Average Error: 0.4 → 0.3
Time: 10.3s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \frac{1}{\frac{6}{{\left(-2 \cdot \log u1\right)}^{0.5}}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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(1 \cdot \frac{1}{\frac{6}{{\left(-2 \cdot \log u1\right)}^{0.5}}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r63408 = 1.0;
        double r63409 = 6.0;
        double r63410 = r63408 / r63409;
        double r63411 = -2.0;
        double r63412 = u1;
        double r63413 = log(r63412);
        double r63414 = r63411 * r63413;
        double r63415 = 0.5;
        double r63416 = pow(r63414, r63415);
        double r63417 = r63410 * r63416;
        double r63418 = 2.0;
        double r63419 = atan2(1.0, 0.0);
        double r63420 = r63418 * r63419;
        double r63421 = u2;
        double r63422 = r63420 * r63421;
        double r63423 = cos(r63422);
        double r63424 = r63417 * r63423;
        double r63425 = r63424 + r63415;
        return r63425;
}
↓
double f(double u1, double u2) {
        double r63426 = 1.0;
        double r63427 = 1.0;
        double r63428 = 6.0;
        double r63429 = -2.0;
        double r63430 = u1;
        double r63431 = log(r63430);
        double r63432 = r63429 * r63431;
        double r63433 = 0.5;
        double r63434 = pow(r63432, r63433);
        double r63435 = r63428 / r63434;
        double r63436 = r63427 / r63435;
        double r63437 = r63426 * r63436;
        double r63438 = 2.0;
        double r63439 = atan2(1.0, 0.0);
        double r63440 = r63438 * r63439;
        double r63441 = u2;
        double r63442 = r63440 * r63441;
        double r63443 = cos(r63442);
        double r63444 = r63437 * r63443;
        double r63445 = r63444 + r63433;
        return r63445;
}

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 div-inv0.4

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

  4. Applied associate-*l*0.4

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

  5. Simplified0.3

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

  7. Applied clear-num0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020062 +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))