Average Error: 0.4 → 0.3
Time: 24.9s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}, 0.5\right)\]
\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(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}, 0.5\right)
double f(double u1, double u2) {
        double r701260 = 1.0;
        double r701261 = 6.0;
        double r701262 = r701260 / r701261;
        double r701263 = -2.0;
        double r701264 = u1;
        double r701265 = log(r701264);
        double r701266 = r701263 * r701265;
        double r701267 = 0.5;
        double r701268 = pow(r701266, r701267);
        double r701269 = r701262 * r701268;
        double r701270 = 2.0;
        double r701271 = atan2(1.0, 0.0);
        double r701272 = r701270 * r701271;
        double r701273 = u2;
        double r701274 = r701272 * r701273;
        double r701275 = cos(r701274);
        double r701276 = r701269 * r701275;
        double r701277 = r701276 + r701267;
        return r701277;
}
↓
double f(double u1, double u2) {
        double r701278 = atan2(1.0, 0.0);
        double r701279 = 2.0;
        double r701280 = r701278 * r701279;
        double r701281 = u2;
        double r701282 = r701280 * r701281;
        double r701283 = cos(r701282);
        double r701284 = -2.0;
        double r701285 = u1;
        double r701286 = log(r701285);
        double r701287 = r701284 * r701286;
        double r701288 = 0.5;
        double r701289 = pow(r701287, r701288);
        double r701290 = 0.16666666666666666;
        double r701291 = sqrt(r701290);
        double r701292 = r701289 * r701291;
        double r701293 = r701292 * r701291;
        double r701294 = fma(r701283, r701293, r701288);
        return r701294;
}

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

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*r*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019154 +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))