Average Error: 0.4 → 0.4
Time: 19.0s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(0.1666666666666666574148081281236954964697 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) \cdot {\left({-2}^{1} \cdot {\left(\log u1\right)}^{1}\right)}^{0.5} + 0.5\]
\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
↓
\left(0.1666666666666666574148081281236954964697 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) \cdot {\left({-2}^{1} \cdot {\left(\log u1\right)}^{1}\right)}^{0.5} + 0.5
double f(double u1, double u2) {
        double r70360 = 1.0;
        double r70361 = 6.0;
        double r70362 = r70360 / r70361;
        double r70363 = -2.0;
        double r70364 = u1;
        double r70365 = log(r70364);
        double r70366 = r70363 * r70365;
        double r70367 = 0.5;
        double r70368 = pow(r70366, r70367);
        double r70369 = r70362 * r70368;
        double r70370 = 2.0;
        double r70371 = atan2(1.0, 0.0);
        double r70372 = r70370 * r70371;
        double r70373 = u2;
        double r70374 = r70372 * r70373;
        double r70375 = cos(r70374);
        double r70376 = r70369 * r70375;
        double r70377 = r70376 + r70367;
        return r70377;
}
↓
double f(double u1, double u2) {
        double r70378 = 0.16666666666666666;
        double r70379 = 2.0;
        double r70380 = atan2(1.0, 0.0);
        double r70381 = r70379 * r70380;
        double r70382 = u2;
        double r70383 = r70381 * r70382;
        double r70384 = cos(r70383);
        double r70385 = r70378 * r70384;
        double r70386 = -2.0;
        double r70387 = 1.0;
        double r70388 = pow(r70386, r70387);
        double r70389 = u1;
        double r70390 = log(r70389);
        double r70391 = pow(r70390, r70387);
        double r70392 = r70388 * r70391;
        double r70393 = 0.5;
        double r70394 = pow(r70392, r70393);
        double r70395 = r70385 * r70394;
        double r70396 = r70395 + r70393;
        return r70396;
}

Error

Bits error versus u1

Bits error versus u2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied div-inv0.4

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

  4. Applied associate-*l*0.4

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

  5. Simplified0.3

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

  6. Taylor expanded around -inf 64.0

    \[\leadsto \color{blue}{0.1666666666666666574148081281236954964697 \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot {\left({-2}^{1} \cdot {\left(\log -1 - \log \left(\frac{-1}{u1}\right)\right)}^{1}\right)}^{0.5}\right) + 0.5}\]

  7. Simplified0.4

    \[\leadsto \color{blue}{0.5 + {\left({\left(0 + \log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5} \cdot \left(0.1666666666666666574148081281236954964697 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)}\]

  8. Final simplification0.4

    \[\leadsto \left(0.1666666666666666574148081281236954964697 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) \cdot {\left({-2}^{1} \cdot {\left(\log u1\right)}^{1}\right)}^{0.5} + 0.5\]

Reproduce

herbie shell --seed 2019350 
(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))