Average Error: 0.4 → 0.3
Time: 12.5s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\left(1 \cdot \frac{{\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 r73430 = 1.0;
        double r73431 = 6.0;
        double r73432 = r73430 / r73431;
        double r73433 = -2.0;
        double r73434 = u1;
        double r73435 = log(r73434);
        double r73436 = r73433 * r73435;
        double r73437 = 0.5;
        double r73438 = pow(r73436, r73437);
        double r73439 = r73432 * r73438;
        double r73440 = 2.0;
        double r73441 = atan2(1.0, 0.0);
        double r73442 = r73440 * r73441;
        double r73443 = u2;
        double r73444 = r73442 * r73443;
        double r73445 = cos(r73444);
        double r73446 = r73439 * r73445;
        double r73447 = r73446 + r73437;
        return r73447;
}
↓
double f(double u1, double u2) {
        double r73448 = 1.0;
        double r73449 = -2.0;
        double r73450 = u1;
        double r73451 = log(r73450);
        double r73452 = r73449 * r73451;
        double r73453 = 0.5;
        double r73454 = pow(r73452, r73453);
        double r73455 = 6.0;
        double r73456 = r73454 / r73455;
        double r73457 = r73448 * r73456;
        double r73458 = 2.0;
        double r73459 = atan2(1.0, 0.0);
        double r73460 = r73458 * r73459;
        double r73461 = u2;
        double r73462 = r73460 * r73461;
        double r73463 = cos(r73462);
        double r73464 = r73457 * r73463;
        double r73465 = r73464 + r73453;
        return r73465;
}

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. Final simplification0.3

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

Reproduce

herbie shell --seed 2020083 
(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))