Average Error: 0.4 → 0.3
Time: 28.8s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left({\left({\left(-\log u1\right)}^{1}\right)}^{0.5} \cdot {\left({-2}^{1} \cdot {-1}^{1}\right)}^{0.5}, \cos \left(\left(\pi \cdot u2\right) \cdot 2\right) \cdot 0.1666666666666666574148081281236954964697, 0.5\right)\]
\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({\left({\left(-\log u1\right)}^{1}\right)}^{0.5} \cdot {\left({-2}^{1} \cdot {-1}^{1}\right)}^{0.5}, \cos \left(\left(\pi \cdot u2\right) \cdot 2\right) \cdot 0.1666666666666666574148081281236954964697, 0.5\right)
double f(double u1, double u2) {
        double r85138 = 1.0;
        double r85139 = 6.0;
        double r85140 = r85138 / r85139;
        double r85141 = -2.0;
        double r85142 = u1;
        double r85143 = log(r85142);
        double r85144 = r85141 * r85143;
        double r85145 = 0.5;
        double r85146 = pow(r85144, r85145);
        double r85147 = r85140 * r85146;
        double r85148 = 2.0;
        double r85149 = atan2(1.0, 0.0);
        double r85150 = r85148 * r85149;
        double r85151 = u2;
        double r85152 = r85150 * r85151;
        double r85153 = cos(r85152);
        double r85154 = r85147 * r85153;
        double r85155 = r85154 + r85145;
        return r85155;
}
↓
double f(double u1, double u2) {
        double r85156 = u1;
        double r85157 = log(r85156);
        double r85158 = -r85157;
        double r85159 = 1.0;
        double r85160 = pow(r85158, r85159);
        double r85161 = 0.5;
        double r85162 = pow(r85160, r85161);
        double r85163 = -2.0;
        double r85164 = pow(r85163, r85159);
        double r85165 = -1.0;
        double r85166 = pow(r85165, r85159);
        double r85167 = r85164 * r85166;
        double r85168 = pow(r85167, r85161);
        double r85169 = r85162 * r85168;
        double r85170 = atan2(1.0, 0.0);
        double r85171 = u2;
        double r85172 = r85170 * r85171;
        double r85173 = 2.0;
        double r85174 = r85172 * r85173;
        double r85175 = cos(r85174);
        double r85176 = 0.16666666666666666;
        double r85177 = r85175 * r85176;
        double r85178 = fma(r85169, r85177, r85161);
        return r85178;
}

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. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{0.1666666666666666574148081281236954964697 \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot {\left({-2}^{1} \cdot \left({-1}^{1} \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1}\right)\right)}^{0.5}\right) + 0.5}\]

  4. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left({-1}^{1} \cdot {-2}^{1}\right) \cdot {\left(-\log u1\right)}^{1}\right)}^{0.5}, 0.1666666666666666574148081281236954964697 \cdot \cos \left(\left(u2 \cdot \pi\right) \cdot 2\right), 0.5\right)}\]
  5. Using strategy rm

  6. Applied unpow-prod-down0.3

    \[\leadsto \mathsf{fma}\left(\color{blue}{{\left({-1}^{1} \cdot {-2}^{1}\right)}^{0.5} \cdot {\left({\left(-\log u1\right)}^{1}\right)}^{0.5}}, 0.1666666666666666574148081281236954964697 \cdot \cos \left(\left(u2 \cdot \pi\right) \cdot 2\right), 0.5\right)\]
  7. Using strategy rm

  8. Applied *-un-lft-identity0.3

    \[\leadsto \mathsf{fma}\left({\left({-1}^{1} \cdot {-2}^{1}\right)}^{0.5} \cdot {\left({\color{blue}{\left(1 \cdot \left(-\log u1\right)\right)}}^{1}\right)}^{0.5}, 0.1666666666666666574148081281236954964697 \cdot \cos \left(\left(u2 \cdot \pi\right) \cdot 2\right), 0.5\right)\]

  9. Applied unpow-prod-down0.3

    \[\leadsto \mathsf{fma}\left({\left({-1}^{1} \cdot {-2}^{1}\right)}^{0.5} \cdot {\color{blue}{\left({1}^{1} \cdot {\left(-\log u1\right)}^{1}\right)}}^{0.5}, 0.1666666666666666574148081281236954964697 \cdot \cos \left(\left(u2 \cdot \pi\right) \cdot 2\right), 0.5\right)\]

  10. Applied unpow-prod-down0.3

    \[\leadsto \mathsf{fma}\left({\left({-1}^{1} \cdot {-2}^{1}\right)}^{0.5} \cdot \color{blue}{\left({\left({1}^{1}\right)}^{0.5} \cdot {\left({\left(-\log u1\right)}^{1}\right)}^{0.5}\right)}, 0.1666666666666666574148081281236954964697 \cdot \cos \left(\left(u2 \cdot \pi\right) \cdot 2\right), 0.5\right)\]

  11. Simplified0.3

    \[\leadsto \mathsf{fma}\left({\left({-1}^{1} \cdot {-2}^{1}\right)}^{0.5} \cdot \left(\color{blue}{1} \cdot {\left({\left(-\log u1\right)}^{1}\right)}^{0.5}\right), 0.1666666666666666574148081281236954964697 \cdot \cos \left(\left(u2 \cdot \pi\right) \cdot 2\right), 0.5\right)\]

  12. Final simplification0.3

    \[\leadsto \mathsf{fma}\left({\left({\left(-\log u1\right)}^{1}\right)}^{0.5} \cdot {\left({-2}^{1} \cdot {-1}^{1}\right)}^{0.5}, \cos \left(\left(\pi \cdot u2\right) \cdot 2\right) \cdot 0.1666666666666666574148081281236954964697, 0.5\right)\]

Reproduce

herbie shell --seed 2019174 +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))