Average Error: 0.4 → 0.4
Time: 31.5s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) + 0.5\]
\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
↓
\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) + 0.5
double f(double u1, double u2) {
        double r84445 = 1.0;
        double r84446 = 6.0;
        double r84447 = r84445 / r84446;
        double r84448 = -2.0;
        double r84449 = u1;
        double r84450 = log(r84449);
        double r84451 = r84448 * r84450;
        double r84452 = 0.5;
        double r84453 = pow(r84451, r84452);
        double r84454 = r84447 * r84453;
        double r84455 = 2.0;
        double r84456 = atan2(1.0, 0.0);
        double r84457 = r84455 * r84456;
        double r84458 = u2;
        double r84459 = r84457 * r84458;
        double r84460 = cos(r84459);
        double r84461 = r84454 * r84460;
        double r84462 = r84461 + r84452;
        return r84462;
}
↓
double f(double u1, double u2) {
        double r84463 = 2.0;
        double r84464 = atan2(1.0, 0.0);
        double r84465 = r84463 * r84464;
        double r84466 = u2;
        double r84467 = r84465 * r84466;
        double r84468 = cos(r84467);
        double r84469 = 1.0;
        double r84470 = 6.0;
        double r84471 = r84469 / r84470;
        double r84472 = -2.0;
        double r84473 = u1;
        double r84474 = log(r84473);
        double r84475 = r84472 * r84474;
        double r84476 = 0.5;
        double r84477 = pow(r84475, r84476);
        double r84478 = r84471 * r84477;
        double r84479 = r84468 * r84478;
        double r84480 = r84479 + r84476;
        return r84480;
}

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*0.4

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

  6. Applied pow10.4

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

  7. Applied pow10.4

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

  8. Applied pow-prod-down0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019326 
(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))