Average Error: 0.4 → 0.3
Time: 33.1s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{\frac{6}{1}}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right)\]
\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{{\left(-2 \cdot \log u1\right)}^{0.5}}{\frac{6}{1}}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right)
double f(double u1, double u2) {
        double r2296088 = 1.0;
        double r2296089 = 6.0;
        double r2296090 = r2296088 / r2296089;
        double r2296091 = -2.0;
        double r2296092 = u1;
        double r2296093 = log(r2296092);
        double r2296094 = r2296091 * r2296093;
        double r2296095 = 0.5;
        double r2296096 = pow(r2296094, r2296095);
        double r2296097 = r2296090 * r2296096;
        double r2296098 = 2.0;
        double r2296099 = atan2(1.0, 0.0);
        double r2296100 = r2296098 * r2296099;
        double r2296101 = u2;
        double r2296102 = r2296100 * r2296101;
        double r2296103 = cos(r2296102);
        double r2296104 = r2296097 * r2296103;
        double r2296105 = r2296104 + r2296095;
        return r2296105;
}
↓
double f(double u1, double u2) {
        double r2296106 = -2.0;
        double r2296107 = u1;
        double r2296108 = log(r2296107);
        double r2296109 = r2296106 * r2296108;
        double r2296110 = 0.5;
        double r2296111 = pow(r2296109, r2296110);
        double r2296112 = 6.0;
        double r2296113 = 1.0;
        double r2296114 = r2296112 / r2296113;
        double r2296115 = r2296111 / r2296114;
        double r2296116 = 2.0;
        double r2296117 = atan2(1.0, 0.0);
        double r2296118 = u2;
        double r2296119 = r2296117 * r2296118;
        double r2296120 = r2296116 * r2296119;
        double r2296121 = cos(r2296120);
        double r2296122 = fma(r2296115, r2296121, r2296110);
        return r2296122;
}

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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019171 +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))