Average Error: 5.4 → 2.1
Time: 27.1s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i = -\infty:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(\left(t \cdot x\right) \cdot \left(y \cdot 18.0\right)\right) \cdot z - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), 27.0 \cdot \left(k \cdot j\right)\right)\right)\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i \le 2.403209667818115 \cdot 10^{+294}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - j \cdot \left(27.0 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(y \cdot \left(\left(z \cdot x\right) \cdot t\right)\right) \cdot 18.0 - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - \sqrt[3]{\left(k \cdot j\right) \cdot \left(27.0 \cdot \left(\left(27.0 \cdot \left(k \cdot j\right)\right) \cdot \left(27.0 \cdot \left(k \cdot j\right)\right)\right)\right)}\\ \end{array}\]
\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k
↓
\begin{array}{l}
\mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i = -\infty:\\
\;\;\;\;\mathsf{fma}\left(b, c, \left(\left(t \cdot x\right) \cdot \left(y \cdot 18.0\right)\right) \cdot z - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), 27.0 \cdot \left(k \cdot j\right)\right)\right)\\

\mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i \le 2.403209667818115 \cdot 10^{+294}:\\
\;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - j \cdot \left(27.0 \cdot k\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(c \cdot b + \left(\left(y \cdot \left(\left(z \cdot x\right) \cdot t\right)\right) \cdot 18.0 - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - \sqrt[3]{\left(k \cdot j\right) \cdot \left(27.0 \cdot \left(\left(27.0 \cdot \left(k \cdot j\right)\right) \cdot \left(27.0 \cdot \left(k \cdot j\right)\right)\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 r39400644 = x;
        double r39400645 = 18.0;
        double r39400646 = r39400644 * r39400645;
        double r39400647 = y;
        double r39400648 = r39400646 * r39400647;
        double r39400649 = z;
        double r39400650 = r39400648 * r39400649;
        double r39400651 = t;
        double r39400652 = r39400650 * r39400651;
        double r39400653 = a;
        double r39400654 = 4.0;
        double r39400655 = r39400653 * r39400654;
        double r39400656 = r39400655 * r39400651;
        double r39400657 = r39400652 - r39400656;
        double r39400658 = b;
        double r39400659 = c;
        double r39400660 = r39400658 * r39400659;
        double r39400661 = r39400657 + r39400660;
        double r39400662 = r39400644 * r39400654;
        double r39400663 = i;
        double r39400664 = r39400662 * r39400663;
        double r39400665 = r39400661 - r39400664;
        double r39400666 = j;
        double r39400667 = 27.0;
        double r39400668 = r39400666 * r39400667;
        double r39400669 = k;
        double r39400670 = r39400668 * r39400669;
        double r39400671 = r39400665 - r39400670;
        return r39400671;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r39400672 = t;
        double r39400673 = x;
        double r39400674 = 18.0;
        double r39400675 = r39400673 * r39400674;
        double r39400676 = y;
        double r39400677 = r39400675 * r39400676;
        double r39400678 = z;
        double r39400679 = r39400677 * r39400678;
        double r39400680 = r39400672 * r39400679;
        double r39400681 = a;
        double r39400682 = 4.0;
        double r39400683 = r39400681 * r39400682;
        double r39400684 = r39400683 * r39400672;
        double r39400685 = r39400680 - r39400684;
        double r39400686 = c;
        double r39400687 = b;
        double r39400688 = r39400686 * r39400687;
        double r39400689 = r39400685 + r39400688;
        double r39400690 = r39400673 * r39400682;
        double r39400691 = i;
        double r39400692 = r39400690 * r39400691;
        double r39400693 = r39400689 - r39400692;
        double r39400694 = -inf.0;
        bool r39400695 = r39400693 <= r39400694;
        double r39400696 = r39400672 * r39400673;
        double r39400697 = r39400676 * r39400674;
        double r39400698 = r39400696 * r39400697;
        double r39400699 = r39400698 * r39400678;
        double r39400700 = r39400673 * r39400691;
        double r39400701 = fma(r39400672, r39400681, r39400700);
        double r39400702 = 27.0;
        double r39400703 = k;
        double r39400704 = j;
        double r39400705 = r39400703 * r39400704;
        double r39400706 = r39400702 * r39400705;
        double r39400707 = fma(r39400682, r39400701, r39400706);
        double r39400708 = r39400699 - r39400707;
        double r39400709 = fma(r39400687, r39400686, r39400708);
        double r39400710 = 2.403209667818115e+294;
        bool r39400711 = r39400693 <= r39400710;
        double r39400712 = r39400702 * r39400703;
        double r39400713 = r39400704 * r39400712;
        double r39400714 = r39400693 - r39400713;
        double r39400715 = r39400678 * r39400673;
        double r39400716 = r39400715 * r39400672;
        double r39400717 = r39400676 * r39400716;
        double r39400718 = r39400717 * r39400674;
        double r39400719 = r39400718 - r39400684;
        double r39400720 = r39400688 + r39400719;
        double r39400721 = r39400720 - r39400692;
        double r39400722 = r39400706 * r39400706;
        double r39400723 = r39400702 * r39400722;
        double r39400724 = r39400705 * r39400723;
        double r39400725 = cbrt(r39400724);
        double r39400726 = r39400721 - r39400725;
        double r39400727 = r39400711 ? r39400714 : r39400726;
        double r39400728 = r39400695 ? r39400709 : r39400727;
        return r39400728;
}

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

Original5.4
Target1.4
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18.0 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) - \left(\left(k \cdot j\right) \cdot 27.0 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.68027943805222:\\ \;\;\;\;\left(\left(18.0 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) + \left(c \cdot b - 27.0 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18.0 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) - \left(\left(k \cdot j\right) \cdot 27.0 - c \cdot b\right)\\ \end{array}\]

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)) < -inf.0

    1. Initial program 60.5

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    2. Simplified12.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, z \cdot \left(\left(t \cdot x\right) \cdot \left(y \cdot 18.0\right)\right) - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), 27.0 \cdot \left(k \cdot j\right)\right)\right)}\]

    if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 2.403209667818115e+294

    1. Initial program 0.3

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    2. Using strategy rm

    3. Applied associate-*l*0.3

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \color{blue}{j \cdot \left(27.0 \cdot k\right)}\]

    if 2.403209667818115e+294 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

    1. Initial program 42.0

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    2. Taylor expanded around inf 27.1

      \[\leadsto \left(\left(\left(\color{blue}{18.0 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    3. Using strategy rm

    4. Applied associate-*r*27.2

      \[\leadsto \left(\left(\left(18.0 \cdot \left(t \cdot \color{blue}{\left(\left(x \cdot z\right) \cdot y\right)}\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    5. Using strategy rm

    6. Applied associate-*r*10.2

      \[\leadsto \left(\left(\left(18.0 \cdot \color{blue}{\left(\left(t \cdot \left(x \cdot z\right)\right) \cdot y\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    7. Using strategy rm

    8. Applied add-cbrt-cube21.3

      \[\leadsto \left(\left(\left(18.0 \cdot \left(\left(t \cdot \left(x \cdot z\right)\right) \cdot y\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot \color{blue}{\sqrt[3]{\left(k \cdot k\right) \cdot k}}\]

    9. Applied add-cbrt-cube21.3

      \[\leadsto \left(\left(\left(18.0 \cdot \left(\left(t \cdot \left(x \cdot z\right)\right) \cdot y\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot \color{blue}{\sqrt[3]{\left(27.0 \cdot 27.0\right) \cdot 27.0}}\right) \cdot \sqrt[3]{\left(k \cdot k\right) \cdot k}\]

    10. Applied add-cbrt-cube34.1

      \[\leadsto \left(\left(\left(18.0 \cdot \left(\left(t \cdot \left(x \cdot z\right)\right) \cdot y\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(\color{blue}{\sqrt[3]{\left(j \cdot j\right) \cdot j}} \cdot \sqrt[3]{\left(27.0 \cdot 27.0\right) \cdot 27.0}\right) \cdot \sqrt[3]{\left(k \cdot k\right) \cdot k}\]

    11. Applied cbrt-unprod34.1

      \[\leadsto \left(\left(\left(18.0 \cdot \left(\left(t \cdot \left(x \cdot z\right)\right) \cdot y\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \color{blue}{\sqrt[3]{\left(\left(j \cdot j\right) \cdot j\right) \cdot \left(\left(27.0 \cdot 27.0\right) \cdot 27.0\right)}} \cdot \sqrt[3]{\left(k \cdot k\right) \cdot k}\]

    12. Applied cbrt-unprod34.5

      \[\leadsto \left(\left(\left(18.0 \cdot \left(\left(t \cdot \left(x \cdot z\right)\right) \cdot y\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \color{blue}{\sqrt[3]{\left(\left(\left(j \cdot j\right) \cdot j\right) \cdot \left(\left(27.0 \cdot 27.0\right) \cdot 27.0\right)\right) \cdot \left(\left(k \cdot k\right) \cdot k\right)}}\]

    13. Simplified21.9

      \[\leadsto \left(\left(\left(18.0 \cdot \left(\left(t \cdot \left(x \cdot z\right)\right) \cdot y\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \sqrt[3]{\color{blue}{\left(\left(\left(27.0 \cdot \left(j \cdot k\right)\right) \cdot \left(27.0 \cdot \left(j \cdot k\right)\right)\right) \cdot 27.0\right) \cdot \left(j \cdot k\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i = -\infty:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(\left(t \cdot x\right) \cdot \left(y \cdot 18.0\right)\right) \cdot z - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), 27.0 \cdot \left(k \cdot j\right)\right)\right)\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i \le 2.403209667818115 \cdot 10^{+294}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - j \cdot \left(27.0 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(y \cdot \left(\left(z \cdot x\right) \cdot t\right)\right) \cdot 18.0 - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - \sqrt[3]{\left(k \cdot j\right) \cdot \left(27.0 \cdot \left(\left(27.0 \cdot \left(k \cdot j\right)\right) \cdot \left(27.0 \cdot \left(k \cdot j\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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)))