Average Error: 0.4 → 0.3
Time: 11.2s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r70321 = 1.0;
        double r70322 = 6.0;
        double r70323 = r70321 / r70322;
        double r70324 = -2.0;
        double r70325 = u1;
        double r70326 = log(r70325);
        double r70327 = r70324 * r70326;
        double r70328 = 0.5;
        double r70329 = pow(r70327, r70328);
        double r70330 = r70323 * r70329;
        double r70331 = 2.0;
        double r70332 = atan2(1.0, 0.0);
        double r70333 = r70331 * r70332;
        double r70334 = u2;
        double r70335 = r70333 * r70334;
        double r70336 = cos(r70335);
        double r70337 = r70330 * r70336;
        double r70338 = r70337 + r70328;
        return r70338;
}
↓
double f(double u1, double u2) {
        double r70339 = 1.0;
        double r70340 = -2.0;
        double r70341 = u1;
        double r70342 = log(r70341);
        double r70343 = r70340 * r70342;
        double r70344 = 0.5;
        double r70345 = pow(r70343, r70344);
        double r70346 = r70339 * r70345;
        double r70347 = 6.0;
        double r70348 = r70346 / r70347;
        double r70349 = 2.0;
        double r70350 = atan2(1.0, 0.0);
        double r70351 = r70349 * r70350;
        double r70352 = u2;
        double r70353 = r70351 * r70352;
        double r70354 = cos(r70353);
        double r70355 = r70348 * r70354;
        double r70356 = r70355 + r70344;
        return r70356;
}

Error

Bits error versus u1

Bits error versus u2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied associate-*l/0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2020064 
(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))