Average Error: 0.4 → 0.4
Time: 34.0s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + 0.5\]
\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
↓
\left(\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + 0.5
double f(double u1, double u2) {
        double r1266153 = 1.0;
        double r1266154 = 6.0;
        double r1266155 = r1266153 / r1266154;
        double r1266156 = -2.0;
        double r1266157 = u1;
        double r1266158 = log(r1266157);
        double r1266159 = r1266156 * r1266158;
        double r1266160 = 0.5;
        double r1266161 = pow(r1266159, r1266160);
        double r1266162 = r1266155 * r1266161;
        double r1266163 = 2.0;
        double r1266164 = atan2(1.0, 0.0);
        double r1266165 = r1266163 * r1266164;
        double r1266166 = u2;
        double r1266167 = r1266165 * r1266166;
        double r1266168 = cos(r1266167);
        double r1266169 = r1266162 * r1266168;
        double r1266170 = r1266169 + r1266160;
        return r1266170;
}
↓
double f(double u1, double u2) {
        double r1266171 = u1;
        double r1266172 = log(r1266171);
        double r1266173 = -2.0;
        double r1266174 = r1266172 * r1266173;
        double r1266175 = 0.5;
        double r1266176 = pow(r1266174, r1266175);
        double r1266177 = 0.16666666666666666;
        double r1266178 = sqrt(r1266177);
        double r1266179 = r1266176 * r1266178;
        double r1266180 = r1266179 * r1266178;
        double r1266181 = 2.0;
        double r1266182 = atan2(1.0, 0.0);
        double r1266183 = u2;
        double r1266184 = r1266182 * r1266183;
        double r1266185 = r1266181 * r1266184;
        double r1266186 = cos(r1266185);
        double r1266187 = r1266180 * r1266186;
        double r1266188 = r1266187 + r1266175;
        return r1266188;
}

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. Simplified0.4

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019139 
(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))