Average Error: 0.4 → 0.3
Time: 26.3s
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 r67181 = 1.0;
        double r67182 = 6.0;
        double r67183 = r67181 / r67182;
        double r67184 = -2.0;
        double r67185 = u1;
        double r67186 = log(r67185);
        double r67187 = r67184 * r67186;
        double r67188 = 0.5;
        double r67189 = pow(r67187, r67188);
        double r67190 = r67183 * r67189;
        double r67191 = 2.0;
        double r67192 = atan2(1.0, 0.0);
        double r67193 = r67191 * r67192;
        double r67194 = u2;
        double r67195 = r67193 * r67194;
        double r67196 = cos(r67195);
        double r67197 = r67190 * r67196;
        double r67198 = r67197 + r67188;
        return r67198;
}
↓
double f(double u1, double u2) {
        double r67199 = 1.0;
        double r67200 = 6.0;
        double r67201 = r67199 / r67200;
        double r67202 = sqrt(r67201);
        double r67203 = -2.0;
        double r67204 = u1;
        double r67205 = log(r67204);
        double r67206 = r67203 * r67205;
        double r67207 = 0.5;
        double r67208 = pow(r67206, r67207);
        double r67209 = r67202 * r67208;
        double r67210 = r67202 * r67209;
        double r67211 = 2.0;
        double r67212 = atan2(1.0, 0.0);
        double r67213 = r67211 * r67212;
        double r67214 = u2;
        double r67215 = r67213 * r67214;
        double r67216 = cos(r67215);
        double r67217 = r67210 * r67216;
        double r67218 = r67217 + r67207;
        return r67218;
}

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*0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019325 
(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))