Average Error: 3.7 → 0.8
Time: 25.5s
Precision: 64
\[\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(y \cdot 9.0\right) \cdot z \le -5.677468110041499 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - \left(\sqrt[3]{9.0} \cdot \left(\left(\left(t \cdot y\right) \cdot \sqrt[3]{9.0}\right) \cdot \sqrt[3]{9.0}\right)\right) \cdot z\\ \mathbf{elif}\;\left(y \cdot 9.0\right) \cdot z \le 3.2463793104744963 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - \left(t \cdot \left(z \cdot y\right)\right) \cdot 9.0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 2.0, a \cdot \left(27.0 \cdot b\right)\right) - \left(\left(\left(t \cdot y\right) \cdot \left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right)\right) \cdot \sqrt[3]{9.0}\right) \cdot z\\ \end{array}\]
\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;\left(y \cdot 9.0\right) \cdot z \le -5.677468110041499 \cdot 10^{+60}:\\
\;\;\;\;\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - \left(\sqrt[3]{9.0} \cdot \left(\left(\left(t \cdot y\right) \cdot \sqrt[3]{9.0}\right) \cdot \sqrt[3]{9.0}\right)\right) \cdot z\\

\mathbf{elif}\;\left(y \cdot 9.0\right) \cdot z \le 3.2463793104744963 \cdot 10^{+302}:\\
\;\;\;\;\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - \left(t \cdot \left(z \cdot y\right)\right) \cdot 9.0\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, 2.0, a \cdot \left(27.0 \cdot b\right)\right) - \left(\left(\left(t \cdot y\right) \cdot \left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right)\right) \cdot \sqrt[3]{9.0}\right) \cdot z\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r30807704 = x;
        double r30807705 = 2.0;
        double r30807706 = r30807704 * r30807705;
        double r30807707 = y;
        double r30807708 = 9.0;
        double r30807709 = r30807707 * r30807708;
        double r30807710 = z;
        double r30807711 = r30807709 * r30807710;
        double r30807712 = t;
        double r30807713 = r30807711 * r30807712;
        double r30807714 = r30807706 - r30807713;
        double r30807715 = a;
        double r30807716 = 27.0;
        double r30807717 = r30807715 * r30807716;
        double r30807718 = b;
        double r30807719 = r30807717 * r30807718;
        double r30807720 = r30807714 + r30807719;
        return r30807720;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r30807721 = y;
        double r30807722 = 9.0;
        double r30807723 = r30807721 * r30807722;
        double r30807724 = z;
        double r30807725 = r30807723 * r30807724;
        double r30807726 = -5.677468110041499e+60;
        bool r30807727 = r30807725 <= r30807726;
        double r30807728 = 27.0;
        double r30807729 = a;
        double r30807730 = r30807728 * r30807729;
        double r30807731 = b;
        double r30807732 = x;
        double r30807733 = 2.0;
        double r30807734 = r30807732 * r30807733;
        double r30807735 = fma(r30807730, r30807731, r30807734);
        double r30807736 = cbrt(r30807722);
        double r30807737 = t;
        double r30807738 = r30807737 * r30807721;
        double r30807739 = r30807738 * r30807736;
        double r30807740 = r30807739 * r30807736;
        double r30807741 = r30807736 * r30807740;
        double r30807742 = r30807741 * r30807724;
        double r30807743 = r30807735 - r30807742;
        double r30807744 = 3.2463793104744963e+302;
        bool r30807745 = r30807725 <= r30807744;
        double r30807746 = r30807724 * r30807721;
        double r30807747 = r30807737 * r30807746;
        double r30807748 = r30807747 * r30807722;
        double r30807749 = r30807735 - r30807748;
        double r30807750 = r30807728 * r30807731;
        double r30807751 = r30807729 * r30807750;
        double r30807752 = fma(r30807732, r30807733, r30807751);
        double r30807753 = r30807736 * r30807736;
        double r30807754 = r30807738 * r30807753;
        double r30807755 = r30807754 * r30807736;
        double r30807756 = r30807755 * r30807724;
        double r30807757 = r30807752 - r30807756;
        double r30807758 = r30807745 ? r30807749 : r30807757;
        double r30807759 = r30807727 ? r30807743 : r30807758;
        return r30807759;
}

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

Original3.7
Target2.6
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + a \cdot \left(27.0 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2.0 - 9.0 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27.0\right) \cdot b\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (* y 9.0) z) < -5.677468110041499e+60

    1. Initial program 10.8

      \[\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\]

    2. Simplified3.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - z \cdot \left(\left(t \cdot y\right) \cdot 9.0\right)}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt3.1

      \[\leadsto \mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - z \cdot \left(\left(t \cdot y\right) \cdot \color{blue}{\left(\left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right) \cdot \sqrt[3]{9.0}\right)}\right)\]

    5. Applied associate-*r*3.1

      \[\leadsto \mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - z \cdot \color{blue}{\left(\left(\left(t \cdot y\right) \cdot \left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right)\right) \cdot \sqrt[3]{9.0}\right)}\]
    6. Using strategy rm

    7. Applied associate-*r*3.1

      \[\leadsto \mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - z \cdot \left(\color{blue}{\left(\left(\left(t \cdot y\right) \cdot \sqrt[3]{9.0}\right) \cdot \sqrt[3]{9.0}\right)} \cdot \sqrt[3]{9.0}\right)\]

    if -5.677468110041499e+60 < (* (* y 9.0) z) < 3.2463793104744963e+302

    1. Initial program 0.4

      \[\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\]

    2. Simplified3.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - z \cdot \left(\left(t \cdot y\right) \cdot 9.0\right)}\]

    3. Taylor expanded around inf 0.4

      \[\leadsto \mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - \color{blue}{9.0 \cdot \left(t \cdot \left(z \cdot y\right)\right)}\]

    if 3.2463793104744963e+302 < (* (* y 9.0) z)

    1. Initial program 58.0

      \[\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\]

    2. Simplified0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - z \cdot \left(\left(t \cdot y\right) \cdot 9.0\right)}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt0.8

      \[\leadsto \mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - z \cdot \left(\left(t \cdot y\right) \cdot \color{blue}{\left(\left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right) \cdot \sqrt[3]{9.0}\right)}\right)\]

    5. Applied associate-*r*0.9

      \[\leadsto \mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - z \cdot \color{blue}{\left(\left(\left(t \cdot y\right) \cdot \left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right)\right) \cdot \sqrt[3]{9.0}\right)}\]

    6. Taylor expanded around 0 0.6

      \[\leadsto \color{blue}{\left(2.0 \cdot x + 27.0 \cdot \left(a \cdot b\right)\right)} - z \cdot \left(\left(\left(t \cdot y\right) \cdot \left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right)\right) \cdot \sqrt[3]{9.0}\right)\]

    7. Simplified0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2.0, a \cdot \left(b \cdot 27.0\right)\right)} - z \cdot \left(\left(\left(t \cdot y\right) \cdot \left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right)\right) \cdot \sqrt[3]{9.0}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 9.0\right) \cdot z \le -5.677468110041499 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - \left(\sqrt[3]{9.0} \cdot \left(\left(\left(t \cdot y\right) \cdot \sqrt[3]{9.0}\right) \cdot \sqrt[3]{9.0}\right)\right) \cdot z\\ \mathbf{elif}\;\left(y \cdot 9.0\right) \cdot z \le 3.2463793104744963 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(27.0 \cdot a, b, x \cdot 2.0\right) - \left(t \cdot \left(z \cdot y\right)\right) \cdot 9.0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 2.0, a \cdot \left(27.0 \cdot b\right)\right) - \left(\left(\left(t \cdot y\right) \cdot \left(\sqrt[3]{9.0} \cdot \sqrt[3]{9.0}\right)\right) \cdot \sqrt[3]{9.0}\right) \cdot z\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b))) (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b)))

  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))