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

Average Error: 5.6 → 1.3

Time: 23.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i = -\infty:\\ \;\;\;\;\left(\left(c \cdot b + \left(x \cdot \left(18 \cdot \left(\left(t \cdot z\right) \cdot y\right)\right) - \left(a \cdot 4\right) \cdot t\right)\right) - 4 \cdot \left(x \cdot i\right)\right) - \left(27 \cdot j\right) \cdot k\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 3.252609788198741362599457824359920825411 \cdot 10^{268}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(k \cdot j\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(x \cdot \left(18 \cdot \left(\left(t \cdot z\right) \cdot y\right)\right) - \left(a \cdot 4\right) \cdot t\right)\right) - 4 \cdot \left(x \cdot i\right)\right) - \left(27 \cdot j\right) \cdot k\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r6184642 = x;
        double r6184643 = 18.0;
        double r6184644 = r6184642 * r6184643;
        double r6184645 = y;
        double r6184646 = r6184644 * r6184645;
        double r6184647 = z;
        double r6184648 = r6184646 * r6184647;
        double r6184649 = t;
        double r6184650 = r6184648 * r6184649;
        double r6184651 = a;
        double r6184652 = 4.0;
        double r6184653 = r6184651 * r6184652;
        double r6184654 = r6184653 * r6184649;
        double r6184655 = r6184650 - r6184654;
        double r6184656 = b;
        double r6184657 = c;
        double r6184658 = r6184656 * r6184657;
        double r6184659 = r6184655 + r6184658;
        double r6184660 = r6184642 * r6184652;
        double r6184661 = i;
        double r6184662 = r6184660 * r6184661;
        double r6184663 = r6184659 - r6184662;
        double r6184664 = j;
        double r6184665 = 27.0;
        double r6184666 = r6184664 * r6184665;
        double r6184667 = k;
        double r6184668 = r6184666 * r6184667;
        double r6184669 = r6184663 - r6184668;
        return r6184669;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r6184670 = t;
        double r6184671 = x;
        double r6184672 = 18.0;
        double r6184673 = r6184671 * r6184672;
        double r6184674 = y;
        double r6184675 = r6184673 * r6184674;
        double r6184676 = z;
        double r6184677 = r6184675 * r6184676;
        double r6184678 = r6184670 * r6184677;
        double r6184679 = a;
        double r6184680 = 4.0;
        double r6184681 = r6184679 * r6184680;
        double r6184682 = r6184681 * r6184670;
        double r6184683 = r6184678 - r6184682;
        double r6184684 = c;
        double r6184685 = b;
        double r6184686 = r6184684 * r6184685;
        double r6184687 = r6184683 + r6184686;
        double r6184688 = r6184671 * r6184680;
        double r6184689 = i;
        double r6184690 = r6184688 * r6184689;
        double r6184691 = r6184687 - r6184690;
        double r6184692 = -inf.0;
        bool r6184693 = r6184691 <= r6184692;
        double r6184694 = r6184670 * r6184676;
        double r6184695 = r6184694 * r6184674;
        double r6184696 = r6184672 * r6184695;
        double r6184697 = r6184671 * r6184696;
        double r6184698 = r6184697 - r6184682;
        double r6184699 = r6184686 + r6184698;
        double r6184700 = r6184671 * r6184689;
        double r6184701 = r6184680 * r6184700;
        double r6184702 = r6184699 - r6184701;
        double r6184703 = 27.0;
        double r6184704 = j;
        double r6184705 = r6184703 * r6184704;
        double r6184706 = k;
        double r6184707 = r6184705 * r6184706;
        double r6184708 = r6184702 - r6184707;
        double r6184709 = 3.2526097881987414e+268;
        bool r6184710 = r6184691 <= r6184709;
        double r6184711 = r6184706 * r6184704;
        double r6184712 = r6184711 * r6184703;
        double r6184713 = r6184691 - r6184712;
        double r6184714 = r6184710 ? r6184713 : r6184708;
        double r6184715 = r6184693 ? r6184708 : r6184714;
        return r6184715;
}

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

Derivation

1. Split input into 2 regimes

2. if (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < -inf.0 or 3.2526097881987414e+268 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 38.2

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

   3. Applied associate-*l* 23.8

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

   5. Applied associate-*l* 9.0

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

   7. Applied associate-*l* 8.6

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

   8. Taylor expanded around 0 8.4

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

   if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 3.2526097881987414e+268

   1. Initial program 0.3

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

   2. Taylor expanded around 0 0.2

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i = -\infty:\\ \;\;\;\;\left(\left(c \cdot b + \left(x \cdot \left(18 \cdot \left(\left(t \cdot z\right) \cdot y\right)\right) - \left(a \cdot 4\right) \cdot t\right)\right) - 4 \cdot \left(x \cdot i\right)\right) - \left(27 \cdot j\right) \cdot k\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 3.252609788198741362599457824359920825411 \cdot 10^{268}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(k \cdot j\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(x \cdot \left(18 \cdot \left(\left(t \cdot z\right) \cdot y\right)\right) - \left(a \cdot 4\right) \cdot t\right)\right) - 4 \cdot \left(x \cdot i\right)\right) - \left(27 \cdot j\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))