Average Error: 0.4 → 0.3
Time: 15.4s
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 r68229 = 1.0;
        double r68230 = 6.0;
        double r68231 = r68229 / r68230;
        double r68232 = -2.0;
        double r68233 = u1;
        double r68234 = log(r68233);
        double r68235 = r68232 * r68234;
        double r68236 = 0.5;
        double r68237 = pow(r68235, r68236);
        double r68238 = r68231 * r68237;
        double r68239 = 2.0;
        double r68240 = atan2(1.0, 0.0);
        double r68241 = r68239 * r68240;
        double r68242 = u2;
        double r68243 = r68241 * r68242;
        double r68244 = cos(r68243);
        double r68245 = r68238 * r68244;
        double r68246 = r68245 + r68236;
        return r68246;
}
↓
double f(double u1, double u2) {
        double r68247 = 1.0;
        double r68248 = -2.0;
        double r68249 = u1;
        double r68250 = log(r68249);
        double r68251 = r68248 * r68250;
        double r68252 = 0.5;
        double r68253 = pow(r68251, r68252);
        double r68254 = r68247 * r68253;
        double r68255 = 6.0;
        double r68256 = r68254 / r68255;
        double r68257 = 2.0;
        double r68258 = atan2(1.0, 0.0);
        double r68259 = r68257 * r68258;
        double r68260 = u2;
        double r68261 = r68259 * r68260;
        double r68262 = cos(r68261);
        double r68263 = r68256 * r68262;
        double r68264 = r68263 + r68252;
        return r68264;
}

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 2020045 +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))