Average Error: 0.4 → 0.3
Time: 33.7s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \frac{{\left(\log u1 \cdot -2\right)}^{0.5}}{6}\right) \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) + 0.5\]
\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(1 \cdot \frac{{\left(\log u1 \cdot -2\right)}^{0.5}}{6}\right) \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) + 0.5
double f(double u1, double u2) {
        double r2561458 = 1.0;
        double r2561459 = 6.0;
        double r2561460 = r2561458 / r2561459;
        double r2561461 = -2.0;
        double r2561462 = u1;
        double r2561463 = log(r2561462);
        double r2561464 = r2561461 * r2561463;
        double r2561465 = 0.5;
        double r2561466 = pow(r2561464, r2561465);
        double r2561467 = r2561460 * r2561466;
        double r2561468 = 2.0;
        double r2561469 = atan2(1.0, 0.0);
        double r2561470 = r2561468 * r2561469;
        double r2561471 = u2;
        double r2561472 = r2561470 * r2561471;
        double r2561473 = cos(r2561472);
        double r2561474 = r2561467 * r2561473;
        double r2561475 = r2561474 + r2561465;
        return r2561475;
}
↓
double f(double u1, double u2) {
        double r2561476 = 1.0;
        double r2561477 = u1;
        double r2561478 = log(r2561477);
        double r2561479 = -2.0;
        double r2561480 = r2561478 * r2561479;
        double r2561481 = 0.5;
        double r2561482 = pow(r2561480, r2561481);
        double r2561483 = 6.0;
        double r2561484 = r2561482 / r2561483;
        double r2561485 = r2561476 * r2561484;
        double r2561486 = u2;
        double r2561487 = 2.0;
        double r2561488 = atan2(1.0, 0.0);
        double r2561489 = r2561487 * r2561488;
        double r2561490 = r2561486 * r2561489;
        double r2561491 = cos(r2561490);
        double r2561492 = r2561485 * r2561491;
        double r2561493 = r2561492 + r2561481;
        return r2561493;
}

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

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 
(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))