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

Average Error: 5.3 → 2.0

Time: 7.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -417746856154403613434896783179265469317100 \lor \neg \left(x \le 2.587025851706818846773690812532575971732 \cdot 10^{67}\right):\\ \;\;\;\;\left(\left(\left(x \cdot \left(\left(\left(18 \cdot y\right) \cdot z\right) \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r773665 = x;
        double r773666 = 18.0;
        double r773667 = r773665 * r773666;
        double r773668 = y;
        double r773669 = r773667 * r773668;
        double r773670 = z;
        double r773671 = r773669 * r773670;
        double r773672 = t;
        double r773673 = r773671 * r773672;
        double r773674 = a;
        double r773675 = 4.0;
        double r773676 = r773674 * r773675;
        double r773677 = r773676 * r773672;
        double r773678 = r773673 - r773677;
        double r773679 = b;
        double r773680 = c;
        double r773681 = r773679 * r773680;
        double r773682 = r773678 + r773681;
        double r773683 = r773665 * r773675;
        double r773684 = i;
        double r773685 = r773683 * r773684;
        double r773686 = r773682 - r773685;
        double r773687 = j;
        double r773688 = 27.0;
        double r773689 = r773687 * r773688;
        double r773690 = k;
        double r773691 = r773689 * r773690;
        double r773692 = r773686 - r773691;
        return r773692;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r773693 = x;
        double r773694 = -4.177468561544036e+41;
        bool r773695 = r773693 <= r773694;
        double r773696 = 2.5870258517068188e+67;
        bool r773697 = r773693 <= r773696;
        double r773698 = !r773697;
        bool r773699 = r773695 || r773698;
        double r773700 = 18.0;
        double r773701 = y;
        double r773702 = r773700 * r773701;
        double r773703 = z;
        double r773704 = r773702 * r773703;
        double r773705 = t;
        double r773706 = r773704 * r773705;
        double r773707 = r773693 * r773706;
        double r773708 = a;
        double r773709 = 4.0;
        double r773710 = r773708 * r773709;
        double r773711 = r773710 * r773705;
        double r773712 = r773707 - r773711;
        double r773713 = b;
        double r773714 = c;
        double r773715 = r773713 * r773714;
        double r773716 = r773712 + r773715;
        double r773717 = r773693 * r773709;
        double r773718 = i;
        double r773719 = r773717 * r773718;
        double r773720 = r773716 - r773719;
        double r773721 = j;
        double r773722 = 27.0;
        double r773723 = r773721 * r773722;
        double r773724 = k;
        double r773725 = r773723 * r773724;
        double r773726 = r773720 - r773725;
        double r773727 = r773693 * r773700;
        double r773728 = r773727 * r773701;
        double r773729 = r773728 * r773703;
        double r773730 = r773729 * r773705;
        double r773731 = r773730 - r773711;
        double r773732 = r773731 + r773715;
        double r773733 = r773732 - r773719;
        double r773734 = r773722 * r773724;
        double r773735 = r773721 * r773734;
        double r773736 = r773733 - r773735;
        double r773737 = r773699 ? r773726 : r773736;
        return r773737;
}

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	1.3
Herbie	2.0

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

Derivation

1. Split input into 2 regimes

2. if x < -4.177468561544036e+41 or 2.5870258517068188e+67 < x

   1. Initial program 14.1

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

   3. Applied associate-*l* 14.1

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

   5. Applied associate-*l* 8.6

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

   7. Applied associate-*l* 1.7

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

   if -4.177468561544036e+41 < x < 2.5870258517068188e+67

   1. Initial program 2.0

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

   3. Applied associate-*l* 2.0

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -417746856154403613434896783179265469317100 \lor \neg \left(x \le 2.587025851706818846773690812532575971732 \cdot 10^{67}\right):\\ \;\;\;\;\left(\left(\left(x \cdot \left(\left(\left(18 \cdot y\right) \cdot z\right) \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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)))