Average Error: 0.4 → 0.4
Time: 34.0s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\frac{1}{{\left(\frac{1}{{\left(\log u1\right)}^{1} \cdot {-2}^{1}}\right)}^{0.5} \cdot 6}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right)\]
\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
↓
\mathsf{fma}\left(\frac{1}{{\left(\frac{1}{{\left(\log u1\right)}^{1} \cdot {-2}^{1}}\right)}^{0.5} \cdot 6}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right)
double f(double u1, double u2) {
        double r1700284 = 1.0;
        double r1700285 = 6.0;
        double r1700286 = r1700284 / r1700285;
        double r1700287 = -2.0;
        double r1700288 = u1;
        double r1700289 = log(r1700288);
        double r1700290 = r1700287 * r1700289;
        double r1700291 = 0.5;
        double r1700292 = pow(r1700290, r1700291);
        double r1700293 = r1700286 * r1700292;
        double r1700294 = 2.0;
        double r1700295 = atan2(1.0, 0.0);
        double r1700296 = r1700294 * r1700295;
        double r1700297 = u2;
        double r1700298 = r1700296 * r1700297;
        double r1700299 = cos(r1700298);
        double r1700300 = r1700293 * r1700299;
        double r1700301 = r1700300 + r1700291;
        return r1700301;
}
↓
double f(double u1, double u2) {
        double r1700302 = 1.0;
        double r1700303 = u1;
        double r1700304 = log(r1700303);
        double r1700305 = 1.0;
        double r1700306 = pow(r1700304, r1700305);
        double r1700307 = -2.0;
        double r1700308 = pow(r1700307, r1700305);
        double r1700309 = r1700306 * r1700308;
        double r1700310 = r1700302 / r1700309;
        double r1700311 = 0.5;
        double r1700312 = pow(r1700310, r1700311);
        double r1700313 = 6.0;
        double r1700314 = r1700312 * r1700313;
        double r1700315 = r1700302 / r1700314;
        double r1700316 = 2.0;
        double r1700317 = atan2(1.0, 0.0);
        double r1700318 = u2;
        double r1700319 = r1700317 * r1700318;
        double r1700320 = r1700316 * r1700319;
        double r1700321 = cos(r1700320);
        double r1700322 = fma(r1700315, r1700321, r1700311);
        return r1700322;
}

Error

Bits error versus u1

Bits error versus u2

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. Simplified0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{\frac{6}{1}}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right)}\]
  3. Using strategy rm

  4. Applied clear-num0.3

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

  5. Taylor expanded around 0 0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019172 +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))