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

Average Error: 5.4 → 4.2

Time: 9.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -6.673005960037381889549124025506898760796:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;z \le 2.783733361079767197510504649520285092119 \cdot 10^{228}:\\ \;\;\;\;\left(\left(\left({\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)\right)}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{elif}\;z \le 3.090990606180461355587497618592842199524 \cdot 10^{278}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left({\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)\right)}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r113723 = x;
        double r113724 = 18.0;
        double r113725 = r113723 * r113724;
        double r113726 = y;
        double r113727 = r113725 * r113726;
        double r113728 = z;
        double r113729 = r113727 * r113728;
        double r113730 = t;
        double r113731 = r113729 * r113730;
        double r113732 = a;
        double r113733 = 4.0;
        double r113734 = r113732 * r113733;
        double r113735 = r113734 * r113730;
        double r113736 = r113731 - r113735;
        double r113737 = b;
        double r113738 = c;
        double r113739 = r113737 * r113738;
        double r113740 = r113736 + r113739;
        double r113741 = r113723 * r113733;
        double r113742 = i;
        double r113743 = r113741 * r113742;
        double r113744 = r113740 - r113743;
        double r113745 = j;
        double r113746 = 27.0;
        double r113747 = r113745 * r113746;
        double r113748 = k;
        double r113749 = r113747 * r113748;
        double r113750 = r113744 - r113749;
        return r113750;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r113751 = z;
        double r113752 = -6.673005960037382;
        bool r113753 = r113751 <= r113752;
        double r113754 = x;
        double r113755 = 18.0;
        double r113756 = y;
        double r113757 = r113755 * r113756;
        double r113758 = r113754 * r113757;
        double r113759 = r113758 * r113751;
        double r113760 = t;
        double r113761 = r113759 * r113760;
        double r113762 = a;
        double r113763 = 4.0;
        double r113764 = r113762 * r113763;
        double r113765 = r113764 * r113760;
        double r113766 = r113761 - r113765;
        double r113767 = b;
        double r113768 = c;
        double r113769 = r113767 * r113768;
        double r113770 = r113766 + r113769;
        double r113771 = r113754 * r113763;
        double r113772 = i;
        double r113773 = r113771 * r113772;
        double r113774 = r113770 - r113773;
        double r113775 = j;
        double r113776 = 27.0;
        double r113777 = k;
        double r113778 = r113776 * r113777;
        double r113779 = r113775 * r113778;
        double r113780 = r113774 - r113779;
        double r113781 = 2.7837333610797672e+228;
        bool r113782 = r113751 <= r113781;
        double r113783 = r113751 * r113756;
        double r113784 = r113754 * r113783;
        double r113785 = r113760 * r113784;
        double r113786 = r113755 * r113785;
        double r113787 = 1.0;
        double r113788 = pow(r113786, r113787);
        double r113789 = r113788 - r113765;
        double r113790 = r113789 + r113769;
        double r113791 = r113790 - r113773;
        double r113792 = r113775 * r113776;
        double r113793 = r113792 * r113777;
        double r113794 = r113791 - r113793;
        double r113795 = 3.0909906061804614e+278;
        bool r113796 = r113751 <= r113795;
        double r113797 = r113754 * r113755;
        double r113798 = r113797 * r113756;
        double r113799 = r113751 * r113760;
        double r113800 = r113798 * r113799;
        double r113801 = r113800 - r113765;
        double r113802 = r113801 + r113769;
        double r113803 = r113802 - r113773;
        double r113804 = r113803 - r113793;
        double r113805 = r113796 ? r113804 : r113794;
        double r113806 = r113782 ? r113794 : r113805;
        double r113807 = r113753 ? r113780 : r113806;
        return r113807;
}

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 3 regimes

2. if z < -6.673005960037382

   1. Initial program 6.7

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

   3. Applied associate-*l* 6.7

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

   5. Applied associate-*l* 6.6

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

   if -6.673005960037382 < z < 2.7837333610797672e+228 or 3.0909906061804614e+278 < z

   1. Initial program 4.8

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

   3. Applied add-cube-cbrt 4.9

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot \color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   4. Applied associate-*r* 4.9

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

   6. Applied pow1 4.9

      \[\leadsto \left(\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{t}\right)}^{1}} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   7. Applied pow1 4.9

      \[\leadsto \left(\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot \left(\sqrt[3]{t} \cdot \color{blue}{{\left(\sqrt[3]{t}\right)}^{1}}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   8. Applied pow1 4.9

      \[\leadsto \left(\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{t}\right)}^{1}} \cdot {\left(\sqrt[3]{t}\right)}^{1}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   9. Applied pow-prod-down 4.9

      \[\leadsto \left(\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot \color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{1}}\right) \cdot {\left(\sqrt[3]{t}\right)}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   10. Applied pow1 4.9

       \[\leadsto \left(\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \color{blue}{{z}^{1}}\right) \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{1}\right) \cdot {\left(\sqrt[3]{t}\right)}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   11. Applied pow1 4.9

       \[\leadsto \left(\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot \color{blue}{{y}^{1}}\right) \cdot {z}^{1}\right) \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{1}\right) \cdot {\left(\sqrt[3]{t}\right)}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   12. Applied pow1 4.9

       \[\leadsto \left(\left(\left(\left(\left(\left(\left(x \cdot \color{blue}{{18}^{1}}\right) \cdot {y}^{1}\right) \cdot {z}^{1}\right) \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{1}\right) \cdot {\left(\sqrt[3]{t}\right)}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   13. Applied pow1 4.9

       \[\leadsto \left(\left(\left(\left(\left(\left(\left(\color{blue}{{x}^{1}} \cdot {18}^{1}\right) \cdot {y}^{1}\right) \cdot {z}^{1}\right) \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{1}\right) \cdot {\left(\sqrt[3]{t}\right)}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   14. Applied pow-prod-down 4.9

       \[\leadsto \left(\left(\left(\left(\left(\left(\color{blue}{{\left(x \cdot 18\right)}^{1}} \cdot {y}^{1}\right) \cdot {z}^{1}\right) \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{1}\right) \cdot {\left(\sqrt[3]{t}\right)}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   15. Applied pow-prod-down 4.9

       \[\leadsto \left(\left(\left(\left(\left(\color{blue}{{\left(\left(x \cdot 18\right) \cdot y\right)}^{1}} \cdot {z}^{1}\right) \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{1}\right) \cdot {\left(\sqrt[3]{t}\right)}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   16. Applied pow-prod-down 4.9

       \[\leadsto \left(\left(\left(\left(\color{blue}{{\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right)}^{1}} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{1}\right) \cdot {\left(\sqrt[3]{t}\right)}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   17. Applied pow-prod-down 4.9

       \[\leadsto \left(\left(\left(\color{blue}{{\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)}^{1}} \cdot {\left(\sqrt[3]{t}\right)}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   18. Applied pow-prod-down 4.9

       \[\leadsto \left(\left(\left(\color{blue}{{\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) \cdot \sqrt[3]{t}\right)}^{1}} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   19. Simplified 3.3

       \[\leadsto \left(\left(\left({\color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)\right)}}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   if 2.7837333610797672e+228 < z < 3.0909906061804614e+278

   1. Initial program 13.2

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

   3. Applied associate-*l* 11.3

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

4. Final simplification 4.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.673005960037381889549124025506898760796:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;z \le 2.783733361079767197510504649520285092119 \cdot 10^{228}:\\ \;\;\;\;\left(\left(\left({\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)\right)}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{elif}\;z \le 3.090990606180461355587497618592842199524 \cdot 10^{278}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left({\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)\right)}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  :precision binary64
  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))