Average Error: 0.4 → 0.4
Time: 11.1s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\log \left(e^{\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}\right), \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\]
\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(\log \left(e^{\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}\right), \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)
double f(double u1, double u2) {
        double r68117 = 1.0;
        double r68118 = 6.0;
        double r68119 = r68117 / r68118;
        double r68120 = -2.0;
        double r68121 = u1;
        double r68122 = log(r68121);
        double r68123 = r68120 * r68122;
        double r68124 = 0.5;
        double r68125 = pow(r68123, r68124);
        double r68126 = r68119 * r68125;
        double r68127 = 2.0;
        double r68128 = atan2(1.0, 0.0);
        double r68129 = r68127 * r68128;
        double r68130 = u2;
        double r68131 = r68129 * r68130;
        double r68132 = cos(r68131);
        double r68133 = r68126 * r68132;
        double r68134 = r68133 + r68124;
        return r68134;
}
↓
double f(double u1, double u2) {
        double r68135 = 1.0;
        double r68136 = 6.0;
        double r68137 = r68135 / r68136;
        double r68138 = -2.0;
        double r68139 = u1;
        double r68140 = log(r68139);
        double r68141 = r68138 * r68140;
        double r68142 = 0.5;
        double r68143 = pow(r68141, r68142);
        double r68144 = r68137 * r68143;
        double r68145 = exp(r68144);
        double r68146 = log(r68145);
        double r68147 = 2.0;
        double r68148 = atan2(1.0, 0.0);
        double r68149 = r68147 * r68148;
        double r68150 = u2;
        double r68151 = r68149 * r68150;
        double r68152 = cos(r68151);
        double r68153 = fma(r68146, r68152, r68142);
        return r68153;
}

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 add-log-exp0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2020060 +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))