Average Error: 0.4 → 0.4
Time: 24.3s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right), \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right), 0.5\right)\]
\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(\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right), \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right), 0.5\right)
double f(double u1, double u2) {
        double r761033 = 1.0;
        double r761034 = 6.0;
        double r761035 = r761033 / r761034;
        double r761036 = -2.0;
        double r761037 = u1;
        double r761038 = log(r761037);
        double r761039 = r761036 * r761038;
        double r761040 = 0.5;
        double r761041 = pow(r761039, r761040);
        double r761042 = r761035 * r761041;
        double r761043 = 2.0;
        double r761044 = atan2(1.0, 0.0);
        double r761045 = r761043 * r761044;
        double r761046 = u2;
        double r761047 = r761045 * r761046;
        double r761048 = cos(r761047);
        double r761049 = r761042 * r761048;
        double r761050 = r761049 + r761040;
        return r761050;
}
↓
double f(double u1, double u2) {
        double r761051 = atan2(1.0, 0.0);
        double r761052 = 2.0;
        double r761053 = r761051 * r761052;
        double r761054 = u2;
        double r761055 = r761053 * r761054;
        double r761056 = cos(r761055);
        double r761057 = -2.0;
        double r761058 = u1;
        double r761059 = log(r761058);
        double r761060 = r761057 * r761059;
        double r761061 = 0.5;
        double r761062 = pow(r761060, r761061);
        double r761063 = 0.16666666666666666;
        double r761064 = r761062 * r761063;
        double r761065 = fma(r761056, r761064, r761061);
        return r761065;
}

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

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

  4. Applied div-inv0.4

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

  5. Simplified0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019128 +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))