Average Error: 0.4 → 0.3
Time: 29.1s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{{\left(\log u1 \cdot -2\right)}^{0.5} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)}{6} + 0.5\]
\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{{\left(\log u1 \cdot -2\right)}^{0.5} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)}{6} + 0.5
double f(double u1, double u2) {
        double r1824033 = 1.0;
        double r1824034 = 6.0;
        double r1824035 = r1824033 / r1824034;
        double r1824036 = -2.0;
        double r1824037 = u1;
        double r1824038 = log(r1824037);
        double r1824039 = r1824036 * r1824038;
        double r1824040 = 0.5;
        double r1824041 = pow(r1824039, r1824040);
        double r1824042 = r1824035 * r1824041;
        double r1824043 = 2.0;
        double r1824044 = atan2(1.0, 0.0);
        double r1824045 = r1824043 * r1824044;
        double r1824046 = u2;
        double r1824047 = r1824045 * r1824046;
        double r1824048 = cos(r1824047);
        double r1824049 = r1824042 * r1824048;
        double r1824050 = r1824049 + r1824040;
        return r1824050;
}
↓
double f(double u1, double u2) {
        double r1824051 = u1;
        double r1824052 = log(r1824051);
        double r1824053 = -2.0;
        double r1824054 = r1824052 * r1824053;
        double r1824055 = 0.5;
        double r1824056 = pow(r1824054, r1824055);
        double r1824057 = u2;
        double r1824058 = 2.0;
        double r1824059 = atan2(1.0, 0.0);
        double r1824060 = r1824058 * r1824059;
        double r1824061 = r1824057 * r1824060;
        double r1824062 = cos(r1824061);
        double r1824063 = r1824056 * r1824062;
        double r1824064 = 6.0;
        double r1824065 = r1824063 / r1824064;
        double r1824066 = r1824065 + r1824055;
        return r1824066;
}

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. Simplified0.4

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

  4. Applied associate-*l/0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019152 
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0 u1 1) (<= 0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))