Average Error: 0.4 → 0.3
Time: 11.1s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\log \left(e^{\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\log \left(e^{\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r69445 = 1.0;
        double r69446 = 6.0;
        double r69447 = r69445 / r69446;
        double r69448 = -2.0;
        double r69449 = u1;
        double r69450 = log(r69449);
        double r69451 = r69448 * r69450;
        double r69452 = 0.5;
        double r69453 = pow(r69451, r69452);
        double r69454 = r69447 * r69453;
        double r69455 = 2.0;
        double r69456 = atan2(1.0, 0.0);
        double r69457 = r69455 * r69456;
        double r69458 = u2;
        double r69459 = r69457 * r69458;
        double r69460 = cos(r69459);
        double r69461 = r69454 * r69460;
        double r69462 = r69461 + r69452;
        return r69462;
}
↓
double f(double u1, double u2) {
        double r69463 = 1.0;
        double r69464 = -2.0;
        double r69465 = u1;
        double r69466 = log(r69465);
        double r69467 = r69464 * r69466;
        double r69468 = 0.5;
        double r69469 = pow(r69467, r69468);
        double r69470 = r69463 * r69469;
        double r69471 = 6.0;
        double r69472 = r69470 / r69471;
        double r69473 = exp(r69472);
        double r69474 = log(r69473);
        double r69475 = 2.0;
        double r69476 = atan2(1.0, 0.0);
        double r69477 = r69475 * r69476;
        double r69478 = u2;
        double r69479 = r69477 * r69478;
        double r69480 = cos(r69479);
        double r69481 = r69474 * r69480;
        double r69482 = r69481 + r69468;
        return r69482;
}

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. Using strategy rm

  5. Applied add-log-exp0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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))