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

Average Error: 5.3 → 4.1

Time: 24.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -209147167716174719629409475032590057472 \lor \neg \left(y \le 36329201731618045933363253213134848\right):\\ \;\;\;\;\left(\left(\left(\left(18 \cdot \left(x \cdot y\right)\right) \cdot \left(t \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r481827 = x;
        double r481828 = 18.0;
        double r481829 = r481827 * r481828;
        double r481830 = y;
        double r481831 = r481829 * r481830;
        double r481832 = z;
        double r481833 = r481831 * r481832;
        double r481834 = t;
        double r481835 = r481833 * r481834;
        double r481836 = a;
        double r481837 = 4.0;
        double r481838 = r481836 * r481837;
        double r481839 = r481838 * r481834;
        double r481840 = r481835 - r481839;
        double r481841 = b;
        double r481842 = c;
        double r481843 = r481841 * r481842;
        double r481844 = r481840 + r481843;
        double r481845 = r481827 * r481837;
        double r481846 = i;
        double r481847 = r481845 * r481846;
        double r481848 = r481844 - r481847;
        double r481849 = j;
        double r481850 = 27.0;
        double r481851 = r481849 * r481850;
        double r481852 = k;
        double r481853 = r481851 * r481852;
        double r481854 = r481848 - r481853;
        return r481854;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r481855 = y;
        double r481856 = -2.0914716771617472e+38;
        bool r481857 = r481855 <= r481856;
        double r481858 = 3.6329201731618046e+34;
        bool r481859 = r481855 <= r481858;
        double r481860 = !r481859;
        bool r481861 = r481857 || r481860;
        double r481862 = 18.0;
        double r481863 = x;
        double r481864 = r481863 * r481855;
        double r481865 = r481862 * r481864;
        double r481866 = t;
        double r481867 = z;
        double r481868 = r481866 * r481867;
        double r481869 = r481865 * r481868;
        double r481870 = a;
        double r481871 = 4.0;
        double r481872 = r481870 * r481871;
        double r481873 = r481872 * r481866;
        double r481874 = r481869 - r481873;
        double r481875 = b;
        double r481876 = c;
        double r481877 = r481875 * r481876;
        double r481878 = r481874 + r481877;
        double r481879 = r481863 * r481871;
        double r481880 = i;
        double r481881 = r481879 * r481880;
        double r481882 = r481878 - r481881;
        double r481883 = j;
        double r481884 = 27.0;
        double r481885 = k;
        double r481886 = r481884 * r481885;
        double r481887 = r481883 * r481886;
        double r481888 = r481882 - r481887;
        double r481889 = r481867 * r481855;
        double r481890 = r481863 * r481889;
        double r481891 = r481866 * r481890;
        double r481892 = r481862 * r481891;
        double r481893 = r481892 - r481873;
        double r481894 = r481893 + r481877;
        double r481895 = r481894 - r481881;
        double r481896 = r481895 - r481887;
        double r481897 = r481861 ? r481888 : r481896;
        return r481897;
}

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

Target

Original	5.3
Target	1.6
Herbie	4.1

\[\begin{array}{l} \mathbf{if}\;t \lt -1.62108153975413982700795070153457058168 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.6802794380522243500308832153677940369:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -2.0914716771617472e+38 or 3.6329201731618046e+34 < y

   1. Initial program 11.7

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

   3. Applied associate-*l* 11.6

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

   5. Applied associate-*l* 8.5

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

   6. Simplified 8.5

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

   7. Taylor expanded around 0 8.5

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

   if -2.0914716771617472e+38 < y < 3.6329201731618046e+34

   1. Initial program 1.5

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

   3. Applied associate-*l* 1.5

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

   5. Applied pow1 1.5

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

   6. Applied pow1 1.5

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

   7. Applied pow1 1.5

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

   8. Applied pow1 1.5

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

   9. Applied pow1 1.5

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

   10. Applied pow-prod-down 1.5

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

   11. Applied pow-prod-down 1.5

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

   12. Applied pow-prod-down 1.5

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

   13. Applied pow-prod-down 1.5

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

   14. Simplified 1.4

       \[\leadsto \left(\left(\left({\color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)\right)}}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -209147167716174719629409475032590057472 \lor \neg \left(y \le 36329201731618045933363253213134848\right):\\ \;\;\;\;\left(\left(\left(\left(18 \cdot \left(x \cdot y\right)\right) \cdot \left(t \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"
  :precision binary64

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b))) (if (< t 165.680279438052224) (+ (- (* (* 18 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4)) (- (* c b) (* 27 (* k j)))) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))