Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, E

Average Error: 5.3 → 1.8

Time: 12.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -6.4288223988477578 \cdot 10^{-37} \lor \neg \left(t \le 2.64647452676486563 \cdot 10^{-55}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r641723 = x;
        double r641724 = 18.0;
        double r641725 = r641723 * r641724;
        double r641726 = y;
        double r641727 = r641725 * r641726;
        double r641728 = z;
        double r641729 = r641727 * r641728;
        double r641730 = t;
        double r641731 = r641729 * r641730;
        double r641732 = a;
        double r641733 = 4.0;
        double r641734 = r641732 * r641733;
        double r641735 = r641734 * r641730;
        double r641736 = r641731 - r641735;
        double r641737 = b;
        double r641738 = c;
        double r641739 = r641737 * r641738;
        double r641740 = r641736 + r641739;
        double r641741 = r641723 * r641733;
        double r641742 = i;
        double r641743 = r641741 * r641742;
        double r641744 = r641740 - r641743;
        double r641745 = j;
        double r641746 = 27.0;
        double r641747 = r641745 * r641746;
        double r641748 = k;
        double r641749 = r641747 * r641748;
        double r641750 = r641744 - r641749;
        return r641750;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r641751 = t;
        double r641752 = -6.428822398847758e-37;
        bool r641753 = r641751 <= r641752;
        double r641754 = 2.6464745267648656e-55;
        bool r641755 = r641751 <= r641754;
        double r641756 = !r641755;
        bool r641757 = r641753 || r641756;
        double r641758 = x;
        double r641759 = 18.0;
        double r641760 = r641758 * r641759;
        double r641761 = y;
        double r641762 = r641760 * r641761;
        double r641763 = z;
        double r641764 = r641762 * r641763;
        double r641765 = a;
        double r641766 = 4.0;
        double r641767 = r641765 * r641766;
        double r641768 = r641764 - r641767;
        double r641769 = b;
        double r641770 = c;
        double r641771 = r641769 * r641770;
        double r641772 = i;
        double r641773 = r641766 * r641772;
        double r641774 = j;
        double r641775 = 27.0;
        double r641776 = k;
        double r641777 = r641775 * r641776;
        double r641778 = r641774 * r641777;
        double r641779 = fma(r641758, r641773, r641778);
        double r641780 = r641771 - r641779;
        double r641781 = fma(r641751, r641768, r641780);
        double r641782 = r641763 * r641751;
        double r641783 = r641761 * r641782;
        double r641784 = r641760 * r641783;
        double r641785 = r641767 * r641751;
        double r641786 = r641784 - r641785;
        double r641787 = r641786 + r641771;
        double r641788 = r641758 * r641766;
        double r641789 = r641788 * r641772;
        double r641790 = r641787 - r641789;
        double r641791 = r641774 * r641775;
        double r641792 = r641791 * r641776;
        double r641793 = r641790 - r641792;
        double r641794 = r641757 ? r641781 : r641793;
        return r641794;
}

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

Original	5.3
Target	1.5
Herbie	1.8

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

Derivation

1. Split input into 2 regimes

2. if t < -6.428822398847758e-37 or 2.6464745267648656e-55 < t

   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. Simplified 2.1

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

   4. Applied associate-*l* 2.2

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

   if -6.428822398847758e-37 < t < 2.6464745267648656e-55

   1. Initial program 8.1

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

   3. Applied associate-*l* 4.3

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

   5. Applied associate-*l* 1.5

      \[\leadsto \left(\left(\left(\color{blue}{\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right)} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.4288223988477578 \cdot 10^{-37} \lor \neg \left(t \le 2.64647452676486563 \cdot 10^{-55}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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)))