Average Error: 0.4 → 0.4
Time: 30.2s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\left(\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(\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
double f(double u1, double u2) {
        double r84476 = 1.0;
        double r84477 = 6.0;
        double r84478 = r84476 / r84477;
        double r84479 = -2.0;
        double r84480 = u1;
        double r84481 = log(r84480);
        double r84482 = r84479 * r84481;
        double r84483 = 0.5;
        double r84484 = pow(r84482, r84483);
        double r84485 = r84478 * r84484;
        double r84486 = 2.0;
        double r84487 = atan2(1.0, 0.0);
        double r84488 = r84486 * r84487;
        double r84489 = u2;
        double r84490 = r84488 * r84489;
        double r84491 = cos(r84490);
        double r84492 = r84485 * r84491;
        double r84493 = r84492 + r84483;
        return r84493;
}
↓
double f(double u1, double u2) {
        double r84494 = 1.0;
        double r84495 = 6.0;
        double r84496 = r84494 / r84495;
        double r84497 = -2.0;
        double r84498 = u1;
        double r84499 = log(r84498);
        double r84500 = r84497 * r84499;
        double r84501 = 0.5;
        double r84502 = pow(r84500, r84501);
        double r84503 = r84496 * r84502;
        double r84504 = 2.0;
        double r84505 = atan2(1.0, 0.0);
        double r84506 = r84504 * r84505;
        double r84507 = u2;
        double r84508 = r84506 * r84507;
        double r84509 = cos(r84508);
        double r84510 = r84503 * r84509;
        double r84511 = r84510 + r84501;
        return r84511;
}

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. Using strategy rm

  6. Applied associate-*r*0.4

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

  7. Simplified0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019326 
(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))