Average Error: 0.4 → 0.3
Time: 31.4s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\left(0.1666666666666666574148081281236954964697 \cdot {\left({-1}^{1} \cdot {-2}^{1}\right)}^{0.5}\right) \cdot {\left({\left(-\log u1\right)}^{1}\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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(\left(0.1666666666666666574148081281236954964697 \cdot {\left({-1}^{1} \cdot {-2}^{1}\right)}^{0.5}\right) \cdot {\left({\left(-\log u1\right)}^{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 r87475 = 1.0;
        double r87476 = 6.0;
        double r87477 = r87475 / r87476;
        double r87478 = -2.0;
        double r87479 = u1;
        double r87480 = log(r87479);
        double r87481 = r87478 * r87480;
        double r87482 = 0.5;
        double r87483 = pow(r87481, r87482);
        double r87484 = r87477 * r87483;
        double r87485 = 2.0;
        double r87486 = atan2(1.0, 0.0);
        double r87487 = r87485 * r87486;
        double r87488 = u2;
        double r87489 = r87487 * r87488;
        double r87490 = cos(r87489);
        double r87491 = r87484 * r87490;
        double r87492 = r87491 + r87482;
        return r87492;
}
↓
double f(double u1, double u2) {
        double r87493 = 0.16666666666666666;
        double r87494 = -1.0;
        double r87495 = 1.0;
        double r87496 = pow(r87494, r87495);
        double r87497 = -2.0;
        double r87498 = pow(r87497, r87495);
        double r87499 = r87496 * r87498;
        double r87500 = 0.5;
        double r87501 = pow(r87499, r87500);
        double r87502 = r87493 * r87501;
        double r87503 = u1;
        double r87504 = log(r87503);
        double r87505 = -r87504;
        double r87506 = pow(r87505, r87495);
        double r87507 = pow(r87506, r87500);
        double r87508 = r87502 * r87507;
        double r87509 = 2.0;
        double r87510 = atan2(1.0, 0.0);
        double r87511 = r87509 * r87510;
        double r87512 = u2;
        double r87513 = r87511 * r87512;
        double r87514 = cos(r87513);
        double r87515 = r87508 * r87514;
        double r87516 = r87515 + r87500;
        return r87516;
}

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 add-sqr-sqrt0.4

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

  4. Applied associate-*l*0.3

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

  5. Taylor expanded around inf 0.4

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

  6. Simplified0.4

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

  8. Applied unpow-prod-down0.3

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

  9. Applied associate-*r*0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019323 
(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))