Average Error: 0.4 → 0.3
Time: 37.6s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \log \left(e^{\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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 \log \left(e^{\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r3345741 = 1.0;
        double r3345742 = 6.0;
        double r3345743 = r3345741 / r3345742;
        double r3345744 = -2.0;
        double r3345745 = u1;
        double r3345746 = log(r3345745);
        double r3345747 = r3345744 * r3345746;
        double r3345748 = 0.5;
        double r3345749 = pow(r3345747, r3345748);
        double r3345750 = r3345743 * r3345749;
        double r3345751 = 2.0;
        double r3345752 = atan2(1.0, 0.0);
        double r3345753 = r3345751 * r3345752;
        double r3345754 = u2;
        double r3345755 = r3345753 * r3345754;
        double r3345756 = cos(r3345755);
        double r3345757 = r3345750 * r3345756;
        double r3345758 = r3345757 + r3345748;
        return r3345758;
}
↓
double f(double u1, double u2) {
        double r3345759 = 1.0;
        double r3345760 = -2.0;
        double r3345761 = u1;
        double r3345762 = log(r3345761);
        double r3345763 = r3345760 * r3345762;
        double r3345764 = 0.5;
        double r3345765 = pow(r3345763, r3345764);
        double r3345766 = 6.0;
        double r3345767 = r3345765 / r3345766;
        double r3345768 = exp(r3345767);
        double r3345769 = log(r3345768);
        double r3345770 = r3345759 * r3345769;
        double r3345771 = 2.0;
        double r3345772 = atan2(1.0, 0.0);
        double r3345773 = r3345771 * r3345772;
        double r3345774 = u2;
        double r3345775 = r3345773 * r3345774;
        double r3345776 = cos(r3345775);
        double r3345777 = r3345770 * r3345776;
        double r3345778 = r3345777 + r3345764;
        return r3345778;
}

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 sqr-pow0.5

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

  9. Applied add-log-exp0.5

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0.0 u1 1.0) (<= 0.0 u2 1.0))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))