Average Error: 0.4 → 0.3
Time: 26.7s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\right)\right)\]
\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{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)\right)\right)
double f(double u1, double u2) {
        double r71663 = 1.0;
        double r71664 = 6.0;
        double r71665 = r71663 / r71664;
        double r71666 = -2.0;
        double r71667 = u1;
        double r71668 = log(r71667);
        double r71669 = r71666 * r71668;
        double r71670 = 0.5;
        double r71671 = pow(r71669, r71670);
        double r71672 = r71665 * r71671;
        double r71673 = 2.0;
        double r71674 = atan2(1.0, 0.0);
        double r71675 = r71673 * r71674;
        double r71676 = u2;
        double r71677 = r71675 * r71676;
        double r71678 = cos(r71677);
        double r71679 = r71672 * r71678;
        double r71680 = r71679 + r71670;
        return r71680;
}
↓
double f(double u1, double u2) {
        double r71681 = 1.0;
        double r71682 = -2.0;
        double r71683 = u1;
        double r71684 = log(r71683);
        double r71685 = r71682 * r71684;
        double r71686 = 0.5;
        double r71687 = pow(r71685, r71686);
        double r71688 = r71681 * r71687;
        double r71689 = 6.0;
        double r71690 = r71688 / r71689;
        double r71691 = 2.0;
        double r71692 = atan2(1.0, 0.0);
        double r71693 = r71691 * r71692;
        double r71694 = u2;
        double r71695 = r71693 * r71694;
        double r71696 = cos(r71695);
        double r71697 = fma(r71690, r71696, r71686);
        double r71698 = expm1(r71697);
        double r71699 = log1p(r71698);
        return r71699;
}

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 *-un-lft-identity0.4

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

  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

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

  8. Applied log1p-expm1-u0.3

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

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019198 +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))