Average Error: 0.4 → 0.4
Time: 10.2s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\frac{1}{6} \cdot {\left(2 \cdot \log \left(\frac{1}{u1}\right)\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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(\frac{1}{6} \cdot {\left(2 \cdot \log \left(\frac{1}{u1}\right)\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r61241 = 1.0;
        double r61242 = 6.0;
        double r61243 = r61241 / r61242;
        double r61244 = -2.0;
        double r61245 = u1;
        double r61246 = log(r61245);
        double r61247 = r61244 * r61246;
        double r61248 = 0.5;
        double r61249 = pow(r61247, r61248);
        double r61250 = r61243 * r61249;
        double r61251 = 2.0;
        double r61252 = atan2(1.0, 0.0);
        double r61253 = r61251 * r61252;
        double r61254 = u2;
        double r61255 = r61253 * r61254;
        double r61256 = cos(r61255);
        double r61257 = r61250 * r61256;
        double r61258 = r61257 + r61248;
        return r61258;
}
↓
double f(double u1, double u2) {
        double r61259 = 1.0;
        double r61260 = 6.0;
        double r61261 = r61259 / r61260;
        double r61262 = 2.0;
        double r61263 = 1.0;
        double r61264 = u1;
        double r61265 = r61263 / r61264;
        double r61266 = log(r61265);
        double r61267 = r61262 * r61266;
        double r61268 = 0.5;
        double r61269 = pow(r61267, r61268);
        double r61270 = r61261 * r61269;
        double r61271 = atan2(1.0, 0.0);
        double r61272 = r61262 * r61271;
        double r61273 = u2;
        double r61274 = r61272 * r61273;
        double r61275 = cos(r61274);
        double r61276 = r61270 * r61275;
        double r61277 = r61276 + r61268;
        return r61277;
}

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. Taylor expanded around inf 0.4

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2019352 
(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))