Average Error: 0.4 → 0.3
Time: 11.3s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)
double f(double u1, double u2) {
        double r72210 = 1.0;
        double r72211 = 6.0;
        double r72212 = r72210 / r72211;
        double r72213 = -2.0;
        double r72214 = u1;
        double r72215 = log(r72214);
        double r72216 = r72213 * r72215;
        double r72217 = 0.5;
        double r72218 = pow(r72216, r72217);
        double r72219 = r72212 * r72218;
        double r72220 = 2.0;
        double r72221 = atan2(1.0, 0.0);
        double r72222 = r72220 * r72221;
        double r72223 = u2;
        double r72224 = r72222 * r72223;
        double r72225 = cos(r72224);
        double r72226 = r72219 * r72225;
        double r72227 = r72226 + r72217;
        return r72227;
}
↓
double f(double u1, double u2) {
        double r72228 = 1.0;
        double r72229 = -2.0;
        double r72230 = u1;
        double r72231 = log(r72230);
        double r72232 = r72229 * r72231;
        double r72233 = 0.5;
        double r72234 = pow(r72232, r72233);
        double r72235 = r72228 * r72234;
        double r72236 = 6.0;
        double r72237 = r72235 / r72236;
        double r72238 = 2.0;
        double r72239 = atan2(1.0, 0.0);
        double r72240 = r72238 * r72239;
        double r72241 = u2;
        double r72242 = r72240 * r72241;
        double r72243 = cos(r72242);
        double r72244 = fma(r72237, r72243, r72233);
        return r72244;
}

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. Simplified0.4

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

  4. Applied associate-*l/0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2020059 +o rules:numerics
(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))