Average Error: 0.4 → 0.4
Time: 35.0s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + 0.5\]
\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(\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + 0.5
double f(double u1, double u2) {
        double r1840475 = 1.0;
        double r1840476 = 6.0;
        double r1840477 = r1840475 / r1840476;
        double r1840478 = -2.0;
        double r1840479 = u1;
        double r1840480 = log(r1840479);
        double r1840481 = r1840478 * r1840480;
        double r1840482 = 0.5;
        double r1840483 = pow(r1840481, r1840482);
        double r1840484 = r1840477 * r1840483;
        double r1840485 = 2.0;
        double r1840486 = atan2(1.0, 0.0);
        double r1840487 = r1840485 * r1840486;
        double r1840488 = u2;
        double r1840489 = r1840487 * r1840488;
        double r1840490 = cos(r1840489);
        double r1840491 = r1840484 * r1840490;
        double r1840492 = r1840491 + r1840482;
        return r1840492;
}
↓
double f(double u1, double u2) {
        double r1840493 = u1;
        double r1840494 = log(r1840493);
        double r1840495 = -2.0;
        double r1840496 = r1840494 * r1840495;
        double r1840497 = 0.5;
        double r1840498 = pow(r1840496, r1840497);
        double r1840499 = 0.16666666666666666;
        double r1840500 = sqrt(r1840499);
        double r1840501 = r1840498 * r1840500;
        double r1840502 = r1840501 * r1840500;
        double r1840503 = 2.0;
        double r1840504 = atan2(1.0, 0.0);
        double r1840505 = u2;
        double r1840506 = r1840504 * r1840505;
        double r1840507 = r1840503 * r1840506;
        double r1840508 = cos(r1840507);
        double r1840509 = r1840502 * r1840508;
        double r1840510 = r1840509 + r1840497;
        return r1840510;
}

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. Simplified0.4

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019164 
(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))