Average Error: 0.4 → 0.3
Time: 30.4s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \frac{{\left(\log u1 \cdot -2\right)}^{0.5}}{6}\right) \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) + 0.5\]
\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(1 \cdot \frac{{\left(\log u1 \cdot -2\right)}^{0.5}}{6}\right) \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) + 0.5
double f(double u1, double u2) {
        double r1992329 = 1.0;
        double r1992330 = 6.0;
        double r1992331 = r1992329 / r1992330;
        double r1992332 = -2.0;
        double r1992333 = u1;
        double r1992334 = log(r1992333);
        double r1992335 = r1992332 * r1992334;
        double r1992336 = 0.5;
        double r1992337 = pow(r1992335, r1992336);
        double r1992338 = r1992331 * r1992337;
        double r1992339 = 2.0;
        double r1992340 = atan2(1.0, 0.0);
        double r1992341 = r1992339 * r1992340;
        double r1992342 = u2;
        double r1992343 = r1992341 * r1992342;
        double r1992344 = cos(r1992343);
        double r1992345 = r1992338 * r1992344;
        double r1992346 = r1992345 + r1992336;
        return r1992346;
}
↓
double f(double u1, double u2) {
        double r1992347 = 1.0;
        double r1992348 = u1;
        double r1992349 = log(r1992348);
        double r1992350 = -2.0;
        double r1992351 = r1992349 * r1992350;
        double r1992352 = 0.5;
        double r1992353 = pow(r1992351, r1992352);
        double r1992354 = 6.0;
        double r1992355 = r1992353 / r1992354;
        double r1992356 = r1992347 * r1992355;
        double r1992357 = u2;
        double r1992358 = 2.0;
        double r1992359 = atan2(1.0, 0.0);
        double r1992360 = r1992358 * r1992359;
        double r1992361 = r1992357 * r1992360;
        double r1992362 = cos(r1992361);
        double r1992363 = r1992356 * r1992362;
        double r1992364 = r1992363 + r1992352;
        return r1992364;
}

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

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0.0 u1 1.0) (<= 0.0 u2 1.0))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))