Average Error: 3.5 → 0.6
Time: 18.4s
Precision: 64
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;b \cdot \left(a \cdot 27\right) + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) = -\infty:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot b\right) \cdot a - y \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)\\ \mathbf{elif}\;b \cdot \left(a \cdot 27\right) + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \le 1.479731018921087016022825291889163898001 \cdot 10^{290}:\\ \;\;\;\;b \cdot \left(a \cdot 27\right) + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \sqrt{27} \cdot \left(\left(b \cdot a\right) \cdot \sqrt{27}\right) - \left(y \cdot \left(z \cdot t\right)\right) \cdot 9\right)\\ \end{array}\]
\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}\;b \cdot \left(a \cdot 27\right) + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) = -\infty:\\
\;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot b\right) \cdot a - y \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)\\

\mathbf{elif}\;b \cdot \left(a \cdot 27\right) + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \le 1.479731018921087016022825291889163898001 \cdot 10^{290}:\\
\;\;\;\;b \cdot \left(a \cdot 27\right) + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r31758859 = x;
        double r31758860 = 2.0;
        double r31758861 = r31758859 * r31758860;
        double r31758862 = y;
        double r31758863 = 9.0;
        double r31758864 = r31758862 * r31758863;
        double r31758865 = z;
        double r31758866 = r31758864 * r31758865;
        double r31758867 = t;
        double r31758868 = r31758866 * r31758867;
        double r31758869 = r31758861 - r31758868;
        double r31758870 = a;
        double r31758871 = 27.0;
        double r31758872 = r31758870 * r31758871;
        double r31758873 = b;
        double r31758874 = r31758872 * r31758873;
        double r31758875 = r31758869 + r31758874;
        return r31758875;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r31758876 = b;
        double r31758877 = a;
        double r31758878 = 27.0;
        double r31758879 = r31758877 * r31758878;
        double r31758880 = r31758876 * r31758879;
        double r31758881 = x;
        double r31758882 = 2.0;
        double r31758883 = r31758881 * r31758882;
        double r31758884 = y;
        double r31758885 = 9.0;
        double r31758886 = r31758884 * r31758885;
        double r31758887 = z;
        double r31758888 = r31758886 * r31758887;
        double r31758889 = t;
        double r31758890 = r31758888 * r31758889;
        double r31758891 = r31758883 - r31758890;
        double r31758892 = r31758880 + r31758891;
        double r31758893 = -inf.0;
        bool r31758894 = r31758892 <= r31758893;
        double r31758895 = r31758878 * r31758876;
        double r31758896 = r31758895 * r31758877;
        double r31758897 = r31758887 * r31758885;
        double r31758898 = r31758897 * r31758889;
        double r31758899 = r31758884 * r31758898;
        double r31758900 = r31758896 - r31758899;
        double r31758901 = fma(r31758882, r31758881, r31758900);
        double r31758902 = 1.479731018921087e+290;
        bool r31758903 = r31758892 <= r31758902;
        double r31758904 = sqrt(r31758878);
        double r31758905 = r31758876 * r31758877;
        double r31758906 = r31758905 * r31758904;
        double r31758907 = r31758904 * r31758906;
        double r31758908 = r31758887 * r31758889;
        double r31758909 = r31758884 * r31758908;
        double r31758910 = r31758909 * r31758885;
        double r31758911 = r31758907 - r31758910;
        double r31758912 = fma(r31758882, r31758881, r31758911);
        double r31758913 = r31758903 ? r31758892 : r31758912;
        double r31758914 = r31758894 ? r31758901 : r31758913;
        return r31758914;
}

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
Target2.6
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811188954625810696587370427881 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)) < -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. Simplified61.8

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

    4. Applied associate-*l*60.3

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

    6. Applied associate-*l*0.6

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

    if -inf.0 < (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)) < 1.479731018921087e+290

    1. Initial program 0.3

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

    if 1.479731018921087e+290 < (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b))

    1. Initial program 29.4

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

    2. Simplified29.4

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

    3. Taylor expanded around inf 28.9

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

    5. Applied add-sqr-sqrt28.9

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

    6. Applied associate-*l*28.9

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

    8. Applied associate-*r*4.8

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(a \cdot 27\right) + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) = -\infty:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(27 \cdot b\right) \cdot a - y \cdot \left(\left(z \cdot 9\right) \cdot t\right)\right)\\ \mathbf{elif}\;b \cdot \left(a \cdot 27\right) + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) \le 1.479731018921087016022825291889163898001 \cdot 10^{290}:\\ \;\;\;\;b \cdot \left(a \cdot 27\right) + \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \sqrt{27} \cdot \left(\left(b \cdot a\right) \cdot \sqrt{27}\right) - \left(y \cdot \left(z \cdot t\right)\right) \cdot 9\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +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)))