Average Error: 0.4 → 0.3
Time: 26.1s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \frac{{\left(-2 \cdot \log u1\right)}^{0.5} \cdot 1}{6} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\]
\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
↓
0.5 + \frac{{\left(-2 \cdot \log u1\right)}^{0.5} \cdot 1}{6} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)
double f(double u1, double u2) {
        double r75306 = 1.0;
        double r75307 = 6.0;
        double r75308 = r75306 / r75307;
        double r75309 = -2.0;
        double r75310 = u1;
        double r75311 = log(r75310);
        double r75312 = r75309 * r75311;
        double r75313 = 0.5;
        double r75314 = pow(r75312, r75313);
        double r75315 = r75308 * r75314;
        double r75316 = 2.0;
        double r75317 = atan2(1.0, 0.0);
        double r75318 = r75316 * r75317;
        double r75319 = u2;
        double r75320 = r75318 * r75319;
        double r75321 = cos(r75320);
        double r75322 = r75315 * r75321;
        double r75323 = r75322 + r75313;
        return r75323;
}
↓
double f(double u1, double u2) {
        double r75324 = 0.5;
        double r75325 = -2.0;
        double r75326 = u1;
        double r75327 = log(r75326);
        double r75328 = r75325 * r75327;
        double r75329 = pow(r75328, r75324);
        double r75330 = 1.0;
        double r75331 = r75329 * r75330;
        double r75332 = 6.0;
        double r75333 = r75331 / r75332;
        double r75334 = u2;
        double r75335 = 2.0;
        double r75336 = atan2(1.0, 0.0);
        double r75337 = r75335 * r75336;
        double r75338 = r75334 * r75337;
        double r75339 = cos(r75338);
        double r75340 = r75333 * r75339;
        double r75341 = r75324 + r75340;
        return r75341;
}

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. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019174 
(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))