Average Error: 0.4 → 0.3
Time: 12.6s
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 r72213 = 1.0;
        double r72214 = 6.0;
        double r72215 = r72213 / r72214;
        double r72216 = -2.0;
        double r72217 = u1;
        double r72218 = log(r72217);
        double r72219 = r72216 * r72218;
        double r72220 = 0.5;
        double r72221 = pow(r72219, r72220);
        double r72222 = r72215 * r72221;
        double r72223 = 2.0;
        double r72224 = atan2(1.0, 0.0);
        double r72225 = r72223 * r72224;
        double r72226 = u2;
        double r72227 = r72225 * r72226;
        double r72228 = cos(r72227);
        double r72229 = r72222 * r72228;
        double r72230 = r72229 + r72220;
        return r72230;
}
↓
double f(double u1, double u2) {
        double r72231 = 1.0;
        double r72232 = -2.0;
        double r72233 = u1;
        double r72234 = log(r72233);
        double r72235 = r72232 * r72234;
        double r72236 = 0.5;
        double r72237 = pow(r72235, r72236);
        double r72238 = r72231 * r72237;
        double r72239 = 6.0;
        double r72240 = r72238 / r72239;
        double r72241 = 2.0;
        double r72242 = atan2(1.0, 0.0);
        double r72243 = r72241 * r72242;
        double r72244 = u2;
        double r72245 = r72243 * r72244;
        double r72246 = cos(r72245);
        double r72247 = r72240 * r72246;
        double r72248 = r72247 + r72236;
        return r72248;
}

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 2019346 
(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))