Average Error: 0.4 → 0.3
Time: 9.9s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\]
\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 + \frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)
double f(double u1, double u2) {
        double r60570 = 1.0;
        double r60571 = 6.0;
        double r60572 = r60570 / r60571;
        double r60573 = -2.0;
        double r60574 = u1;
        double r60575 = log(r60574);
        double r60576 = r60573 * r60575;
        double r60577 = 0.5;
        double r60578 = pow(r60576, r60577);
        double r60579 = r60572 * r60578;
        double r60580 = 2.0;
        double r60581 = atan2(1.0, 0.0);
        double r60582 = r60580 * r60581;
        double r60583 = u2;
        double r60584 = r60582 * r60583;
        double r60585 = cos(r60584);
        double r60586 = r60579 * r60585;
        double r60587 = r60586 + r60577;
        return r60587;
}
↓
double f(double u1, double u2) {
        double r60588 = 0.5;
        double r60589 = 1.0;
        double r60590 = -2.0;
        double r60591 = u1;
        double r60592 = log(r60591);
        double r60593 = r60590 * r60592;
        double r60594 = pow(r60593, r60588);
        double r60595 = r60589 * r60594;
        double r60596 = 6.0;
        double r60597 = r60595 / r60596;
        double r60598 = 2.0;
        double r60599 = atan2(1.0, 0.0);
        double r60600 = r60598 * r60599;
        double r60601 = u2;
        double r60602 = r60600 * r60601;
        double r60603 = cos(r60602);
        double r60604 = r60597 * r60603;
        double r60605 = r60588 + r60604;
        return r60605;
}

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 +-commutative0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(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))