Average Error: 0.4 → 0.3
Time: 1.3m
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{{\left(\log u1 \cdot -2\right)}^{0.5} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}{6} + 0.5\]
\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(\left(2 \cdot \pi\right) \cdot u2\right)}{6} + 0.5
double f(double u1, double u2) {
        double r7740717 = 1.0;
        double r7740718 = 6.0;
        double r7740719 = r7740717 / r7740718;
        double r7740720 = -2.0;
        double r7740721 = u1;
        double r7740722 = log(r7740721);
        double r7740723 = r7740720 * r7740722;
        double r7740724 = 0.5;
        double r7740725 = pow(r7740723, r7740724);
        double r7740726 = r7740719 * r7740725;
        double r7740727 = 2.0;
        double r7740728 = atan2(1.0, 0.0);
        double r7740729 = r7740727 * r7740728;
        double r7740730 = u2;
        double r7740731 = r7740729 * r7740730;
        double r7740732 = cos(r7740731);
        double r7740733 = r7740726 * r7740732;
        double r7740734 = r7740733 + r7740724;
        return r7740734;
}
↓
double f(double u1, double u2) {
        double r7740735 = u1;
        double r7740736 = log(r7740735);
        double r7740737 = -2.0;
        double r7740738 = r7740736 * r7740737;
        double r7740739 = 0.5;
        double r7740740 = pow(r7740738, r7740739);
        double r7740741 = 2.0;
        double r7740742 = atan2(1.0, 0.0);
        double r7740743 = r7740741 * r7740742;
        double r7740744 = u2;
        double r7740745 = r7740743 * r7740744;
        double r7740746 = cos(r7740745);
        double r7740747 = r7740740 * r7740746;
        double r7740748 = 6.0;
        double r7740749 = r7740747 / r7740748;
        double r7740750 = r7740749 + r7740739;
        return r7740750;
}

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 sqr-pow0.6

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

  5. Applied associate-*r*0.5

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

  7. Applied associate-*l/0.5

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

  8. Applied associate-*l/0.5

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

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019119 
(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))