Average Error: 0.4 → 0.4
Time: 11.4s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\frac{\frac{1}{6}}{{\left(\frac{1}{{\left(\log u1\right)}^{1} \cdot {-2}^{1}}\right)}^{0.5}}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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{\frac{1}{6}}{{\left(\frac{1}{{\left(\log u1\right)}^{1} \cdot {-2}^{1}}\right)}^{0.5}}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)
double f(double u1, double u2) {
        double r68234 = 1.0;
        double r68235 = 6.0;
        double r68236 = r68234 / r68235;
        double r68237 = -2.0;
        double r68238 = u1;
        double r68239 = log(r68238);
        double r68240 = r68237 * r68239;
        double r68241 = 0.5;
        double r68242 = pow(r68240, r68241);
        double r68243 = r68236 * r68242;
        double r68244 = 2.0;
        double r68245 = atan2(1.0, 0.0);
        double r68246 = r68244 * r68245;
        double r68247 = u2;
        double r68248 = r68246 * r68247;
        double r68249 = cos(r68248);
        double r68250 = r68243 * r68249;
        double r68251 = r68250 + r68241;
        return r68251;
}
↓
double f(double u1, double u2) {
        double r68252 = 1.0;
        double r68253 = 6.0;
        double r68254 = r68252 / r68253;
        double r68255 = 1.0;
        double r68256 = u1;
        double r68257 = log(r68256);
        double r68258 = pow(r68257, r68252);
        double r68259 = -2.0;
        double r68260 = pow(r68259, r68252);
        double r68261 = r68258 * r68260;
        double r68262 = r68255 / r68261;
        double r68263 = 0.5;
        double r68264 = pow(r68262, r68263);
        double r68265 = r68254 / r68264;
        double r68266 = 2.0;
        double r68267 = atan2(1.0, 0.0);
        double r68268 = r68266 * r68267;
        double r68269 = u2;
        double r68270 = r68268 * r68269;
        double r68271 = cos(r68270);
        double r68272 = fma(r68265, r68271, r68263);
        return r68272;
}

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. Using strategy rm

  6. Applied associate-/l*0.3

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

  7. Taylor expanded around 0 0.4

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

  9. Applied associate-/r*0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020065 +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))