Average Error: 0.4 → 0.3
Time: 34.0s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \sqrt{\frac{1}{6}}, 0.5\right)\]
\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), \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \sqrt{\frac{1}{6}}, 0.5\right)
double f(double u1, double u2) {
        double r1635163 = 1.0;
        double r1635164 = 6.0;
        double r1635165 = r1635163 / r1635164;
        double r1635166 = -2.0;
        double r1635167 = u1;
        double r1635168 = log(r1635167);
        double r1635169 = r1635166 * r1635168;
        double r1635170 = 0.5;
        double r1635171 = pow(r1635169, r1635170);
        double r1635172 = r1635165 * r1635171;
        double r1635173 = 2.0;
        double r1635174 = atan2(1.0, 0.0);
        double r1635175 = r1635173 * r1635174;
        double r1635176 = u2;
        double r1635177 = r1635175 * r1635176;
        double r1635178 = cos(r1635177);
        double r1635179 = r1635172 * r1635178;
        double r1635180 = r1635179 + r1635170;
        return r1635180;
}
↓
double f(double u1, double u2) {
        double r1635181 = atan2(1.0, 0.0);
        double r1635182 = 2.0;
        double r1635183 = r1635181 * r1635182;
        double r1635184 = u2;
        double r1635185 = r1635183 * r1635184;
        double r1635186 = cos(r1635185);
        double r1635187 = 0.16666666666666666;
        double r1635188 = sqrt(r1635187);
        double r1635189 = -2.0;
        double r1635190 = u1;
        double r1635191 = log(r1635190);
        double r1635192 = r1635189 * r1635191;
        double r1635193 = 0.5;
        double r1635194 = pow(r1635192, r1635193);
        double r1635195 = r1635188 * r1635194;
        double r1635196 = r1635195 * r1635188;
        double r1635197 = fma(r1635186, r1635196, r1635193);
        return r1635197;
}

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

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019163 +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))