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

Average Error: 5.5 → 2.1

Time: 20.3s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -6.580977851776064168940237967642256471683 \cdot 10^{71}:\\ \;\;\;\;\left(\left(\left(18 \cdot \left(\left(t \cdot x\right) \cdot z\right)\right) \cdot y + \left(-a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \mathbf{elif}\;y \le 8.516527818885921284360578709208870828863 \cdot 10^{-88}:\\ \;\;\;\;\left(t \cdot \left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right) - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot \left(\left(\left(t \cdot x\right) \cdot z\right) \cdot y\right) + \left(-a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(27 \cdot \left(k \cdot j\right) + \left(x \cdot 4\right) \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r125915 = x;
        double r125916 = 18.0;
        double r125917 = r125915 * r125916;
        double r125918 = y;
        double r125919 = r125917 * r125918;
        double r125920 = z;
        double r125921 = r125919 * r125920;
        double r125922 = t;
        double r125923 = r125921 * r125922;
        double r125924 = a;
        double r125925 = 4.0;
        double r125926 = r125924 * r125925;
        double r125927 = r125926 * r125922;
        double r125928 = r125923 - r125927;
        double r125929 = b;
        double r125930 = c;
        double r125931 = r125929 * r125930;
        double r125932 = r125928 + r125931;
        double r125933 = r125915 * r125925;
        double r125934 = i;
        double r125935 = r125933 * r125934;
        double r125936 = r125932 - r125935;
        double r125937 = j;
        double r125938 = 27.0;
        double r125939 = r125937 * r125938;
        double r125940 = k;
        double r125941 = r125939 * r125940;
        double r125942 = r125936 - r125941;
        return r125942;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r125943 = y;
        double r125944 = -6.580977851776064e+71;
        bool r125945 = r125943 <= r125944;
        double r125946 = 18.0;
        double r125947 = t;
        double r125948 = x;
        double r125949 = r125947 * r125948;
        double r125950 = z;
        double r125951 = r125949 * r125950;
        double r125952 = r125946 * r125951;
        double r125953 = r125952 * r125943;
        double r125954 = a;
        double r125955 = 4.0;
        double r125956 = r125954 * r125955;
        double r125957 = -r125956;
        double r125958 = r125957 * r125947;
        double r125959 = r125953 + r125958;
        double r125960 = b;
        double r125961 = c;
        double r125962 = r125960 * r125961;
        double r125963 = r125959 + r125962;
        double r125964 = r125948 * r125955;
        double r125965 = i;
        double r125966 = r125964 * r125965;
        double r125967 = j;
        double r125968 = 27.0;
        double r125969 = r125967 * r125968;
        double r125970 = k;
        double r125971 = r125969 * r125970;
        double r125972 = r125966 + r125971;
        double r125973 = r125963 - r125972;
        double r125974 = 8.516527818885921e-88;
        bool r125975 = r125943 <= r125974;
        double r125976 = r125950 * r125943;
        double r125977 = r125948 * r125976;
        double r125978 = r125946 * r125977;
        double r125979 = r125978 - r125956;
        double r125980 = r125947 * r125979;
        double r125981 = r125980 + r125962;
        double r125982 = r125981 - r125972;
        double r125983 = r125951 * r125943;
        double r125984 = r125946 * r125983;
        double r125985 = r125984 + r125958;
        double r125986 = r125985 + r125962;
        double r125987 = r125970 * r125967;
        double r125988 = r125968 * r125987;
        double r125989 = r125988 + r125966;
        double r125990 = r125986 - r125989;
        double r125991 = r125975 ? r125982 : r125990;
        double r125992 = r125945 ? r125973 : r125991;
        return r125992;
}

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 y < -6.580977851776064e+71

   1. Initial program 12.6

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

   2. Simplified 12.6

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

   4. Applied sub-neg 12.6

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

   5. Applied distribute-lft-in 12.6

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

   6. Simplified 16.1

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

   7. Simplified 16.1

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

   9. Applied associate-*r* 13.6

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

   11. Applied associate-*r* 2.2

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

   13. Applied associate-*r* 2.2

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

   if -6.580977851776064e+71 < y < 8.516527818885921e-88

   1. Initial program 1.7

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

   2. Simplified 1.7

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

   3. Taylor expanded around inf 1.7

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

   if 8.516527818885921e-88 < y

   1. Initial program 8.8

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

   2. Simplified 8.8

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

   4. Applied sub-neg 8.8

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

   5. Applied distribute-lft-in 8.8

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

   6. Simplified 10.6

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

   7. Simplified 10.6

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

   9. Applied associate-*r* 9.7

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

   11. Applied associate-*r* 2.9

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

   13. Applied pow1 2.9

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

   14. Applied pow1 2.9

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

   15. Applied pow1 2.9

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

   16. Applied pow-prod-down 2.9

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

   17. Applied pow-prod-down 2.9

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

   18. Simplified 2.9

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6.580977851776064168940237967642256471683 \cdot 10^{71}:\\ \;\;\;\;\left(\left(\left(18 \cdot \left(\left(t \cdot x\right) \cdot z\right)\right) \cdot y + \left(-a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \mathbf{elif}\;y \le 8.516527818885921284360578709208870828863 \cdot 10^{-88}:\\ \;\;\;\;\left(t \cdot \left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right) - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot \left(\left(\left(t \cdot x\right) \cdot z\right) \cdot y\right) + \left(-a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(27 \cdot \left(k \cdot j\right) + \left(x \cdot 4\right) \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(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)))