Average Error: 0.4 → 0.4
Time: 32.2s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), {\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{0.5} \cdot \frac{1}{6}, 0.5\right)\]
\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({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{0.5} \cdot \frac{1}{6}, 0.5\right)
double f(double u1, double u2) {
        double r1501320 = 1.0;
        double r1501321 = 6.0;
        double r1501322 = r1501320 / r1501321;
        double r1501323 = -2.0;
        double r1501324 = u1;
        double r1501325 = log(r1501324);
        double r1501326 = r1501323 * r1501325;
        double r1501327 = 0.5;
        double r1501328 = pow(r1501326, r1501327);
        double r1501329 = r1501322 * r1501328;
        double r1501330 = 2.0;
        double r1501331 = atan2(1.0, 0.0);
        double r1501332 = r1501330 * r1501331;
        double r1501333 = u2;
        double r1501334 = r1501332 * r1501333;
        double r1501335 = cos(r1501334);
        double r1501336 = r1501329 * r1501335;
        double r1501337 = r1501336 + r1501327;
        return r1501337;
}
↓
double f(double u1, double u2) {
        double r1501338 = atan2(1.0, 0.0);
        double r1501339 = 2.0;
        double r1501340 = r1501338 * r1501339;
        double r1501341 = u2;
        double r1501342 = r1501340 * r1501341;
        double r1501343 = cos(r1501342);
        double r1501344 = -2.0;
        double r1501345 = 1.0;
        double r1501346 = pow(r1501344, r1501345);
        double r1501347 = u1;
        double r1501348 = log(r1501347);
        double r1501349 = pow(r1501348, r1501345);
        double r1501350 = r1501346 * r1501349;
        double r1501351 = 0.5;
        double r1501352 = pow(r1501350, r1501351);
        double r1501353 = 0.16666666666666666;
        double r1501354 = r1501352 * r1501353;
        double r1501355 = fma(r1501343, r1501354, r1501351);
        return r1501355;
}

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

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

  4. Applied sqr-pow0.6

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

  5. Applied associate-*r*0.5

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

  6. Taylor expanded around 0 0.4

    \[\leadsto \mathsf{fma}\left(\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right), \color{blue}{\frac{1}{6} \cdot {\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{0.5}}, 0.5\right)\]

  7. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right), {\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{0.5} \cdot \frac{1}{6}, 0.5\right)\]

Reproduce

herbie shell --seed 2019149 +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))