Average Error: 0.4 → 0.3
Time: 11.9s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r76179 = 1.0;
        double r76180 = 6.0;
        double r76181 = r76179 / r76180;
        double r76182 = -2.0;
        double r76183 = u1;
        double r76184 = log(r76183);
        double r76185 = r76182 * r76184;
        double r76186 = 0.5;
        double r76187 = pow(r76185, r76186);
        double r76188 = r76181 * r76187;
        double r76189 = 2.0;
        double r76190 = atan2(1.0, 0.0);
        double r76191 = r76189 * r76190;
        double r76192 = u2;
        double r76193 = r76191 * r76192;
        double r76194 = cos(r76193);
        double r76195 = r76188 * r76194;
        double r76196 = r76195 + r76186;
        return r76196;
}
↓
double f(double u1, double u2) {
        double r76197 = 1.0;
        double r76198 = -2.0;
        double r76199 = u1;
        double r76200 = log(r76199);
        double r76201 = r76198 * r76200;
        double r76202 = 0.5;
        double r76203 = pow(r76201, r76202);
        double r76204 = r76197 * r76203;
        double r76205 = 6.0;
        double r76206 = r76204 / r76205;
        double r76207 = 2.0;
        double r76208 = atan2(1.0, 0.0);
        double r76209 = r76207 * r76208;
        double r76210 = u2;
        double r76211 = r76209 * r76210;
        double r76212 = cos(r76211);
        double r76213 = r76206 * r76212;
        double r76214 = r76213 + r76202;
        return r76214;
}

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 associate-*l/0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2020027 +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))