Average Error: 5.6 → 2.7
Time: 1.2m
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7702228306921125838848:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(z \cdot \left(\left(y \cdot x\right) \cdot 18\right)\right) \cdot t - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;t \le 1.806883127258244205731970851622029403045 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\left(18 \cdot t\right) \cdot x\right), z, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), \left(j \cdot k\right) \cdot 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right)\right) \cdot t - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - k \cdot \left(j \cdot 27\right)\\ \end{array}\]
\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k
↓
\begin{array}{l}
\mathbf{if}\;t \le -7702228306921125838848:\\
\;\;\;\;\left(\left(b \cdot c + \left(\left(z \cdot \left(\left(y \cdot x\right) \cdot 18\right)\right) \cdot t - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\

\mathbf{elif}\;t \le 1.806883127258244205731970851622029403045 \cdot 10^{-164}:\\
\;\;\;\;\mathsf{fma}\left(y \cdot \left(\left(18 \cdot t\right) \cdot x\right), z, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), \left(j \cdot k\right) \cdot 27\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(b \cdot c + \left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right)\right) \cdot t - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - k \cdot \left(j \cdot 27\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r35727768 = x;
        double r35727769 = 18.0;
        double r35727770 = r35727768 * r35727769;
        double r35727771 = y;
        double r35727772 = r35727770 * r35727771;
        double r35727773 = z;
        double r35727774 = r35727772 * r35727773;
        double r35727775 = t;
        double r35727776 = r35727774 * r35727775;
        double r35727777 = a;
        double r35727778 = 4.0;
        double r35727779 = r35727777 * r35727778;
        double r35727780 = r35727779 * r35727775;
        double r35727781 = r35727776 - r35727780;
        double r35727782 = b;
        double r35727783 = c;
        double r35727784 = r35727782 * r35727783;
        double r35727785 = r35727781 + r35727784;
        double r35727786 = r35727768 * r35727778;
        double r35727787 = i;
        double r35727788 = r35727786 * r35727787;
        double r35727789 = r35727785 - r35727788;
        double r35727790 = j;
        double r35727791 = 27.0;
        double r35727792 = r35727790 * r35727791;
        double r35727793 = k;
        double r35727794 = r35727792 * r35727793;
        double r35727795 = r35727789 - r35727794;
        return r35727795;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r35727796 = t;
        double r35727797 = -7.702228306921126e+21;
        bool r35727798 = r35727796 <= r35727797;
        double r35727799 = b;
        double r35727800 = c;
        double r35727801 = r35727799 * r35727800;
        double r35727802 = z;
        double r35727803 = y;
        double r35727804 = x;
        double r35727805 = r35727803 * r35727804;
        double r35727806 = 18.0;
        double r35727807 = r35727805 * r35727806;
        double r35727808 = r35727802 * r35727807;
        double r35727809 = r35727808 * r35727796;
        double r35727810 = a;
        double r35727811 = 4.0;
        double r35727812 = r35727810 * r35727811;
        double r35727813 = r35727796 * r35727812;
        double r35727814 = r35727809 - r35727813;
        double r35727815 = r35727801 + r35727814;
        double r35727816 = r35727804 * r35727811;
        double r35727817 = i;
        double r35727818 = r35727816 * r35727817;
        double r35727819 = r35727815 - r35727818;
        double r35727820 = j;
        double r35727821 = 27.0;
        double r35727822 = k;
        double r35727823 = r35727821 * r35727822;
        double r35727824 = r35727820 * r35727823;
        double r35727825 = r35727819 - r35727824;
        double r35727826 = 1.8068831272582442e-164;
        bool r35727827 = r35727796 <= r35727826;
        double r35727828 = r35727806 * r35727796;
        double r35727829 = r35727828 * r35727804;
        double r35727830 = r35727803 * r35727829;
        double r35727831 = r35727817 * r35727804;
        double r35727832 = fma(r35727796, r35727810, r35727831);
        double r35727833 = r35727820 * r35727822;
        double r35727834 = r35727833 * r35727821;
        double r35727835 = fma(r35727811, r35727832, r35727834);
        double r35727836 = r35727801 - r35727835;
        double r35727837 = fma(r35727830, r35727802, r35727836);
        double r35727838 = r35727804 * r35727806;
        double r35727839 = r35727803 * r35727802;
        double r35727840 = r35727838 * r35727839;
        double r35727841 = r35727840 * r35727796;
        double r35727842 = r35727841 - r35727813;
        double r35727843 = r35727801 + r35727842;
        double r35727844 = r35727843 - r35727818;
        double r35727845 = r35727820 * r35727821;
        double r35727846 = r35727822 * r35727845;
        double r35727847 = r35727844 - r35727846;
        double r35727848 = r35727827 ? r35727837 : r35727847;
        double r35727849 = r35727798 ? r35727825 : r35727848;
        return r35727849;
}

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

Bits error versus c

Bits error versus i

Bits error versus j

Bits error versus k

Target

Original5.6
Target1.7
Herbie2.7
\[\begin{array}{l} \mathbf{if}\;t \lt -1.62108153975413982700795070153457058168 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.6802794380522243500308832153677940369:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -7.702228306921126e+21

    1. Initial program 2.1

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

    2. Taylor expanded around 0 2.1

      \[\leadsto \left(\left(\left(\left(\color{blue}{\left(18 \cdot \left(x \cdot y\right)\right)} \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
    3. Using strategy rm

    4. Applied associate-*l*2.1

      \[\leadsto \left(\left(\left(\left(\left(18 \cdot \left(x \cdot y\right)\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \color{blue}{j \cdot \left(27 \cdot k\right)}\]

    if -7.702228306921126e+21 < t < 1.8068831272582442e-164

    1. Initial program 8.2

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

    2. Simplified4.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(y \cdot x\right) \cdot 18\right) \cdot t, z, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), 27 \cdot \left(k \cdot j\right)\right)\right)}\]
    3. Using strategy rm

    4. Applied associate-*l*4.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot x\right) \cdot \left(18 \cdot t\right)}, z, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), 27 \cdot \left(k \cdot j\right)\right)\right)\]
    5. Using strategy rm

    6. Applied associate-*l*1.6

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(x \cdot \left(18 \cdot t\right)\right)}, z, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), 27 \cdot \left(k \cdot j\right)\right)\right)\]

    if 1.8068831272582442e-164 < t

    1. Initial program 3.8

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
    2. Using strategy rm

    3. Applied associate-*l*4.5

      \[\leadsto \left(\left(\left(\color{blue}{\left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right)\right)} \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7702228306921125838848:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(z \cdot \left(\left(y \cdot x\right) \cdot 18\right)\right) \cdot t - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;t \le 1.806883127258244205731970851622029403045 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\left(18 \cdot t\right) \cdot x\right), z, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), \left(j \cdot k\right) \cdot 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right)\right) \cdot t - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - k \cdot \left(j \cdot 27\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18.0 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4.0)) (- (* c b) (* 27.0 (* k j)))) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))