Average Error: 0.4 → 0.4
Time: 32.3s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r90139 = 1.0;
        double r90140 = 6.0;
        double r90141 = r90139 / r90140;
        double r90142 = -2.0;
        double r90143 = u1;
        double r90144 = log(r90143);
        double r90145 = r90142 * r90144;
        double r90146 = 0.5;
        double r90147 = pow(r90145, r90146);
        double r90148 = r90141 * r90147;
        double r90149 = 2.0;
        double r90150 = atan2(1.0, 0.0);
        double r90151 = r90149 * r90150;
        double r90152 = u2;
        double r90153 = r90151 * r90152;
        double r90154 = cos(r90153);
        double r90155 = r90148 * r90154;
        double r90156 = r90155 + r90146;
        return r90156;
}
↓
double f(double u1, double u2) {
        double r90157 = 1.0;
        double r90158 = 6.0;
        double r90159 = r90157 / r90158;
        double r90160 = sqrt(r90159);
        double r90161 = -2.0;
        double r90162 = u1;
        double r90163 = log(r90162);
        double r90164 = r90161 * r90163;
        double r90165 = 0.5;
        double r90166 = pow(r90164, r90165);
        double r90167 = r90160 * r90166;
        double r90168 = r90160 * r90167;
        double r90169 = 2.0;
        double r90170 = atan2(1.0, 0.0);
        double r90171 = r90169 * r90170;
        double r90172 = u2;
        double r90173 = r90171 * r90172;
        double r90174 = cos(r90173);
        double r90175 = r90168 * r90174;
        double r90176 = r90175 + r90165;
        return r90176;
}

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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019326 +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))