Average Error: 0.4 → 0.3
Time: 13.4s
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 r82382 = 1.0;
        double r82383 = 6.0;
        double r82384 = r82382 / r82383;
        double r82385 = -2.0;
        double r82386 = u1;
        double r82387 = log(r82386);
        double r82388 = r82385 * r82387;
        double r82389 = 0.5;
        double r82390 = pow(r82388, r82389);
        double r82391 = r82384 * r82390;
        double r82392 = 2.0;
        double r82393 = atan2(1.0, 0.0);
        double r82394 = r82392 * r82393;
        double r82395 = u2;
        double r82396 = r82394 * r82395;
        double r82397 = cos(r82396);
        double r82398 = r82391 * r82397;
        double r82399 = r82398 + r82389;
        return r82399;
}
↓
double f(double u1, double u2) {
        double r82400 = 1.0;
        double r82401 = -2.0;
        double r82402 = u1;
        double r82403 = log(r82402);
        double r82404 = r82401 * r82403;
        double r82405 = 0.5;
        double r82406 = pow(r82404, r82405);
        double r82407 = r82400 * r82406;
        double r82408 = 6.0;
        double r82409 = r82407 / r82408;
        double r82410 = 2.0;
        double r82411 = atan2(1.0, 0.0);
        double r82412 = r82410 * r82411;
        double r82413 = u2;
        double r82414 = r82412 * r82413;
        double r82415 = cos(r82414);
        double r82416 = r82409 * r82415;
        double r82417 = r82416 + r82405;
        return r82417;
}

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 2020024 +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))