Average Error: 3.5 → 0.6
Time: 4.3s
Precision: 64
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot 9 \le -2.12931374382048943 \cdot 10^{31}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right)\\ \mathbf{elif}\;y \cdot 9 \le 26838212.4218946211:\\ \;\;\;\;\mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right)\\ \end{array}\]
\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;y \cdot 9 \le -2.12931374382048943 \cdot 10^{31}:\\
\;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right)\\

\mathbf{elif}\;y \cdot 9 \le 26838212.4218946211:\\
\;\;\;\;\mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r733669 = x;
        double r733670 = 2.0;
        double r733671 = r733669 * r733670;
        double r733672 = y;
        double r733673 = 9.0;
        double r733674 = r733672 * r733673;
        double r733675 = z;
        double r733676 = r733674 * r733675;
        double r733677 = t;
        double r733678 = r733676 * r733677;
        double r733679 = r733671 - r733678;
        double r733680 = a;
        double r733681 = 27.0;
        double r733682 = r733680 * r733681;
        double r733683 = b;
        double r733684 = r733682 * r733683;
        double r733685 = r733679 + r733684;
        return r733685;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r733686 = y;
        double r733687 = 9.0;
        double r733688 = r733686 * r733687;
        double r733689 = -2.1293137438204894e+31;
        bool r733690 = r733688 <= r733689;
        double r733691 = a;
        double r733692 = 27.0;
        double r733693 = b;
        double r733694 = r733692 * r733693;
        double r733695 = x;
        double r733696 = 2.0;
        double r733697 = r733695 * r733696;
        double r733698 = z;
        double r733699 = r733687 * r733698;
        double r733700 = t;
        double r733701 = r733699 * r733700;
        double r733702 = r733686 * r733701;
        double r733703 = r733697 - r733702;
        double r733704 = fma(r733691, r733694, r733703);
        double r733705 = 26838212.42189462;
        bool r733706 = r733688 <= r733705;
        double r733707 = r733691 * r733693;
        double r733708 = r733692 * r733707;
        double r733709 = r733698 * r733686;
        double r733710 = r733700 * r733709;
        double r733711 = r733687 * r733710;
        double r733712 = r733708 - r733711;
        double r733713 = fma(r733696, r733695, r733712);
        double r733714 = r733698 * r733700;
        double r733715 = r733688 * r733714;
        double r733716 = r733697 - r733715;
        double r733717 = fma(r733691, r733694, r733716);
        double r733718 = r733706 ? r733713 : r733717;
        double r733719 = r733690 ? r733704 : r733718;
        return r733719;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original3.5
Target2.5
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y 9.0) < -2.1293137438204894e+31

    1. Initial program 8.1

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

    2. Simplified8.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\]
    3. Using strategy rm

    4. Applied associate-*l*1.0

      \[\leadsto \mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right)\]
    5. Using strategy rm

    6. Applied associate-*l*0.8

      \[\leadsto \mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \color{blue}{y \cdot \left(9 \cdot \left(z \cdot t\right)\right)}\right)\]
    7. Using strategy rm

    8. Applied associate-*r*0.9

      \[\leadsto \mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - y \cdot \color{blue}{\left(\left(9 \cdot z\right) \cdot t\right)}\right)\]

    if -2.1293137438204894e+31 < (* y 9.0) < 26838212.42189462

    1. Initial program 0.5

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

    2. Simplified0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\]

    3. Taylor expanded around inf 0.5

      \[\leadsto \color{blue}{\left(2 \cdot x + 27 \cdot \left(a \cdot b\right)\right) - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)}\]

    4. Simplified0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)}\]

    if 26838212.42189462 < (* y 9.0)

    1. Initial program 8.0

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

    2. Simplified8.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\]
    3. Using strategy rm

    4. Applied associate-*l*0.9

      \[\leadsto \mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 9 \le -2.12931374382048943 \cdot 10^{31}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right)\\ \mathbf{elif}\;y \cdot 9 \le 26838212.4218946211:\\ \;\;\;\;\mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2) (* (* (* y 9) z) t)) (* a (* 27 b))) (+ (- (* x 2) (* 9 (* y (* t z)))) (* (* a 27) b)))

  (+ (- (* x 2) (* (* (* y 9) z) t)) (* (* a 27) b)))