Average Error: 0.4 → 0.3
Time: 2.1m
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 r74272 = 1.0;
        double r74273 = 6.0;
        double r74274 = r74272 / r74273;
        double r74275 = -2.0;
        double r74276 = u1;
        double r74277 = log(r74276);
        double r74278 = r74275 * r74277;
        double r74279 = 0.5;
        double r74280 = pow(r74278, r74279);
        double r74281 = r74274 * r74280;
        double r74282 = 2.0;
        double r74283 = atan2(1.0, 0.0);
        double r74284 = r74282 * r74283;
        double r74285 = u2;
        double r74286 = r74284 * r74285;
        double r74287 = cos(r74286);
        double r74288 = r74281 * r74287;
        double r74289 = r74288 + r74279;
        return r74289;
}
↓
double f(double u1, double u2) {
        double r74290 = 1.0;
        double r74291 = 6.0;
        double r74292 = r74290 / r74291;
        double r74293 = sqrt(r74292);
        double r74294 = -2.0;
        double r74295 = u1;
        double r74296 = log(r74295);
        double r74297 = r74294 * r74296;
        double r74298 = 0.5;
        double r74299 = pow(r74297, r74298);
        double r74300 = r74293 * r74299;
        double r74301 = r74293 * r74300;
        double r74302 = 2.0;
        double r74303 = atan2(1.0, 0.0);
        double r74304 = r74302 * r74303;
        double r74305 = u2;
        double r74306 = r74304 * r74305;
        double r74307 = cos(r74306);
        double r74308 = r74301 * r74307;
        double r74309 = r74308 + r74298;
        return r74309;
}

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 2019202 
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0.0 u1 1) (<= 0.0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))