Average Error: 0.4 → 0.5
Time: 12.9s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[e^{\log \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\left(\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
↓
e^{\log \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
double f(double u1, double u2) {
        double r76273 = 1.0;
        double r76274 = 6.0;
        double r76275 = r76273 / r76274;
        double r76276 = -2.0;
        double r76277 = u1;
        double r76278 = log(r76277);
        double r76279 = r76276 * r76278;
        double r76280 = 0.5;
        double r76281 = pow(r76279, r76280);
        double r76282 = r76275 * r76281;
        double r76283 = 2.0;
        double r76284 = atan2(1.0, 0.0);
        double r76285 = r76283 * r76284;
        double r76286 = u2;
        double r76287 = r76285 * r76286;
        double r76288 = cos(r76287);
        double r76289 = r76282 * r76288;
        double r76290 = r76289 + r76280;
        return r76290;
}
↓
double f(double u1, double u2) {
        double r76291 = 1.0;
        double r76292 = 6.0;
        double r76293 = r76291 / r76292;
        double r76294 = -2.0;
        double r76295 = u1;
        double r76296 = log(r76295);
        double r76297 = r76294 * r76296;
        double r76298 = 0.5;
        double r76299 = pow(r76297, r76298);
        double r76300 = r76293 * r76299;
        double r76301 = log(r76300);
        double r76302 = exp(r76301);
        double r76303 = 2.0;
        double r76304 = atan2(1.0, 0.0);
        double r76305 = r76303 * r76304;
        double r76306 = u2;
        double r76307 = r76305 * r76306;
        double r76308 = cos(r76307);
        double r76309 = r76302 * r76308;
        double r76310 = r76309 + r76298;
        return r76310;
}

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-exp-log64.0

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

  4. Applied add-exp-log64.0

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

  5. Applied prod-exp64.0

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

  6. Applied pow-exp64.0

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

  7. Applied add-exp-log64.0

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

  8. Applied add-exp-log64.0

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

  9. Applied div-exp64.0

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

  10. Applied prod-exp64.0

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

  11. Simplified0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019356 
(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))