Average Error: 5.6 → 1.3
Time: 20.0s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i = -\infty:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(x \cdot 18\right), t \cdot y, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 9.44691043772436753 \cdot 10^{300}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b, c, t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right)\right) - 4 \cdot \left(i \cdot x\right)\right) + \left(x \cdot 4\right) \cdot \left(\left(-i\right) + i\right)\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - \mathsf{fma}\left(x \cdot 4, i, \left(a \cdot 4\right) \cdot t - b \cdot c\right)\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]
\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(\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 = -\infty:\\
\;\;\;\;\mathsf{fma}\left(z \cdot \left(x \cdot 18\right), t \cdot y, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\\

\mathbf{elif}\;\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 \le 9.44691043772436753 \cdot 10^{300}:\\
\;\;\;\;\left(\left(\mathsf{fma}\left(b, c, t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right)\right) - 4 \cdot \left(i \cdot x\right)\right) + \left(x \cdot 4\right) \cdot \left(\left(-i\right) + i\right)\right) - \left(j \cdot 27\right) \cdot k\\

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

\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 r828630 = x;
        double r828631 = 18.0;
        double r828632 = r828630 * r828631;
        double r828633 = y;
        double r828634 = r828632 * r828633;
        double r828635 = z;
        double r828636 = r828634 * r828635;
        double r828637 = t;
        double r828638 = r828636 * r828637;
        double r828639 = a;
        double r828640 = 4.0;
        double r828641 = r828639 * r828640;
        double r828642 = r828641 * r828637;
        double r828643 = r828638 - r828642;
        double r828644 = b;
        double r828645 = c;
        double r828646 = r828644 * r828645;
        double r828647 = r828643 + r828646;
        double r828648 = r828630 * r828640;
        double r828649 = i;
        double r828650 = r828648 * r828649;
        double r828651 = r828647 - r828650;
        double r828652 = j;
        double r828653 = 27.0;
        double r828654 = r828652 * r828653;
        double r828655 = k;
        double r828656 = r828654 * r828655;
        double r828657 = r828651 - r828656;
        return r828657;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r828658 = x;
        double r828659 = 18.0;
        double r828660 = r828658 * r828659;
        double r828661 = y;
        double r828662 = r828660 * r828661;
        double r828663 = z;
        double r828664 = r828662 * r828663;
        double r828665 = t;
        double r828666 = r828664 * r828665;
        double r828667 = a;
        double r828668 = 4.0;
        double r828669 = r828667 * r828668;
        double r828670 = r828669 * r828665;
        double r828671 = r828666 - r828670;
        double r828672 = b;
        double r828673 = c;
        double r828674 = r828672 * r828673;
        double r828675 = r828671 + r828674;
        double r828676 = r828658 * r828668;
        double r828677 = i;
        double r828678 = r828676 * r828677;
        double r828679 = r828675 - r828678;
        double r828680 = -inf.0;
        bool r828681 = r828679 <= r828680;
        double r828682 = r828663 * r828660;
        double r828683 = r828665 * r828661;
        double r828684 = r828658 * r828677;
        double r828685 = fma(r828665, r828667, r828684);
        double r828686 = j;
        double r828687 = 27.0;
        double r828688 = r828686 * r828687;
        double r828689 = k;
        double r828690 = r828688 * r828689;
        double r828691 = fma(r828668, r828685, r828690);
        double r828692 = r828674 - r828691;
        double r828693 = fma(r828682, r828683, r828692);
        double r828694 = 9.446910437724368e+300;
        bool r828695 = r828679 <= r828694;
        double r828696 = r828664 - r828669;
        double r828697 = r828665 * r828696;
        double r828698 = fma(r828672, r828673, r828697);
        double r828699 = r828677 * r828658;
        double r828700 = r828668 * r828699;
        double r828701 = r828698 - r828700;
        double r828702 = -r828677;
        double r828703 = r828702 + r828677;
        double r828704 = r828676 * r828703;
        double r828705 = r828701 + r828704;
        double r828706 = r828705 - r828690;
        double r828707 = r828663 * r828665;
        double r828708 = r828661 * r828707;
        double r828709 = r828660 * r828708;
        double r828710 = r828670 - r828674;
        double r828711 = fma(r828676, r828677, r828710);
        double r828712 = r828709 - r828711;
        double r828713 = r828712 - r828690;
        double r828714 = r828695 ? r828706 : r828713;
        double r828715 = r828681 ? r828693 : r828714;
        return r828715;
}

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
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.680279438052224:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

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)) < -inf.0

    1. Initial program 64.0

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

    2. Simplified14.3

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

    if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 9.446910437724368e+300

    1. Initial program 0.4

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

    3. Applied associate-*l*3.3

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

    5. Applied add-sqr-sqrt34.3

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

    6. Applied prod-diff34.3

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

    7. Simplified0.4

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

    8. Simplified0.4

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

    if 9.446910437724368e+300 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

    1. Initial program 49.4

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

    3. Applied associate-*l*28.1

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

    5. Applied associate-*l*7.0

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

    7. Applied associate-+l-7.0

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

    8. Applied associate--l-7.0

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

    9. Simplified7.0

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i = -\infty:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(x \cdot 18\right), t \cdot y, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 9.44691043772436753 \cdot 10^{300}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b, c, t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right)\right) - 4 \cdot \left(i \cdot x\right)\right) + \left(x \cdot 4\right) \cdot \left(\left(-i\right) + i\right)\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - \mathsf{fma}\left(x \cdot 4, i, \left(a \cdot 4\right) \cdot t - b \cdot c\right)\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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"
  :precision binary64

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4)) (- (* c b) (* 27 (* k j)))) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))