Average Error: 3.8 → 0.6
Time: 10.7s
Precision: 64
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z = -\infty:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(x, 2, -\left(z \cdot t\right) \cdot \left(y \cdot 9\right)\right)\right)\\ \mathbf{elif}\;\left(y \cdot 9\right) \cdot z \le 1.07579554652432578 \cdot 10^{266}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right) + \left(\left(a \cdot 27\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) \cdot \sqrt[3]{b}\\ \end{array}\]
\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}\;\left(y \cdot 9\right) \cdot z = -\infty:\\
\;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(x, 2, -\left(z \cdot t\right) \cdot \left(y \cdot 9\right)\right)\right)\\

\mathbf{elif}\;\left(y \cdot 9\right) \cdot z \le 1.07579554652432578 \cdot 10^{266}:\\
\;\;\;\;\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 - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right) + \left(\left(a \cdot 27\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) \cdot \sqrt[3]{b}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r981732 = x;
        double r981733 = 2.0;
        double r981734 = r981732 * r981733;
        double r981735 = y;
        double r981736 = 9.0;
        double r981737 = r981735 * r981736;
        double r981738 = z;
        double r981739 = r981737 * r981738;
        double r981740 = t;
        double r981741 = r981739 * r981740;
        double r981742 = r981734 - r981741;
        double r981743 = a;
        double r981744 = 27.0;
        double r981745 = r981743 * r981744;
        double r981746 = b;
        double r981747 = r981745 * r981746;
        double r981748 = r981742 + r981747;
        return r981748;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r981749 = y;
        double r981750 = 9.0;
        double r981751 = r981749 * r981750;
        double r981752 = z;
        double r981753 = r981751 * r981752;
        double r981754 = -inf.0;
        bool r981755 = r981753 <= r981754;
        double r981756 = 27.0;
        double r981757 = a;
        double r981758 = b;
        double r981759 = r981757 * r981758;
        double r981760 = x;
        double r981761 = 2.0;
        double r981762 = t;
        double r981763 = r981752 * r981762;
        double r981764 = r981763 * r981751;
        double r981765 = -r981764;
        double r981766 = fma(r981760, r981761, r981765);
        double r981767 = fma(r981756, r981759, r981766);
        double r981768 = 1.0757955465243258e+266;
        bool r981769 = r981753 <= r981768;
        double r981770 = r981760 * r981761;
        double r981771 = r981753 * r981762;
        double r981772 = r981770 - r981771;
        double r981773 = r981756 * r981758;
        double r981774 = r981757 * r981773;
        double r981775 = r981772 + r981774;
        double r981776 = r981750 * r981752;
        double r981777 = r981776 * r981762;
        double r981778 = r981749 * r981777;
        double r981779 = r981770 - r981778;
        double r981780 = r981757 * r981756;
        double r981781 = cbrt(r981758);
        double r981782 = r981781 * r981781;
        double r981783 = r981780 * r981782;
        double r981784 = r981783 * r981781;
        double r981785 = r981779 + r981784;
        double r981786 = r981769 ? r981775 : r981785;
        double r981787 = r981755 ? r981767 : r981786;
        return r981787;
}

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

Original3.8
Target2.6
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) z) < -inf.0

    1. Initial program 64.0

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

    3. Applied associate-*l*1.5

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

    4. Taylor expanded around inf 62.1

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

    5. Simplified1.5

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

    if -inf.0 < (* (* y 9.0) z) < 1.0757955465243258e+266

    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. Using strategy rm

    3. Applied associate-*l*0.5

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

    if 1.0757955465243258e+266 < (* (* y 9.0) z)

    1. Initial program 46.2

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

    3. Applied associate-*l*2.2

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

    5. Applied associate-*l*0.5

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

    7. Applied associate-*r*0.5

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

    9. Applied add-cube-cbrt0.6

      \[\leadsto \left(x \cdot 2 - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right) + \left(a \cdot 27\right) \cdot \color{blue}{\left(\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}\right)}\]

    10. Applied associate-*r*0.6

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z = -\infty:\\ \;\;\;\;\mathsf{fma}\left(27, a \cdot b, \mathsf{fma}\left(x, 2, -\left(z \cdot t\right) \cdot \left(y \cdot 9\right)\right)\right)\\ \mathbf{elif}\;\left(y \cdot 9\right) \cdot z \le 1.07579554652432578 \cdot 10^{266}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right) + \left(\left(a \cdot 27\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) \cdot \sqrt[3]{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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)))