Average Error: 0.4 → 0.3
Time: 10.2s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\log \left(e^{\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(\left(2 \cdot \pi\right) \cdot \sqrt{u2}\right) \cdot \sqrt{u2}\right) + 0.5}\right)\]
\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
↓
\log \left(e^{\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(\left(2 \cdot \pi\right) \cdot \sqrt{u2}\right) \cdot \sqrt{u2}\right) + 0.5}\right)
double f(double u1, double u2) {
        double r59652 = 1.0;
        double r59653 = 6.0;
        double r59654 = r59652 / r59653;
        double r59655 = -2.0;
        double r59656 = u1;
        double r59657 = log(r59656);
        double r59658 = r59655 * r59657;
        double r59659 = 0.5;
        double r59660 = pow(r59658, r59659);
        double r59661 = r59654 * r59660;
        double r59662 = 2.0;
        double r59663 = atan2(1.0, 0.0);
        double r59664 = r59662 * r59663;
        double r59665 = u2;
        double r59666 = r59664 * r59665;
        double r59667 = cos(r59666);
        double r59668 = r59661 * r59667;
        double r59669 = r59668 + r59659;
        return r59669;
}
↓
double f(double u1, double u2) {
        double r59670 = 1.0;
        double r59671 = -2.0;
        double r59672 = u1;
        double r59673 = log(r59672);
        double r59674 = r59671 * r59673;
        double r59675 = 0.5;
        double r59676 = pow(r59674, r59675);
        double r59677 = 6.0;
        double r59678 = r59676 / r59677;
        double r59679 = r59670 * r59678;
        double r59680 = 2.0;
        double r59681 = atan2(1.0, 0.0);
        double r59682 = r59680 * r59681;
        double r59683 = u2;
        double r59684 = sqrt(r59683);
        double r59685 = r59682 * r59684;
        double r59686 = r59685 * r59684;
        double r59687 = cos(r59686);
        double r59688 = r59679 * r59687;
        double r59689 = r59688 + r59675;
        double r59690 = exp(r59689);
        double r59691 = log(r59690);
        return r59691;
}

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 add-log-exp0.3

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

  8. Applied add-log-exp0.3

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

  9. Applied sum-log0.3

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

  10. Simplified0.3

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

  12. Applied add-sqr-sqrt0.3

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

  13. Applied associate-*r*0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2020025 
(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))