Average Error: 0.4 → 0.4
Time: 32.7s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r1550425 = 1.0;
        double r1550426 = 6.0;
        double r1550427 = r1550425 / r1550426;
        double r1550428 = -2.0;
        double r1550429 = u1;
        double r1550430 = log(r1550429);
        double r1550431 = r1550428 * r1550430;
        double r1550432 = 0.5;
        double r1550433 = pow(r1550431, r1550432);
        double r1550434 = r1550427 * r1550433;
        double r1550435 = 2.0;
        double r1550436 = atan2(1.0, 0.0);
        double r1550437 = r1550435 * r1550436;
        double r1550438 = u2;
        double r1550439 = r1550437 * r1550438;
        double r1550440 = cos(r1550439);
        double r1550441 = r1550434 * r1550440;
        double r1550442 = r1550441 + r1550432;
        return r1550442;
}
↓
double f(double u1, double u2) {
        double r1550443 = u1;
        double r1550444 = log(r1550443);
        double r1550445 = -2.0;
        double r1550446 = r1550444 * r1550445;
        double r1550447 = 0.5;
        double r1550448 = pow(r1550446, r1550447);
        double r1550449 = 0.16666666666666666;
        double r1550450 = r1550448 * r1550449;
        double r1550451 = 2.0;
        double r1550452 = atan2(1.0, 0.0);
        double r1550453 = r1550451 * r1550452;
        double r1550454 = u2;
        double r1550455 = r1550453 * r1550454;
        double r1550456 = cos(r1550455);
        double r1550457 = r1550450 * r1550456;
        double r1550458 = r1550457 + r1550447;
        return r1550458;
}

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

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.4

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

  7. Applied associate-*r*0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019137 
(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))