Average Error: 0.4 → 0.3
Time: 35.8s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, 0.5\right)\]
\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(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, 0.5\right)
double f(double u1, double u2) {
        double r2087228 = 1.0;
        double r2087229 = 6.0;
        double r2087230 = r2087228 / r2087229;
        double r2087231 = -2.0;
        double r2087232 = u1;
        double r2087233 = log(r2087232);
        double r2087234 = r2087231 * r2087233;
        double r2087235 = 0.5;
        double r2087236 = pow(r2087234, r2087235);
        double r2087237 = r2087230 * r2087236;
        double r2087238 = 2.0;
        double r2087239 = atan2(1.0, 0.0);
        double r2087240 = r2087238 * r2087239;
        double r2087241 = u2;
        double r2087242 = r2087240 * r2087241;
        double r2087243 = cos(r2087242);
        double r2087244 = r2087237 * r2087243;
        double r2087245 = r2087244 + r2087235;
        return r2087245;
}
↓
double f(double u1, double u2) {
        double r2087246 = atan2(1.0, 0.0);
        double r2087247 = 2.0;
        double r2087248 = r2087246 * r2087247;
        double r2087249 = u2;
        double r2087250 = r2087248 * r2087249;
        double r2087251 = cos(r2087250);
        double r2087252 = -2.0;
        double r2087253 = u1;
        double r2087254 = log(r2087253);
        double r2087255 = r2087252 * r2087254;
        double r2087256 = 0.5;
        double r2087257 = pow(r2087255, r2087256);
        double r2087258 = 6.0;
        double r2087259 = r2087257 / r2087258;
        double r2087260 = fma(r2087251, r2087259, r2087256);
        return r2087260;
}

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

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

  3. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0 u1 1) (<= 0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))