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

Average Error: 5.8 → 3.6

Time: 25.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k \le -5.277708727494282798806419574218455720168 \cdot 10^{305} \lor \neg \left(\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k \le 4.669261252917985076120297268222720136991 \cdot 10^{297}\right):\\ \;\;\;\;-\left(\left(\left(x \cdot i\right) \cdot 4 - c \cdot b\right) + 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r514882 = x;
        double r514883 = 18.0;
        double r514884 = r514882 * r514883;
        double r514885 = y;
        double r514886 = r514884 * r514885;
        double r514887 = z;
        double r514888 = r514886 * r514887;
        double r514889 = t;
        double r514890 = r514888 * r514889;
        double r514891 = a;
        double r514892 = 4.0;
        double r514893 = r514891 * r514892;
        double r514894 = r514893 * r514889;
        double r514895 = r514890 - r514894;
        double r514896 = b;
        double r514897 = c;
        double r514898 = r514896 * r514897;
        double r514899 = r514895 + r514898;
        double r514900 = r514882 * r514892;
        double r514901 = i;
        double r514902 = r514900 * r514901;
        double r514903 = r514899 - r514902;
        double r514904 = j;
        double r514905 = 27.0;
        double r514906 = r514904 * r514905;
        double r514907 = k;
        double r514908 = r514906 * r514907;
        double r514909 = r514903 - r514908;
        return r514909;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r514910 = t;
        double r514911 = x;
        double r514912 = 18.0;
        double r514913 = r514911 * r514912;
        double r514914 = y;
        double r514915 = r514913 * r514914;
        double r514916 = z;
        double r514917 = r514915 * r514916;
        double r514918 = r514910 * r514917;
        double r514919 = a;
        double r514920 = 4.0;
        double r514921 = r514919 * r514920;
        double r514922 = r514921 * r514910;
        double r514923 = r514918 - r514922;
        double r514924 = c;
        double r514925 = b;
        double r514926 = r514924 * r514925;
        double r514927 = r514923 + r514926;
        double r514928 = r514911 * r514920;
        double r514929 = i;
        double r514930 = r514928 * r514929;
        double r514931 = r514927 - r514930;
        double r514932 = 27.0;
        double r514933 = j;
        double r514934 = r514932 * r514933;
        double r514935 = k;
        double r514936 = r514934 * r514935;
        double r514937 = r514931 - r514936;
        double r514938 = -5.277708727494283e+305;
        bool r514939 = r514937 <= r514938;
        double r514940 = 4.669261252917985e+297;
        bool r514941 = r514937 <= r514940;
        double r514942 = !r514941;
        bool r514943 = r514939 || r514942;
        double r514944 = r514911 * r514929;
        double r514945 = r514944 * r514920;
        double r514946 = r514945 - r514926;
        double r514947 = r514935 * r514933;
        double r514948 = r514932 * r514947;
        double r514949 = r514946 + r514948;
        double r514950 = -r514949;
        double r514951 = r514943 ? r514950 : r514937;
        return r514951;
}

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	5.8
Target	1.6
Herbie	3.6

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

Derivation

1. Split input into 2 regimes

2. if (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)) < -5.277708727494283e+305 or 4.669261252917985e+297 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k))

   1. Initial program 50.5

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

   2. Simplified 31.7

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

   4. Applied associate-*r* 31.6

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

   5. Taylor expanded around 0 30.2

      \[\leadsto \color{blue}{0} - \left(\left(j \cdot k\right) \cdot 27 + \left(\left(x \cdot i\right) \cdot 4 - b \cdot c\right)\right)\]

   if -5.277708727494283e+305 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)) < 4.669261252917985e+297

   1. Initial program 0.3

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

4. Final simplification 3.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k \le -5.277708727494282798806419574218455720168 \cdot 10^{305} \lor \neg \left(\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k \le 4.669261252917985076120297268222720136991 \cdot 10^{297}\right):\\ \;\;\;\;-\left(\left(\left(x \cdot i\right) \cdot 4 - c \cdot b\right) + 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18.0 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4.0)) (- (* c b) (* 27.0 (* k j)))) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))