Average Error: 0.4 → 0.4
Time: 38.1s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + {\left(\log u1 \cdot -2\right)}^{0.5} \cdot \left(\frac{1}{6} \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right)\]
\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 + {\left(\log u1 \cdot -2\right)}^{0.5} \cdot \left(\frac{1}{6} \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right)
double f(double u1, double u2) {
        double r2049316 = 1.0;
        double r2049317 = 6.0;
        double r2049318 = r2049316 / r2049317;
        double r2049319 = -2.0;
        double r2049320 = u1;
        double r2049321 = log(r2049320);
        double r2049322 = r2049319 * r2049321;
        double r2049323 = 0.5;
        double r2049324 = pow(r2049322, r2049323);
        double r2049325 = r2049318 * r2049324;
        double r2049326 = 2.0;
        double r2049327 = atan2(1.0, 0.0);
        double r2049328 = r2049326 * r2049327;
        double r2049329 = u2;
        double r2049330 = r2049328 * r2049329;
        double r2049331 = cos(r2049330);
        double r2049332 = r2049325 * r2049331;
        double r2049333 = r2049332 + r2049323;
        return r2049333;
}
↓
double f(double u1, double u2) {
        double r2049334 = 0.5;
        double r2049335 = u1;
        double r2049336 = log(r2049335);
        double r2049337 = -2.0;
        double r2049338 = r2049336 * r2049337;
        double r2049339 = pow(r2049338, r2049334);
        double r2049340 = 1.0;
        double r2049341 = 6.0;
        double r2049342 = r2049340 / r2049341;
        double r2049343 = u2;
        double r2049344 = atan2(1.0, 0.0);
        double r2049345 = 2.0;
        double r2049346 = r2049344 * r2049345;
        double r2049347 = r2049343 * r2049346;
        double r2049348 = cos(r2049347);
        double r2049349 = r2049342 * r2049348;
        double r2049350 = r2049339 * r2049349;
        double r2049351 = r2049334 + r2049350;
        return r2049351;
}

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 flip-+0.4

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

  5. Applied *-un-lft-identity0.4

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

  6. Applied *-un-lft-identity0.4

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

  7. Applied times-frac0.4

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019171 
(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))