Average Error: 0.4 → 0.3
Time: 14.7s
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 r74095 = 1.0;
        double r74096 = 6.0;
        double r74097 = r74095 / r74096;
        double r74098 = -2.0;
        double r74099 = u1;
        double r74100 = log(r74099);
        double r74101 = r74098 * r74100;
        double r74102 = 0.5;
        double r74103 = pow(r74101, r74102);
        double r74104 = r74097 * r74103;
        double r74105 = 2.0;
        double r74106 = atan2(1.0, 0.0);
        double r74107 = r74105 * r74106;
        double r74108 = u2;
        double r74109 = r74107 * r74108;
        double r74110 = cos(r74109);
        double r74111 = r74104 * r74110;
        double r74112 = r74111 + r74102;
        return r74112;
}
↓
double f(double u1, double u2) {
        double r74113 = 1.0;
        double r74114 = -2.0;
        double r74115 = u1;
        double r74116 = log(r74115);
        double r74117 = r74114 * r74116;
        double r74118 = 0.5;
        double r74119 = pow(r74117, r74118);
        double r74120 = r74113 * r74119;
        double r74121 = 6.0;
        double r74122 = r74120 / r74121;
        double r74123 = 2.0;
        double r74124 = atan2(1.0, 0.0);
        double r74125 = r74123 * r74124;
        double r74126 = u2;
        double r74127 = r74125 * r74126;
        double r74128 = cos(r74127);
        double r74129 = fma(r74122, r74128, r74118);
        return r74129;
}

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 2020046 +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))