Average Error: 0.4 → 0.4
Time: 12.6s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.166666666666666657 \cdot \left({\left({-1}^{1} \cdot \left({-2}^{1} \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1}\right)\right)}^{0.5} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) + 0.5\]
\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
↓
0.166666666666666657 \cdot \left({\left({-1}^{1} \cdot \left({-2}^{1} \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1}\right)\right)}^{0.5} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) + 0.5
double f(double u1, double u2) {
        double r75122 = 1.0;
        double r75123 = 6.0;
        double r75124 = r75122 / r75123;
        double r75125 = -2.0;
        double r75126 = u1;
        double r75127 = log(r75126);
        double r75128 = r75125 * r75127;
        double r75129 = 0.5;
        double r75130 = pow(r75128, r75129);
        double r75131 = r75124 * r75130;
        double r75132 = 2.0;
        double r75133 = atan2(1.0, 0.0);
        double r75134 = r75132 * r75133;
        double r75135 = u2;
        double r75136 = r75134 * r75135;
        double r75137 = cos(r75136);
        double r75138 = r75131 * r75137;
        double r75139 = r75138 + r75129;
        return r75139;
}
↓
double f(double u1, double u2) {
        double r75140 = 0.16666666666666666;
        double r75141 = -1.0;
        double r75142 = 1.0;
        double r75143 = pow(r75141, r75142);
        double r75144 = -2.0;
        double r75145 = pow(r75144, r75142);
        double r75146 = 1.0;
        double r75147 = u1;
        double r75148 = r75146 / r75147;
        double r75149 = log(r75148);
        double r75150 = pow(r75149, r75142);
        double r75151 = r75145 * r75150;
        double r75152 = r75143 * r75151;
        double r75153 = 0.5;
        double r75154 = pow(r75152, r75153);
        double r75155 = 2.0;
        double r75156 = u2;
        double r75157 = atan2(1.0, 0.0);
        double r75158 = r75156 * r75157;
        double r75159 = r75155 * r75158;
        double r75160 = cos(r75159);
        double r75161 = r75154 * r75160;
        double r75162 = r75140 * r75161;
        double r75163 = r75162 + r75153;
        return r75163;
}

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. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{0.166666666666666657 \cdot \left({\left({-1}^{1} \cdot \left({-2}^{1} \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1}\right)\right)}^{0.5} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) + 0.5}\]

  3. Final simplification0.4

    \[\leadsto 0.166666666666666657 \cdot \left({\left({-1}^{1} \cdot \left({-2}^{1} \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1}\right)\right)}^{0.5} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) + 0.5\]

Reproduce

herbie shell --seed 2020033 
(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))