Average Error: 0.4 → 0.3
Time: 25.4s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\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
double f(double u1, double u2) {
        double r74501 = 1.0;
        double r74502 = 6.0;
        double r74503 = r74501 / r74502;
        double r74504 = -2.0;
        double r74505 = u1;
        double r74506 = log(r74505);
        double r74507 = r74504 * r74506;
        double r74508 = 0.5;
        double r74509 = pow(r74507, r74508);
        double r74510 = r74503 * r74509;
        double r74511 = 2.0;
        double r74512 = atan2(1.0, 0.0);
        double r74513 = r74511 * r74512;
        double r74514 = u2;
        double r74515 = r74513 * r74514;
        double r74516 = cos(r74515);
        double r74517 = r74510 * r74516;
        double r74518 = r74517 + r74508;
        return r74518;
}
↓
double f(double u1, double u2) {
        double r74519 = 1.0;
        double r74520 = 6.0;
        double r74521 = r74519 / r74520;
        double r74522 = sqrt(r74521);
        double r74523 = -2.0;
        double r74524 = u1;
        double r74525 = log(r74524);
        double r74526 = r74523 * r74525;
        double r74527 = 0.5;
        double r74528 = pow(r74526, r74527);
        double r74529 = r74522 * r74528;
        double r74530 = r74522 * r74529;
        double r74531 = 2.0;
        double r74532 = atan2(1.0, 0.0);
        double r74533 = r74531 * r74532;
        double r74534 = u2;
        double r74535 = r74533 * r74534;
        double r74536 = cos(r74535);
        double r74537 = r74530 * r74536;
        double r74538 = r74537 + r74527;
        return r74538;
}

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 add-sqr-sqrt0.4

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

  4. Applied associate-*l*0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019347 
(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))