Average Error: 0.4 → 0.3
Time: 32.2s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \frac{{\left(\log u1 \cdot -2\right)}^{0.5}}{6}\right) \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) + 0.5\]
\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(1 \cdot \frac{{\left(\log u1 \cdot -2\right)}^{0.5}}{6}\right) \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) + 0.5
double f(double u1, double u2) {
        double r1971250 = 1.0;
        double r1971251 = 6.0;
        double r1971252 = r1971250 / r1971251;
        double r1971253 = -2.0;
        double r1971254 = u1;
        double r1971255 = log(r1971254);
        double r1971256 = r1971253 * r1971255;
        double r1971257 = 0.5;
        double r1971258 = pow(r1971256, r1971257);
        double r1971259 = r1971252 * r1971258;
        double r1971260 = 2.0;
        double r1971261 = atan2(1.0, 0.0);
        double r1971262 = r1971260 * r1971261;
        double r1971263 = u2;
        double r1971264 = r1971262 * r1971263;
        double r1971265 = cos(r1971264);
        double r1971266 = r1971259 * r1971265;
        double r1971267 = r1971266 + r1971257;
        return r1971267;
}
↓
double f(double u1, double u2) {
        double r1971268 = 1.0;
        double r1971269 = u1;
        double r1971270 = log(r1971269);
        double r1971271 = -2.0;
        double r1971272 = r1971270 * r1971271;
        double r1971273 = 0.5;
        double r1971274 = pow(r1971272, r1971273);
        double r1971275 = 6.0;
        double r1971276 = r1971274 / r1971275;
        double r1971277 = r1971268 * r1971276;
        double r1971278 = u2;
        double r1971279 = 2.0;
        double r1971280 = atan2(1.0, 0.0);
        double r1971281 = r1971279 * r1971280;
        double r1971282 = r1971278 * r1971281;
        double r1971283 = cos(r1971282);
        double r1971284 = r1971277 * r1971283;
        double r1971285 = r1971284 + r1971273;
        return r1971285;
}

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 div-inv0.4

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

  4. Applied associate-*l*0.4

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0.0 u1 1.0) (<= 0.0 u2 1.0))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))