Average Error: 0.4 → 0.3
Time: 9.8s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{1 \cdot {\left(2 \cdot \log \left(\frac{1}{u1}\right)\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\frac{1 \cdot {\left(2 \cdot \log \left(\frac{1}{u1}\right)\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r60575 = 1.0;
        double r60576 = 6.0;
        double r60577 = r60575 / r60576;
        double r60578 = -2.0;
        double r60579 = u1;
        double r60580 = log(r60579);
        double r60581 = r60578 * r60580;
        double r60582 = 0.5;
        double r60583 = pow(r60581, r60582);
        double r60584 = r60577 * r60583;
        double r60585 = 2.0;
        double r60586 = atan2(1.0, 0.0);
        double r60587 = r60585 * r60586;
        double r60588 = u2;
        double r60589 = r60587 * r60588;
        double r60590 = cos(r60589);
        double r60591 = r60584 * r60590;
        double r60592 = r60591 + r60582;
        return r60592;
}
↓
double f(double u1, double u2) {
        double r60593 = 1.0;
        double r60594 = 2.0;
        double r60595 = 1.0;
        double r60596 = u1;
        double r60597 = r60595 / r60596;
        double r60598 = log(r60597);
        double r60599 = r60594 * r60598;
        double r60600 = 0.5;
        double r60601 = pow(r60599, r60600);
        double r60602 = r60593 * r60601;
        double r60603 = 6.0;
        double r60604 = r60602 / r60603;
        double r60605 = atan2(1.0, 0.0);
        double r60606 = r60594 * r60605;
        double r60607 = u2;
        double r60608 = r60606 * r60607;
        double r60609 = cos(r60608);
        double r60610 = r60604 * r60609;
        double r60611 = r60610 + r60600;
        return r60611;
}

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. Taylor expanded around inf 0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2020060 
(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))