Average Error: 0.4 → 0.3
Time: 12.4s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r75546 = 1.0;
        double r75547 = 6.0;
        double r75548 = r75546 / r75547;
        double r75549 = -2.0;
        double r75550 = u1;
        double r75551 = log(r75550);
        double r75552 = r75549 * r75551;
        double r75553 = 0.5;
        double r75554 = pow(r75552, r75553);
        double r75555 = r75548 * r75554;
        double r75556 = 2.0;
        double r75557 = atan2(1.0, 0.0);
        double r75558 = r75556 * r75557;
        double r75559 = u2;
        double r75560 = r75558 * r75559;
        double r75561 = cos(r75560);
        double r75562 = r75555 * r75561;
        double r75563 = r75562 + r75553;
        return r75563;
}
↓
double f(double u1, double u2) {
        double r75564 = 1.0;
        double r75565 = -2.0;
        double r75566 = u1;
        double r75567 = log(r75566);
        double r75568 = r75565 * r75567;
        double r75569 = 0.5;
        double r75570 = pow(r75568, r75569);
        double r75571 = 6.0;
        double r75572 = r75570 / r75571;
        double r75573 = r75564 * r75572;
        double r75574 = 2.0;
        double r75575 = atan2(1.0, 0.0);
        double r75576 = r75574 * r75575;
        double r75577 = u2;
        double r75578 = r75576 * r75577;
        double r75579 = cos(r75578);
        double r75580 = r75573 * r75579;
        double r75581 = r75580 + r75569;
        return r75581;
}

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 div-inv0.4

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

  4. Applied associate-*l*0.4

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2020056 +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))