Average Error: 0.4 → 0.4
Time: 11.2s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(0.166666666666666657 \cdot {\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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(0.166666666666666657 \cdot {\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r66535 = 1.0;
        double r66536 = 6.0;
        double r66537 = r66535 / r66536;
        double r66538 = -2.0;
        double r66539 = u1;
        double r66540 = log(r66539);
        double r66541 = r66538 * r66540;
        double r66542 = 0.5;
        double r66543 = pow(r66541, r66542);
        double r66544 = r66537 * r66543;
        double r66545 = 2.0;
        double r66546 = atan2(1.0, 0.0);
        double r66547 = r66545 * r66546;
        double r66548 = u2;
        double r66549 = r66547 * r66548;
        double r66550 = cos(r66549);
        double r66551 = r66544 * r66550;
        double r66552 = r66551 + r66542;
        return r66552;
}
↓
double f(double u1, double u2) {
        double r66553 = 0.16666666666666666;
        double r66554 = u1;
        double r66555 = log(r66554);
        double r66556 = 1.0;
        double r66557 = pow(r66555, r66556);
        double r66558 = -2.0;
        double r66559 = pow(r66558, r66556);
        double r66560 = r66557 * r66559;
        double r66561 = 0.5;
        double r66562 = pow(r66560, r66561);
        double r66563 = r66553 * r66562;
        double r66564 = 2.0;
        double r66565 = atan2(1.0, 0.0);
        double r66566 = r66564 * r66565;
        double r66567 = u2;
        double r66568 = r66566 * r66567;
        double r66569 = cos(r66568);
        double r66570 = r66563 * r66569;
        double r66571 = r66570 + r66561;
        return r66571;
}

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 associate-*l/0.3

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

  5. Applied associate-/l*0.3

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

  6. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{\left(0.166666666666666657 \cdot {\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  7. Final simplification0.4

    \[\leadsto \left(0.166666666666666657 \cdot {\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

Reproduce

herbie shell --seed 2020062 
(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))