Average Error: 5.6 → 0.7
Time: 37.1s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k \le -2.768204954998920714545692361756405988778 \cdot 10^{305}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(\left(y \cdot \left(z \cdot \left(t \cdot x\right)\right)\right) \cdot \sqrt{18}\right) \cdot \sqrt{18} - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)\right)\\ \mathbf{elif}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k \le 4.506487107324018323312747251398502442091 \cdot 10^{306}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(y \cdot 18\right) \cdot \left(z \cdot \left(t \cdot x\right)\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(\sqrt{27} \cdot \left(k \cdot j\right)\right) \cdot \sqrt{27}\right)\right)\\ \end{array}\]
\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}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k \le -2.768204954998920714545692361756405988778 \cdot 10^{305}:\\
\;\;\;\;\mathsf{fma}\left(b, c, \left(\left(y \cdot \left(z \cdot \left(t \cdot x\right)\right)\right) \cdot \sqrt{18}\right) \cdot \sqrt{18} - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)\right)\\

\mathbf{elif}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k \le 4.506487107324018323312747251398502442091 \cdot 10^{306}:\\
\;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(b, c, \left(y \cdot 18\right) \cdot \left(z \cdot \left(t \cdot x\right)\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(\sqrt{27} \cdot \left(k \cdot j\right)\right) \cdot \sqrt{27}\right)\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 r35145645 = x;
        double r35145646 = 18.0;
        double r35145647 = r35145645 * r35145646;
        double r35145648 = y;
        double r35145649 = r35145647 * r35145648;
        double r35145650 = z;
        double r35145651 = r35145649 * r35145650;
        double r35145652 = t;
        double r35145653 = r35145651 * r35145652;
        double r35145654 = a;
        double r35145655 = 4.0;
        double r35145656 = r35145654 * r35145655;
        double r35145657 = r35145656 * r35145652;
        double r35145658 = r35145653 - r35145657;
        double r35145659 = b;
        double r35145660 = c;
        double r35145661 = r35145659 * r35145660;
        double r35145662 = r35145658 + r35145661;
        double r35145663 = r35145645 * r35145655;
        double r35145664 = i;
        double r35145665 = r35145663 * r35145664;
        double r35145666 = r35145662 - r35145665;
        double r35145667 = j;
        double r35145668 = 27.0;
        double r35145669 = r35145667 * r35145668;
        double r35145670 = k;
        double r35145671 = r35145669 * r35145670;
        double r35145672 = r35145666 - r35145671;
        return r35145672;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r35145673 = t;
        double r35145674 = x;
        double r35145675 = 18.0;
        double r35145676 = r35145674 * r35145675;
        double r35145677 = y;
        double r35145678 = r35145676 * r35145677;
        double r35145679 = z;
        double r35145680 = r35145678 * r35145679;
        double r35145681 = r35145673 * r35145680;
        double r35145682 = a;
        double r35145683 = 4.0;
        double r35145684 = r35145682 * r35145683;
        double r35145685 = r35145684 * r35145673;
        double r35145686 = r35145681 - r35145685;
        double r35145687 = c;
        double r35145688 = b;
        double r35145689 = r35145687 * r35145688;
        double r35145690 = r35145686 + r35145689;
        double r35145691 = r35145674 * r35145683;
        double r35145692 = i;
        double r35145693 = r35145691 * r35145692;
        double r35145694 = r35145690 - r35145693;
        double r35145695 = 27.0;
        double r35145696 = j;
        double r35145697 = r35145695 * r35145696;
        double r35145698 = k;
        double r35145699 = r35145697 * r35145698;
        double r35145700 = r35145694 - r35145699;
        double r35145701 = -2.7682049549989207e+305;
        bool r35145702 = r35145700 <= r35145701;
        double r35145703 = r35145673 * r35145674;
        double r35145704 = r35145679 * r35145703;
        double r35145705 = r35145677 * r35145704;
        double r35145706 = sqrt(r35145675);
        double r35145707 = r35145705 * r35145706;
        double r35145708 = r35145707 * r35145706;
        double r35145709 = r35145674 * r35145692;
        double r35145710 = fma(r35145673, r35145682, r35145709);
        double r35145711 = r35145698 * r35145696;
        double r35145712 = r35145711 * r35145695;
        double r35145713 = fma(r35145683, r35145710, r35145712);
        double r35145714 = r35145708 - r35145713;
        double r35145715 = fma(r35145688, r35145687, r35145714);
        double r35145716 = 4.5064871073240183e+306;
        bool r35145717 = r35145700 <= r35145716;
        double r35145718 = r35145677 * r35145675;
        double r35145719 = r35145718 * r35145704;
        double r35145720 = sqrt(r35145695);
        double r35145721 = r35145720 * r35145711;
        double r35145722 = r35145721 * r35145720;
        double r35145723 = fma(r35145683, r35145710, r35145722);
        double r35145724 = r35145719 - r35145723;
        double r35145725 = fma(r35145688, r35145687, r35145724);
        double r35145726 = r35145717 ? r35145700 : r35145725;
        double r35145727 = r35145702 ? r35145715 : r35145726;
        return r35145727;
}

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
Target1.6
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;t \lt -1.62108153975413982700795070153457058168 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.6802794380522243500308832153677940369:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)) < -2.7682049549989207e+305

    1. Initial program 58.2

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

    2. Simplified12.5

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

    4. Applied associate-*r*5.8

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

    6. Applied associate-*r*4.9

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

    8. Applied add-sqr-sqrt5.1

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

    9. Applied associate-*r*5.1

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

    if -2.7682049549989207e+305 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)) < 4.5064871073240183e+306

    1. Initial program 0.2

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

    if 4.5064871073240183e+306 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k))

    1. Initial program 60.7

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

    2. Simplified12.1

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

    4. Applied associate-*r*5.7

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

    6. Applied add-sqr-sqrt5.7

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

    7. Applied associate-*l*5.7

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k \le -2.768204954998920714545692361756405988778 \cdot 10^{305}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(\left(y \cdot \left(z \cdot \left(t \cdot x\right)\right)\right) \cdot \sqrt{18}\right) \cdot \sqrt{18} - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot j\right) \cdot 27\right)\right)\\ \mathbf{elif}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k \le 4.506487107324018323312747251398502442091 \cdot 10^{306}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(y \cdot 18\right) \cdot \left(z \cdot \left(t \cdot x\right)\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(\sqrt{27} \cdot \left(k \cdot j\right)\right) \cdot \sqrt{27}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +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)))