Average Error: 0.4 → 0.3
Time: 43.3s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) + 0.5\]
\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
↓
\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) + 0.5
double f(double u1, double u2) {
        double r1661383 = 1.0;
        double r1661384 = 6.0;
        double r1661385 = r1661383 / r1661384;
        double r1661386 = -2.0;
        double r1661387 = u1;
        double r1661388 = log(r1661387);
        double r1661389 = r1661386 * r1661388;
        double r1661390 = 0.5;
        double r1661391 = pow(r1661389, r1661390);
        double r1661392 = r1661385 * r1661391;
        double r1661393 = 2.0;
        double r1661394 = atan2(1.0, 0.0);
        double r1661395 = r1661393 * r1661394;
        double r1661396 = u2;
        double r1661397 = r1661395 * r1661396;
        double r1661398 = cos(r1661397);
        double r1661399 = r1661392 * r1661398;
        double r1661400 = r1661399 + r1661390;
        return r1661400;
}
↓
double f(double u1, double u2) {
        double r1661401 = -2.0;
        double r1661402 = u1;
        double r1661403 = log(r1661402);
        double r1661404 = r1661401 * r1661403;
        double r1661405 = 0.5;
        double r1661406 = pow(r1661404, r1661405);
        double r1661407 = 6.0;
        double r1661408 = r1661406 / r1661407;
        double r1661409 = u2;
        double r1661410 = 2.0;
        double r1661411 = atan2(1.0, 0.0);
        double r1661412 = r1661410 * r1661411;
        double r1661413 = r1661409 * r1661412;
        double r1661414 = cos(r1661413);
        double r1661415 = r1661408 * r1661414;
        double r1661416 = r1661415 + r1661405;
        return r1661416;
}

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 div-inv0.4

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

  5. Applied associate-*l*0.4

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019151 
(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))