Average Error: 0.4 → 0.3
Time: 1.7m
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{{\left(\log u1 \cdot -2\right)}^{0.5} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)}{6} + 0.5\]
\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{{\left(\log u1 \cdot -2\right)}^{0.5} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)}{6} + 0.5
double f(double u1, double u2) {
        double r10101305 = 1.0;
        double r10101306 = 6.0;
        double r10101307 = r10101305 / r10101306;
        double r10101308 = -2.0;
        double r10101309 = u1;
        double r10101310 = log(r10101309);
        double r10101311 = r10101308 * r10101310;
        double r10101312 = 0.5;
        double r10101313 = pow(r10101311, r10101312);
        double r10101314 = r10101307 * r10101313;
        double r10101315 = 2.0;
        double r10101316 = atan2(1.0, 0.0);
        double r10101317 = r10101315 * r10101316;
        double r10101318 = u2;
        double r10101319 = r10101317 * r10101318;
        double r10101320 = cos(r10101319);
        double r10101321 = r10101314 * r10101320;
        double r10101322 = r10101321 + r10101312;
        return r10101322;
}
↓
double f(double u1, double u2) {
        double r10101323 = u1;
        double r10101324 = log(r10101323);
        double r10101325 = -2.0;
        double r10101326 = r10101324 * r10101325;
        double r10101327 = 0.5;
        double r10101328 = pow(r10101326, r10101327);
        double r10101329 = u2;
        double r10101330 = 2.0;
        double r10101331 = atan2(1.0, 0.0);
        double r10101332 = r10101330 * r10101331;
        double r10101333 = r10101329 * r10101332;
        double r10101334 = cos(r10101333);
        double r10101335 = r10101328 * r10101334;
        double r10101336 = 6.0;
        double r10101337 = r10101335 / r10101336;
        double r10101338 = r10101337 + r10101327;
        return r10101338;
}

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

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

  4. Applied associate-*l/0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019104 
(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))