Average Error: 0.4 → 0.3
Time: 17.3s
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 r68263 = 1.0;
        double r68264 = 6.0;
        double r68265 = r68263 / r68264;
        double r68266 = -2.0;
        double r68267 = u1;
        double r68268 = log(r68267);
        double r68269 = r68266 * r68268;
        double r68270 = 0.5;
        double r68271 = pow(r68269, r68270);
        double r68272 = r68265 * r68271;
        double r68273 = 2.0;
        double r68274 = atan2(1.0, 0.0);
        double r68275 = r68273 * r68274;
        double r68276 = u2;
        double r68277 = r68275 * r68276;
        double r68278 = cos(r68277);
        double r68279 = r68272 * r68278;
        double r68280 = r68279 + r68270;
        return r68280;
}
↓
double f(double u1, double u2) {
        double r68281 = 1.0;
        double r68282 = -2.0;
        double r68283 = u1;
        double r68284 = log(r68283);
        double r68285 = r68282 * r68284;
        double r68286 = 0.5;
        double r68287 = pow(r68285, r68286);
        double r68288 = r68281 * r68287;
        double r68289 = 6.0;
        double r68290 = r68288 / r68289;
        double r68291 = 2.0;
        double r68292 = atan2(1.0, 0.0);
        double r68293 = r68291 * r68292;
        double r68294 = u2;
        double r68295 = r68293 * r68294;
        double r68296 = cos(r68295);
        double r68297 = r68290 * r68296;
        double r68298 = r68297 + r68286;
        return r68298;
}

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 2020043 
(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))