Average Error: 0.4 → 0.4
Time: 18.7s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \left(0.1666666666666666574148081281236954964697 \cdot {\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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 \left(0.1666666666666666574148081281236954964697 \cdot {\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r75493 = 1.0;
        double r75494 = 6.0;
        double r75495 = r75493 / r75494;
        double r75496 = -2.0;
        double r75497 = u1;
        double r75498 = log(r75497);
        double r75499 = r75496 * r75498;
        double r75500 = 0.5;
        double r75501 = pow(r75499, r75500);
        double r75502 = r75495 * r75501;
        double r75503 = 2.0;
        double r75504 = atan2(1.0, 0.0);
        double r75505 = r75503 * r75504;
        double r75506 = u2;
        double r75507 = r75505 * r75506;
        double r75508 = cos(r75507);
        double r75509 = r75502 * r75508;
        double r75510 = r75509 + r75500;
        return r75510;
}
↓
double f(double u1, double u2) {
        double r75511 = 1.0;
        double r75512 = 0.16666666666666666;
        double r75513 = u1;
        double r75514 = log(r75513);
        double r75515 = pow(r75514, r75511);
        double r75516 = -2.0;
        double r75517 = pow(r75516, r75511);
        double r75518 = r75515 * r75517;
        double r75519 = 0.5;
        double r75520 = pow(r75518, r75519);
        double r75521 = r75512 * r75520;
        double r75522 = r75511 * r75521;
        double r75523 = 2.0;
        double r75524 = atan2(1.0, 0.0);
        double r75525 = r75523 * r75524;
        double r75526 = u2;
        double r75527 = r75525 * r75526;
        double r75528 = cos(r75527);
        double r75529 = r75522 * r75528;
        double r75530 = r75529 + r75519;
        return r75530;
}

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. Taylor expanded around 0 0.4

    \[\leadsto \left(1 \cdot \color{blue}{\left(0.1666666666666666574148081281236954964697 \cdot {\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  7. Final simplification0.4

    \[\leadsto \left(1 \cdot \left(0.1666666666666666574148081281236954964697 \cdot {\left({\left(\log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

Reproduce

herbie shell --seed 2019308 
(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))