Average Error: 0.4 → 0.4
Time: 27.8s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r72465 = 1.0;
        double r72466 = 6.0;
        double r72467 = r72465 / r72466;
        double r72468 = -2.0;
        double r72469 = u1;
        double r72470 = log(r72469);
        double r72471 = r72468 * r72470;
        double r72472 = 0.5;
        double r72473 = pow(r72471, r72472);
        double r72474 = r72467 * r72473;
        double r72475 = 2.0;
        double r72476 = atan2(1.0, 0.0);
        double r72477 = r72475 * r72476;
        double r72478 = u2;
        double r72479 = r72477 * r72478;
        double r72480 = cos(r72479);
        double r72481 = r72474 * r72480;
        double r72482 = r72481 + r72472;
        return r72482;
}
↓
double f(double u1, double u2) {
        double r72483 = 1.0;
        double r72484 = 6.0;
        double r72485 = r72483 / r72484;
        double r72486 = sqrt(r72485);
        double r72487 = -2.0;
        double r72488 = u1;
        double r72489 = log(r72488);
        double r72490 = r72487 * r72489;
        double r72491 = 0.5;
        double r72492 = pow(r72490, r72491);
        double r72493 = r72486 * r72492;
        double r72494 = r72486 * r72493;
        double r72495 = 2.0;
        double r72496 = atan2(1.0, 0.0);
        double r72497 = r72495 * r72496;
        double r72498 = u2;
        double r72499 = r72497 * r72498;
        double r72500 = cos(r72499);
        double r72501 = r72494 * r72500;
        double r72502 = r72501 + r72491;
        return r72502;
}

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 add-sqr-sqrt0.4

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

  4. Applied associate-*l*0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019303 
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (<= 0.0 u1 1) (<= 0.0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))