Average Error: 0.4 → 0.3
Time: 38.7s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + 0.5\]
\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 r3081508 = 1.0;
        double r3081509 = 6.0;
        double r3081510 = r3081508 / r3081509;
        double r3081511 = -2.0;
        double r3081512 = u1;
        double r3081513 = log(r3081512);
        double r3081514 = r3081511 * r3081513;
        double r3081515 = 0.5;
        double r3081516 = pow(r3081514, r3081515);
        double r3081517 = r3081510 * r3081516;
        double r3081518 = 2.0;
        double r3081519 = atan2(1.0, 0.0);
        double r3081520 = r3081518 * r3081519;
        double r3081521 = u2;
        double r3081522 = r3081520 * r3081521;
        double r3081523 = cos(r3081522);
        double r3081524 = r3081517 * r3081523;
        double r3081525 = r3081524 + r3081515;
        return r3081525;
}
↓
double f(double u1, double u2) {
        double r3081526 = u1;
        double r3081527 = log(r3081526);
        double r3081528 = -2.0;
        double r3081529 = r3081527 * r3081528;
        double r3081530 = 0.5;
        double r3081531 = pow(r3081529, r3081530);
        double r3081532 = 0.16666666666666666;
        double r3081533 = sqrt(r3081532);
        double r3081534 = r3081531 * r3081533;
        double r3081535 = r3081534 * r3081533;
        double r3081536 = 2.0;
        double r3081537 = atan2(1.0, 0.0);
        double r3081538 = u2;
        double r3081539 = r3081537 * r3081538;
        double r3081540 = r3081536 * r3081539;
        double r3081541 = cos(r3081540);
        double r3081542 = r3081535 * r3081541;
        double r3081543 = r3081542 + r3081530;
        return r3081543;
}

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*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019162 
(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))