Average Error: 0.4 → 0.4
Time: 38.7s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{\frac{1}{6}} \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right)\right) \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) + 0.5\]
\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(\sqrt{\frac{1}{6}} \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right)\right) \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) + 0.5
double f(double u1, double u2) {
        double r2065365 = 1.0;
        double r2065366 = 6.0;
        double r2065367 = r2065365 / r2065366;
        double r2065368 = -2.0;
        double r2065369 = u1;
        double r2065370 = log(r2065369);
        double r2065371 = r2065368 * r2065370;
        double r2065372 = 0.5;
        double r2065373 = pow(r2065371, r2065372);
        double r2065374 = r2065367 * r2065373;
        double r2065375 = 2.0;
        double r2065376 = atan2(1.0, 0.0);
        double r2065377 = r2065375 * r2065376;
        double r2065378 = u2;
        double r2065379 = r2065377 * r2065378;
        double r2065380 = cos(r2065379);
        double r2065381 = r2065374 * r2065380;
        double r2065382 = r2065381 + r2065372;
        return r2065382;
}
↓
double f(double u1, double u2) {
        double r2065383 = 0.16666666666666666;
        double r2065384 = sqrt(r2065383);
        double r2065385 = -2.0;
        double r2065386 = u1;
        double r2065387 = log(r2065386);
        double r2065388 = r2065385 * r2065387;
        double r2065389 = 0.5;
        double r2065390 = pow(r2065388, r2065389);
        double r2065391 = r2065390 * r2065384;
        double r2065392 = r2065384 * r2065391;
        double r2065393 = u2;
        double r2065394 = 2.0;
        double r2065395 = atan2(1.0, 0.0);
        double r2065396 = r2065394 * r2065395;
        double r2065397 = r2065393 * r2065396;
        double r2065398 = cos(r2065397);
        double r2065399 = r2065392 * r2065398;
        double r2065400 = r2065399 + r2065389;
        return r2065400;
}

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

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019129 
(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))