Average Error: 0.4 → 0.4
Time: 33.8s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\]
\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
↓
0.5 + \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)
double f(double u1, double u2) {
        double r1266695 = 1.0;
        double r1266696 = 6.0;
        double r1266697 = r1266695 / r1266696;
        double r1266698 = -2.0;
        double r1266699 = u1;
        double r1266700 = log(r1266699);
        double r1266701 = r1266698 * r1266700;
        double r1266702 = 0.5;
        double r1266703 = pow(r1266701, r1266702);
        double r1266704 = r1266697 * r1266703;
        double r1266705 = 2.0;
        double r1266706 = atan2(1.0, 0.0);
        double r1266707 = r1266705 * r1266706;
        double r1266708 = u2;
        double r1266709 = r1266707 * r1266708;
        double r1266710 = cos(r1266709);
        double r1266711 = r1266704 * r1266710;
        double r1266712 = r1266711 + r1266702;
        return r1266712;
}
↓
double f(double u1, double u2) {
        double r1266713 = 0.5;
        double r1266714 = 0.16666666666666666;
        double r1266715 = -2.0;
        double r1266716 = u1;
        double r1266717 = log(r1266716);
        double r1266718 = r1266715 * r1266717;
        double r1266719 = pow(r1266718, r1266713);
        double r1266720 = r1266714 * r1266719;
        double r1266721 = u2;
        double r1266722 = 2.0;
        double r1266723 = atan2(1.0, 0.0);
        double r1266724 = r1266722 * r1266723;
        double r1266725 = r1266721 * r1266724;
        double r1266726 = cos(r1266725);
        double r1266727 = r1266720 * r1266726;
        double r1266728 = r1266713 + r1266727;
        return r1266728;
}

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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(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))