Average Error: 0.4 → 0.4
Time: 33.5s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + 0.5\]
\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({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + 0.5
double f(double u1, double u2) {
        double r2101451 = 1.0;
        double r2101452 = 6.0;
        double r2101453 = r2101451 / r2101452;
        double r2101454 = -2.0;
        double r2101455 = u1;
        double r2101456 = log(r2101455);
        double r2101457 = r2101454 * r2101456;
        double r2101458 = 0.5;
        double r2101459 = pow(r2101457, r2101458);
        double r2101460 = r2101453 * r2101459;
        double r2101461 = 2.0;
        double r2101462 = atan2(1.0, 0.0);
        double r2101463 = r2101461 * r2101462;
        double r2101464 = u2;
        double r2101465 = r2101463 * r2101464;
        double r2101466 = cos(r2101465);
        double r2101467 = r2101460 * r2101466;
        double r2101468 = r2101467 + r2101458;
        return r2101468;
}
↓
double f(double u1, double u2) {
        double r2101469 = -2.0;
        double r2101470 = u1;
        double r2101471 = log(r2101470);
        double r2101472 = r2101469 * r2101471;
        double r2101473 = 0.5;
        double r2101474 = pow(r2101472, r2101473);
        double r2101475 = 0.16666666666666666;
        double r2101476 = r2101474 * r2101475;
        double r2101477 = 2.0;
        double r2101478 = atan2(1.0, 0.0);
        double r2101479 = u2;
        double r2101480 = r2101478 * r2101479;
        double r2101481 = r2101477 * r2101480;
        double r2101482 = cos(r2101481);
        double r2101483 = r2101476 * r2101482;
        double r2101484 = r2101483 + r2101473;
        return r2101484;
}

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

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.3

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

  7. Applied associate-*r*0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019163 
(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))