Average Error: 0.4 → 0.3
Time: 30.2s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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 r86386 = 1.0;
        double r86387 = 6.0;
        double r86388 = r86386 / r86387;
        double r86389 = -2.0;
        double r86390 = u1;
        double r86391 = log(r86390);
        double r86392 = r86389 * r86391;
        double r86393 = 0.5;
        double r86394 = pow(r86392, r86393);
        double r86395 = r86388 * r86394;
        double r86396 = 2.0;
        double r86397 = atan2(1.0, 0.0);
        double r86398 = r86396 * r86397;
        double r86399 = u2;
        double r86400 = r86398 * r86399;
        double r86401 = cos(r86400);
        double r86402 = r86395 * r86401;
        double r86403 = r86402 + r86393;
        return r86403;
}
↓
double f(double u1, double u2) {
        double r86404 = 1.0;
        double r86405 = -2.0;
        double r86406 = u1;
        double r86407 = log(r86406);
        double r86408 = r86405 * r86407;
        double r86409 = 0.5;
        double r86410 = pow(r86408, r86409);
        double r86411 = 6.0;
        double r86412 = r86410 / r86411;
        double r86413 = r86404 * r86412;
        double r86414 = 2.0;
        double r86415 = atan2(1.0, 0.0);
        double r86416 = r86414 * r86415;
        double r86417 = u2;
        double r86418 = r86416 * r86417;
        double r86419 = cos(r86418);
        double r86420 = r86413 * r86419;
        double r86421 = r86420 + r86409;
        return r86421;
}

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 2019212 
(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))