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

Average Error: 5.6 → 3.5

Time: 26.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -7.2131406689924034 \cdot 10^{45}:\\ \;\;\;\;\left(\left(\left(t \cdot {\left(\sqrt[3]{\sqrt[3]{z}}\right)}^{3}\right) \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \left(-a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \mathbf{elif}\;z \le 1.86064292666481138 \cdot 10^{-39}:\\ \;\;\;\;\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) + t \cdot \left(-a \cdot 4\right)\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \sqrt{z}\right) \cdot \sqrt{z} - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r163800 = x;
        double r163801 = 18.0;
        double r163802 = r163800 * r163801;
        double r163803 = y;
        double r163804 = r163802 * r163803;
        double r163805 = z;
        double r163806 = r163804 * r163805;
        double r163807 = t;
        double r163808 = r163806 * r163807;
        double r163809 = a;
        double r163810 = 4.0;
        double r163811 = r163809 * r163810;
        double r163812 = r163811 * r163807;
        double r163813 = r163808 - r163812;
        double r163814 = b;
        double r163815 = c;
        double r163816 = r163814 * r163815;
        double r163817 = r163813 + r163816;
        double r163818 = r163800 * r163810;
        double r163819 = i;
        double r163820 = r163818 * r163819;
        double r163821 = r163817 - r163820;
        double r163822 = j;
        double r163823 = 27.0;
        double r163824 = r163822 * r163823;
        double r163825 = k;
        double r163826 = r163824 * r163825;
        double r163827 = r163821 - r163826;
        return r163827;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r163828 = z;
        double r163829 = -7.213140668992403e+45;
        bool r163830 = r163828 <= r163829;
        double r163831 = t;
        double r163832 = cbrt(r163828);
        double r163833 = cbrt(r163832);
        double r163834 = 3.0;
        double r163835 = pow(r163833, r163834);
        double r163836 = r163831 * r163835;
        double r163837 = x;
        double r163838 = 18.0;
        double r163839 = r163837 * r163838;
        double r163840 = y;
        double r163841 = r163839 * r163840;
        double r163842 = r163832 * r163832;
        double r163843 = r163841 * r163842;
        double r163844 = r163836 * r163843;
        double r163845 = a;
        double r163846 = 4.0;
        double r163847 = r163845 * r163846;
        double r163848 = -r163847;
        double r163849 = r163848 * r163831;
        double r163850 = r163844 + r163849;
        double r163851 = b;
        double r163852 = c;
        double r163853 = r163851 * r163852;
        double r163854 = r163850 + r163853;
        double r163855 = r163837 * r163846;
        double r163856 = i;
        double r163857 = r163855 * r163856;
        double r163858 = j;
        double r163859 = 27.0;
        double r163860 = k;
        double r163861 = r163859 * r163860;
        double r163862 = r163858 * r163861;
        double r163863 = r163857 + r163862;
        double r163864 = r163854 - r163863;
        double r163865 = 1.8606429266648114e-39;
        bool r163866 = r163828 <= r163865;
        double r163867 = r163828 * r163840;
        double r163868 = r163837 * r163867;
        double r163869 = r163831 * r163868;
        double r163870 = r163838 * r163869;
        double r163871 = r163831 * r163848;
        double r163872 = r163870 + r163871;
        double r163873 = r163872 + r163853;
        double r163874 = r163873 - r163863;
        double r163875 = sqrt(r163828);
        double r163876 = r163841 * r163875;
        double r163877 = r163876 * r163875;
        double r163878 = r163877 - r163847;
        double r163879 = r163831 * r163878;
        double r163880 = r163879 + r163853;
        double r163881 = r163858 * r163859;
        double r163882 = r163881 * r163860;
        double r163883 = r163857 + r163882;
        double r163884 = r163880 - r163883;
        double r163885 = r163866 ? r163874 : r163884;
        double r163886 = r163830 ? r163864 : r163885;
        return r163886;
}

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 < -7.213140668992403e+45

   1. Initial program 6.6

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

   2. Simplified 6.6

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

   4. Applied associate-*l* 6.5

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

   6. Applied add-cube-cbrt 6.7

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

   7. Applied associate-*r* 6.7

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

   9. Applied add-cube-cbrt 6.8

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

   11. Applied sub-neg 6.8

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

   12. Applied distribute-lft-in 6.8

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

   13. Simplified 5.2

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

   14. Simplified 5.2

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

   if -7.213140668992403e+45 < z < 1.8606429266648114e-39

   1. Initial program 4.9

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

   2. Simplified 4.9

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

   4. Applied associate-*l* 4.8

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

   6. Applied sub-neg 4.8

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

   7. Applied distribute-lft-in 4.8

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

   8. Simplified 1.4

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

   if 1.8606429266648114e-39 < z

   1. Initial program 6.5

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

   2. Simplified 6.5

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

   4. Applied add-sqr-sqrt 6.5

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

   5. Applied associate-*r* 6.5

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

4. Final simplification 3.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.2131406689924034 \cdot 10^{45}:\\ \;\;\;\;\left(\left(\left(t \cdot {\left(\sqrt[3]{\sqrt[3]{z}}\right)}^{3}\right) \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \left(-a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \mathbf{elif}\;z \le 1.86064292666481138 \cdot 10^{-39}:\\ \;\;\;\;\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) + t \cdot \left(-a \cdot 4\right)\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \sqrt{z}\right) \cdot \sqrt{z} - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))