Average Error: 0.4 → 0.4
Time: 25.1s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + \frac{1}{2}\]
\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)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + \frac{1}{2}
double f(double u1, double u2) {
        double r74487 = 1.0;
        double r74488 = 6.0;
        double r74489 = r74487 / r74488;
        double r74490 = -2.0;
        double r74491 = u1;
        double r74492 = log(r74491);
        double r74493 = r74490 * r74492;
        double r74494 = 0.5;
        double r74495 = pow(r74493, r74494);
        double r74496 = r74489 * r74495;
        double r74497 = 2.0;
        double r74498 = atan2(1.0, 0.0);
        double r74499 = r74497 * r74498;
        double r74500 = u2;
        double r74501 = r74499 * r74500;
        double r74502 = cos(r74501);
        double r74503 = r74496 * r74502;
        double r74504 = r74503 + r74494;
        return r74504;
}
↓
double f(double u1, double u2) {
        double r74505 = 1.0;
        double r74506 = 6.0;
        double r74507 = r74505 / r74506;
        double r74508 = sqrt(r74507);
        double r74509 = -2.0;
        double r74510 = u1;
        double r74511 = log(r74510);
        double r74512 = r74509 * r74511;
        double r74513 = 2.0;
        double r74514 = r74505 / r74513;
        double r74515 = pow(r74512, r74514);
        double r74516 = r74508 * r74515;
        double r74517 = r74508 * r74516;
        double r74518 = atan2(1.0, 0.0);
        double r74519 = r74513 * r74518;
        double r74520 = u2;
        double r74521 = r74519 * r74520;
        double r74522 = cos(r74521);
        double r74523 = r74517 * r74522;
        double r74524 = r74523 + r74514;
        return r74524;
}

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. Simplified0.4

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

  6. Final simplification0.4

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

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))