Average Error: 0.4 → 0.3
Time: 11.6s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{fma}\left(1 \cdot \frac{1}{\frac{6}{{\left(-2 \cdot \log u1\right)}^{0.5}}}, \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right)\right), 0.5\right)\]
\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
↓
\mathsf{fma}\left(1 \cdot \frac{1}{\frac{6}{{\left(-2 \cdot \log u1\right)}^{0.5}}}, \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right)\right), 0.5\right)
double f(double u1, double u2) {
        double r62673 = 1.0;
        double r62674 = 6.0;
        double r62675 = r62673 / r62674;
        double r62676 = -2.0;
        double r62677 = u1;
        double r62678 = log(r62677);
        double r62679 = r62676 * r62678;
        double r62680 = 0.5;
        double r62681 = pow(r62679, r62680);
        double r62682 = r62675 * r62681;
        double r62683 = 2.0;
        double r62684 = atan2(1.0, 0.0);
        double r62685 = r62683 * r62684;
        double r62686 = u2;
        double r62687 = r62685 * r62686;
        double r62688 = cos(r62687);
        double r62689 = r62682 * r62688;
        double r62690 = r62689 + r62680;
        return r62690;
}
↓
double f(double u1, double u2) {
        double r62691 = 1.0;
        double r62692 = 1.0;
        double r62693 = 6.0;
        double r62694 = -2.0;
        double r62695 = u1;
        double r62696 = log(r62695);
        double r62697 = r62694 * r62696;
        double r62698 = 0.5;
        double r62699 = pow(r62697, r62698);
        double r62700 = r62693 / r62699;
        double r62701 = r62692 / r62700;
        double r62702 = r62691 * r62701;
        double r62703 = 2.0;
        double r62704 = atan2(1.0, 0.0);
        double r62705 = r62703 * r62704;
        double r62706 = u2;
        double r62707 = r62705 * r62706;
        double r62708 = log1p(r62707);
        double r62709 = expm1(r62708);
        double r62710 = cos(r62709);
        double r62711 = fma(r62702, r62710, r62698);
        return r62711;
}

Error

Bits error versus u1

Bits error versus u2

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

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

  4. Applied div-inv0.4

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

  5. Applied associate-*l*0.4

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

  6. Simplified0.3

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

  8. Applied expm1-log1p-u0.3

    \[\leadsto \mathsf{fma}\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right)\right)}, 0.5\right)\]
  9. Using strategy rm

  10. Applied clear-num0.3

    \[\leadsto \mathsf{fma}\left(1 \cdot \color{blue}{\frac{1}{\frac{6}{{\left(-2 \cdot \log u1\right)}^{0.5}}}}, \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right)\right), 0.5\right)\]

  11. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(1 \cdot \frac{1}{\frac{6}{{\left(-2 \cdot \log u1\right)}^{0.5}}}, \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right)\right), 0.5\right)\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(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))