Average Error: 0.4 → 0.4
Time: 35.9s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \left({\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{0.5} \cdot \frac{1}{6}\right) + 0.5\]
\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
↓
\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \left({\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{0.5} \cdot \frac{1}{6}\right) + 0.5
double f(double u1, double u2) {
        double r2367180 = 1.0;
        double r2367181 = 6.0;
        double r2367182 = r2367180 / r2367181;
        double r2367183 = -2.0;
        double r2367184 = u1;
        double r2367185 = log(r2367184);
        double r2367186 = r2367183 * r2367185;
        double r2367187 = 0.5;
        double r2367188 = pow(r2367186, r2367187);
        double r2367189 = r2367182 * r2367188;
        double r2367190 = 2.0;
        double r2367191 = atan2(1.0, 0.0);
        double r2367192 = r2367190 * r2367191;
        double r2367193 = u2;
        double r2367194 = r2367192 * r2367193;
        double r2367195 = cos(r2367194);
        double r2367196 = r2367189 * r2367195;
        double r2367197 = r2367196 + r2367187;
        return r2367197;
}
↓
double f(double u1, double u2) {
        double r2367198 = 2.0;
        double r2367199 = atan2(1.0, 0.0);
        double r2367200 = u2;
        double r2367201 = r2367199 * r2367200;
        double r2367202 = r2367198 * r2367201;
        double r2367203 = cos(r2367202);
        double r2367204 = u1;
        double r2367205 = log(r2367204);
        double r2367206 = 1.0;
        double r2367207 = pow(r2367205, r2367206);
        double r2367208 = -2.0;
        double r2367209 = pow(r2367208, r2367206);
        double r2367210 = r2367207 * r2367209;
        double r2367211 = 0.5;
        double r2367212 = pow(r2367210, r2367211);
        double r2367213 = 0.16666666666666666;
        double r2367214 = r2367212 * r2367213;
        double r2367215 = r2367203 * r2367214;
        double r2367216 = r2367215 + r2367211;
        return r2367216;
}

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*0.3

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

  6. Taylor expanded around 0 0.4

    \[\leadsto 0.5 + \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \color{blue}{\left({\left(\sqrt{\frac{1}{6}}\right)}^{2} \cdot {\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{0.5}\right)}\]

  7. Simplified0.4

    \[\leadsto 0.5 + \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot {\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{0.5}\right)}\]

  8. Final simplification0.4

    \[\leadsto \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \left({\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{0.5} \cdot \frac{1}{6}\right) + 0.5\]

Reproduce

herbie shell --seed 2019158 
(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))