 = r1060379 * r1060385;
        double r1060387 = 2.0;
        double r1060388 = atan2(1.0, 0.0);
        double r1060389 = r1060387 * r1060388;
        double r1060390 = u2;
        double r1060391 = r1060389 * r1060390;
        double r1060392 = cos(r1060391);
        double r1060393 = r1060386 * r1060392;
        double r1060394 = r1060393 + r1060384;
        return r1060394;
}
↓
double f(double u1, double u2) {
        double r1060395 = atan2(1.0, 0.0);
        double r1060396 = 2.0;
        double r1060397 = r1060395 * r1060396;
        double r1060398 = u2;
        double r1060399 = r1060397 * r1060398;
        double r1060400 = cos(r1060399);
        double r1060401 = 1.0;
        double r1060402 = -2.0;
        double r1060403 = 1.0;
        double r1060404 = pow(r1060402, r1060403);
        double r1060405 = u1;
        double r1060406 = log(r1060405);
        double r1060407 = pow(r1060406, r1060403);
        double r1060408 = r1060404 * r1060407;
        double r1060409 = r1060401 / r1060408;
        double r1060410 = 0.5;
        double r1060411 = pow(r1060409, r1060410);
        double r1060412 = 6.0;
        double r1060413 = r1060411 * r1060412;
        double r1060414 = r1060401 / r1060413;
        double r1060415 = fma(r1060400, r1060414, r1060410);
        return r1060415;
}

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 clear-num0.4

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

  5. Taylor expanded around 0 0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019129 +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))