Average Error: 0.4 → 0.3
Time: 12.9s
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 r81387 = 1.0;
        double r81388 = 6.0;
        double r81389 = r81387 / r81388;
        double r81390 = -2.0;
        double r81391 = u1;
        double r81392 = log(r81391);
        double r81393 = r81390 * r81392;
        double r81394 = 0.5;
        double r81395 = pow(r81393, r81394);
        double r81396 = r81389 * r81395;
        double r81397 = 2.0;
        double r81398 = atan2(1.0, 0.0);
        double r81399 = r81397 * r81398;
        double r81400 = u2;
        double r81401 = r81399 * r81400;
        double r81402 = cos(r81401);
        double r81403 = r81396 * r81402;
        double r81404 = r81403 + r81394;
        return r81404;
}
↓
double f(double u1, double u2) {
        double r81405 = 1.0;
        double r81406 = -2.0;
        double r81407 = u1;
        double r81408 = log(r81407);
        double r81409 = r81406 * r81408;
        double r81410 = 0.5;
        double r81411 = pow(r81409, r81410);
        double r81412 = 6.0;
        double r81413 = r81411 / r81412;
        double r81414 = r81405 * r81413;
        double r81415 = 2.0;
        double r81416 = atan2(1.0, 0.0);
        double r81417 = r81415 * r81416;
        double r81418 = u2;
        double r81419 = r81417 * r81418;
        double r81420 = cos(r81419);
        double r81421 = r81414 * r81420;
        double r81422 = r81421 + r81410;
        return r81422;
}

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 2020039 +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))