Average Error: 0.4 → 0.3
Time: 30.5s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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 u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r72403 = 1.0;
        double r72404 = 6.0;
        double r72405 = r72403 / r72404;
        double r72406 = -2.0;
        double r72407 = u1;
        double r72408 = log(r72407);
        double r72409 = r72406 * r72408;
        double r72410 = 0.5;
        double r72411 = pow(r72409, r72410);
        double r72412 = r72405 * r72411;
        double r72413 = 2.0;
        double r72414 = atan2(1.0, 0.0);
        double r72415 = r72413 * r72414;
        double r72416 = u2;
        double r72417 = r72415 * r72416;
        double r72418 = cos(r72417);
        double r72419 = r72412 * r72418;
        double r72420 = r72419 + r72410;
        return r72420;
}
↓
double f(double u1, double u2) {
        double r72421 = 1.0;
        double r72422 = -2.0;
        double r72423 = u1;
        double r72424 = log(r72423);
        double r72425 = r72422 * r72424;
        double r72426 = 0.5;
        double r72427 = pow(r72425, r72426);
        double r72428 = r72421 * r72427;
        double r72429 = 6.0;
        double r72430 = r72428 / r72429;
        double r72431 = 2.0;
        double r72432 = atan2(1.0, 0.0);
        double r72433 = r72431 * r72432;
        double r72434 = u2;
        double r72435 = r72433 * r72434;
        double r72436 = cos(r72435);
        double r72437 = r72430 * r72436;
        double r72438 = r72437 + r72426;
        return r72438;
}

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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019322 
(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))