Average Error: 0.4 → 0.4
Time: 53.2s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r7163307 = 1.0;
        double r7163308 = 6.0;
        double r7163309 = r7163307 / r7163308;
        double r7163310 = -2.0;
        double r7163311 = u1;
        double r7163312 = log(r7163311);
        double r7163313 = r7163310 * r7163312;
        double r7163314 = 0.5;
        double r7163315 = pow(r7163313, r7163314);
        double r7163316 = r7163309 * r7163315;
        double r7163317 = 2.0;
        double r7163318 = atan2(1.0, 0.0);
        double r7163319 = r7163317 * r7163318;
        double r7163320 = u2;
        double r7163321 = r7163319 * r7163320;
        double r7163322 = cos(r7163321);
        double r7163323 = r7163316 * r7163322;
        double r7163324 = r7163323 + r7163314;
        return r7163324;
}
↓
double f(double u1, double u2) {
        double r7163325 = 1.0;
        double r7163326 = 6.0;
        double r7163327 = r7163325 / r7163326;
        double r7163328 = sqrt(r7163327);
        double r7163329 = -2.0;
        double r7163330 = u1;
        double r7163331 = log(r7163330);
        double r7163332 = r7163329 * r7163331;
        double r7163333 = 0.5;
        double r7163334 = pow(r7163332, r7163333);
        double r7163335 = r7163328 * r7163334;
        double r7163336 = r7163328 * r7163335;
        double r7163337 = 2.0;
        double r7163338 = atan2(1.0, 0.0);
        double r7163339 = r7163337 * r7163338;
        double r7163340 = u2;
        double r7163341 = r7163339 * r7163340;
        double r7163342 = cos(r7163341);
        double r7163343 = r7163336 * r7163342;
        double r7163344 = r7163343 + r7163333;
        return r7163344;
}

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 add-sqr-sqrt0.4

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

  4. Applied associate-*l*0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019173 
(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))