Average Error: 0.4 → 0.3
Time: 2.8m
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{\frac{1}{6}} \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right)\right) \cdot \cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + 0.5\]
\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(\sqrt{\frac{1}{6}} \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right)\right) \cdot \cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + 0.5
double f(double u1, double u2) {
        double r15986260 = 1.0;
        double r15986261 = 6.0;
        double r15986262 = r15986260 / r15986261;
        double r15986263 = -2.0;
        double r15986264 = u1;
        double r15986265 = log(r15986264);
        double r15986266 = r15986263 * r15986265;
        double r15986267 = 0.5;
        double r15986268 = pow(r15986266, r15986267);
        double r15986269 = r15986262 * r15986268;
        double r15986270 = 2.0;
        double r15986271 = atan2(1.0, 0.0);
        double r15986272 = r15986270 * r15986271;
        double r15986273 = u2;
        double r15986274 = r15986272 * r15986273;
        double r15986275 = cos(r15986274);
        double r15986276 = r15986269 * r15986275;
        double r15986277 = r15986276 + r15986267;
        return r15986277;
}
↓
double f(double u1, double u2) {
        double r15986278 = 0.16666666666666666;
        double r15986279 = sqrt(r15986278);
        double r15986280 = -2.0;
        double r15986281 = u1;
        double r15986282 = log(r15986281);
        double r15986283 = r15986280 * r15986282;
        double r15986284 = 0.5;
        double r15986285 = pow(r15986283, r15986284);
        double r15986286 = r15986285 * r15986279;
        double r15986287 = r15986279 * r15986286;
        double r15986288 = atan2(1.0, 0.0);
        double r15986289 = 2.0;
        double r15986290 = u2;
        double r15986291 = r15986289 * r15986290;
        double r15986292 = r15986288 * r15986291;
        double r15986293 = cos(r15986292);
        double r15986294 = r15986287 * r15986293;
        double r15986295 = r15986294 + r15986284;
        return r15986295;
}

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. Simplified0.4

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019125 
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0 u1 1) (<= 0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))