Average Error: 0.4 → 0.3
Time: 27.9s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \left(\left(\sqrt{\frac{1}{6}} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\]
\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
↓
0.5 + \left(\left(\sqrt{\frac{1}{6}} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)
double f(double u1, double u2) {
        double r1481528 = 1.0;
        double r1481529 = 6.0;
        double r1481530 = r1481528 / r1481529;
        double r1481531 = -2.0;
        double r1481532 = u1;
        double r1481533 = log(r1481532);
        double r1481534 = r1481531 * r1481533;
        double r1481535 = 0.5;
        double r1481536 = pow(r1481534, r1481535);
        double r1481537 = r1481530 * r1481536;
        double r1481538 = 2.0;
        double r1481539 = atan2(1.0, 0.0);
        double r1481540 = r1481538 * r1481539;
        double r1481541 = u2;
        double r1481542 = r1481540 * r1481541;
        double r1481543 = cos(r1481542);
        double r1481544 = r1481537 * r1481543;
        double r1481545 = r1481544 + r1481535;
        return r1481545;
}
↓
double f(double u1, double u2) {
        double r1481546 = 0.5;
        double r1481547 = 1.0;
        double r1481548 = 6.0;
        double r1481549 = r1481547 / r1481548;
        double r1481550 = sqrt(r1481549);
        double r1481551 = u1;
        double r1481552 = log(r1481551);
        double r1481553 = -2.0;
        double r1481554 = r1481552 * r1481553;
        double r1481555 = pow(r1481554, r1481546);
        double r1481556 = r1481550 * r1481555;
        double r1481557 = r1481556 * r1481550;
        double r1481558 = u2;
        double r1481559 = atan2(1.0, 0.0);
        double r1481560 = 2.0;
        double r1481561 = r1481559 * r1481560;
        double r1481562 = r1481558 * r1481561;
        double r1481563 = cos(r1481562);
        double r1481564 = r1481557 * r1481563;
        double r1481565 = r1481546 + r1481564;
        return r1481565;
}

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

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

Reproduce

herbie shell --seed 2019192 
(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))