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

Average Error: 5.4 → 1.1

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i \le -9.96911953249632 \cdot 10^{+296}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(\left(t \cdot z\right) \cdot y\right) \cdot \left(x \cdot 18.0\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i \le 8.229283477973083 \cdot 10^{+288}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot k\right) \cdot 27.0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(\left(t \cdot z\right) \cdot y\right) \cdot \left(x \cdot 18.0\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r21890720 = x;
        double r21890721 = 18.0;
        double r21890722 = r21890720 * r21890721;
        double r21890723 = y;
        double r21890724 = r21890722 * r21890723;
        double r21890725 = z;
        double r21890726 = r21890724 * r21890725;
        double r21890727 = t;
        double r21890728 = r21890726 * r21890727;
        double r21890729 = a;
        double r21890730 = 4.0;
        double r21890731 = r21890729 * r21890730;
        double r21890732 = r21890731 * r21890727;
        double r21890733 = r21890728 - r21890732;
        double r21890734 = b;
        double r21890735 = c;
        double r21890736 = r21890734 * r21890735;
        double r21890737 = r21890733 + r21890736;
        double r21890738 = r21890720 * r21890730;
        double r21890739 = i;
        double r21890740 = r21890738 * r21890739;
        double r21890741 = r21890737 - r21890740;
        double r21890742 = j;
        double r21890743 = 27.0;
        double r21890744 = r21890742 * r21890743;
        double r21890745 = k;
        double r21890746 = r21890744 * r21890745;
        double r21890747 = r21890741 - r21890746;
        return r21890747;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r21890748 = t;
        double r21890749 = x;
        double r21890750 = 18.0;
        double r21890751 = r21890749 * r21890750;
        double r21890752 = y;
        double r21890753 = r21890751 * r21890752;
        double r21890754 = z;
        double r21890755 = r21890753 * r21890754;
        double r21890756 = r21890748 * r21890755;
        double r21890757 = a;
        double r21890758 = 4.0;
        double r21890759 = r21890757 * r21890758;
        double r21890760 = r21890759 * r21890748;
        double r21890761 = r21890756 - r21890760;
        double r21890762 = c;
        double r21890763 = b;
        double r21890764 = r21890762 * r21890763;
        double r21890765 = r21890761 + r21890764;
        double r21890766 = r21890749 * r21890758;
        double r21890767 = i;
        double r21890768 = r21890766 * r21890767;
        double r21890769 = r21890765 - r21890768;
        double r21890770 = -9.96911953249632e+296;
        bool r21890771 = r21890769 <= r21890770;
        double r21890772 = r21890748 * r21890754;
        double r21890773 = r21890772 * r21890752;
        double r21890774 = r21890773 * r21890751;
        double r21890775 = r21890774 - r21890760;
        double r21890776 = r21890764 + r21890775;
        double r21890777 = r21890776 - r21890768;
        double r21890778 = j;
        double r21890779 = 27.0;
        double r21890780 = r21890778 * r21890779;
        double r21890781 = k;
        double r21890782 = r21890780 * r21890781;
        double r21890783 = r21890777 - r21890782;
        double r21890784 = 8.229283477973083e+288;
        bool r21890785 = r21890769 <= r21890784;
        double r21890786 = r21890778 * r21890781;
        double r21890787 = r21890786 * r21890779;
        double r21890788 = r21890769 - r21890787;
        double r21890789 = r21890785 ? r21890788 : r21890783;
        double r21890790 = r21890771 ? r21890783 : r21890789;
        return r21890790;
}

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)) < -9.96911953249632e+296 or 8.229283477973083e+288 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 39.8

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

   3. Applied associate-*l* 23.6

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

   5. Applied associate-*l* 7.1

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

   if -9.96911953249632e+296 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 8.229283477973083e+288

   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. Taylor expanded around -inf 0.2

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \color{blue}{27.0 \cdot \left(j \cdot k\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i \le -9.96911953249632 \cdot 10^{+296}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(\left(t \cdot z\right) \cdot y\right) \cdot \left(x \cdot 18.0\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i \le 8.229283477973083 \cdot 10^{+288}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot k\right) \cdot 27.0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(\left(t \cdot z\right) \cdot y\right) \cdot \left(x \cdot 18.0\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(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)))