Average Error: 0.4 → 0.3
Time: 12.3s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r75466 = 1.0;
        double r75467 = 6.0;
        double r75468 = r75466 / r75467;
        double r75469 = -2.0;
        double r75470 = u1;
        double r75471 = log(r75470);
        double r75472 = r75469 * r75471;
        double r75473 = 0.5;
        double r75474 = pow(r75472, r75473);
        double r75475 = r75468 * r75474;
        double r75476 = 2.0;
        double r75477 = atan2(1.0, 0.0);
        double r75478 = r75476 * r75477;
        double r75479 = u2;
        double r75480 = r75478 * r75479;
        double r75481 = cos(r75480);
        double r75482 = r75475 * r75481;
        double r75483 = r75482 + r75473;
        return r75483;
}
↓
double f(double u1, double u2) {
        double r75484 = 1.0;
        double r75485 = -2.0;
        double r75486 = u1;
        double r75487 = log(r75486);
        double r75488 = r75485 * r75487;
        double r75489 = 0.5;
        double r75490 = pow(r75488, r75489);
        double r75491 = r75484 * r75490;
        double r75492 = 6.0;
        double r75493 = r75491 / r75492;
        double r75494 = 2.0;
        double r75495 = atan2(1.0, 0.0);
        double r75496 = r75494 * r75495;
        double r75497 = u2;
        double r75498 = r75496 * r75497;
        double r75499 = cos(r75498);
        double r75500 = r75493 * r75499;
        double r75501 = r75500 + r75489;
        return r75501;
}

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 associate-*l/0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(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))