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

Average Error: 5.4 → 2.9

Time: 21.8s

Precision: 64

Math

\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i = -\infty:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot \left(t \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i \le 1.736218430949862 \cdot 10^{+238}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - j \cdot \left(27.0 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(18.0 \cdot \left(\left(t \cdot \left(z \cdot x\right)\right) \cdot y\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r37538752 = x;
        double r37538753 = 18.0;
        double r37538754 = r37538752 * r37538753;
        double r37538755 = y;
        double r37538756 = r37538754 * r37538755;
        double r37538757 = z;
        double r37538758 = r37538756 * r37538757;
        double r37538759 = t;
        double r37538760 = r37538758 * r37538759;
        double r37538761 = a;
        double r37538762 = 4.0;
        double r37538763 = r37538761 * r37538762;
        double r37538764 = r37538763 * r37538759;
        double r37538765 = r37538760 - r37538764;
        double r37538766 = b;
        double r37538767 = c;
        double r37538768 = r37538766 * r37538767;
        double r37538769 = r37538765 + r37538768;
        double r37538770 = r37538752 * r37538762;
        double r37538771 = i;
        double r37538772 = r37538770 * r37538771;
        double r37538773 = r37538769 - r37538772;
        double r37538774 = j;
        double r37538775 = 27.0;
        double r37538776 = r37538774 * r37538775;
        double r37538777 = k;
        double r37538778 = r37538776 * r37538777;
        double r37538779 = r37538773 - r37538778;
        return r37538779;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r37538780 = t;
        double r37538781 = x;
        double r37538782 = 18.0;
        double r37538783 = r37538781 * r37538782;
        double r37538784 = y;
        double r37538785 = r37538783 * r37538784;
        double r37538786 = z;
        double r37538787 = r37538785 * r37538786;
        double r37538788 = r37538780 * r37538787;
        double r37538789 = a;
        double r37538790 = 4.0;
        double r37538791 = r37538789 * r37538790;
        double r37538792 = r37538791 * r37538780;
        double r37538793 = r37538788 - r37538792;
        double r37538794 = c;
        double r37538795 = b;
        double r37538796 = r37538794 * r37538795;
        double r37538797 = r37538793 + r37538796;
        double r37538798 = r37538781 * r37538790;
        double r37538799 = i;
        double r37538800 = r37538798 * r37538799;
        double r37538801 = r37538797 - r37538800;
        double r37538802 = -inf.0;
        bool r37538803 = r37538801 <= r37538802;
        double r37538804 = r37538780 * r37538786;
        double r37538805 = r37538785 * r37538804;
        double r37538806 = r37538805 - r37538792;
        double r37538807 = r37538806 + r37538796;
        double r37538808 = r37538807 - r37538800;
        double r37538809 = 27.0;
        double r37538810 = j;
        double r37538811 = k;
        double r37538812 = r37538810 * r37538811;
        double r37538813 = r37538809 * r37538812;
        double r37538814 = r37538808 - r37538813;
        double r37538815 = 1.736218430949862e+238;
        bool r37538816 = r37538801 <= r37538815;
        double r37538817 = r37538809 * r37538811;
        double r37538818 = r37538810 * r37538817;
        double r37538819 = r37538801 - r37538818;
        double r37538820 = r37538786 * r37538781;
        double r37538821 = r37538780 * r37538820;
        double r37538822 = r37538821 * r37538784;
        double r37538823 = r37538782 * r37538822;
        double r37538824 = r37538823 - r37538792;
        double r37538825 = r37538824 + r37538796;
        double r37538826 = r37538825 - r37538800;
        double r37538827 = r37538826 - r37538813;
        double r37538828 = r37538816 ? r37538819 : r37538827;
        double r37538829 = r37538803 ? r37538814 : r37538828;
        return r37538829;
}

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	5.4
Target	1.4
Herbie	2.9

\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18.0 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) - \left(\left(k \cdot j\right) \cdot 27.0 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.68027943805222:\\ \;\;\;\;\left(\left(18.0 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) + \left(c \cdot b - 27.0 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18.0 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) - \left(\left(k \cdot j\right) \cdot 27.0 - c \cdot b\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < -inf.0

   1. Initial program 60.5

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

   2. Taylor expanded around 0 60.5

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \color{blue}{27.0 \cdot \left(j \cdot k\right)}\]
   3. Using strategy rm

   4. Applied associate-*l* 32.5

      \[\leadsto \left(\left(\left(\color{blue}{\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot \left(z \cdot t\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\]

   if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 1.736218430949862e+238

   1. Initial program 0.3

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
   2. Using strategy rm

   3. Applied associate-*l* 0.3

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \color{blue}{j \cdot \left(27.0 \cdot k\right)}\]

   if 1.736218430949862e+238 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 18.4

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

   2. Taylor expanded around 0 18.3

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \color{blue}{27.0 \cdot \left(j \cdot k\right)}\]

   3. Taylor expanded around inf 13.4

      \[\leadsto \left(\left(\left(\color{blue}{18.0 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\]
   4. Using strategy rm

   5. Applied associate-*r* 13.8

      \[\leadsto \left(\left(\left(18.0 \cdot \left(t \cdot \color{blue}{\left(\left(x \cdot z\right) \cdot y\right)}\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\]
   6. Using strategy rm

   7. Applied associate-*r* 8.6

      \[\leadsto \left(\left(\left(18.0 \cdot \color{blue}{\left(\left(t \cdot \left(x \cdot z\right)\right) \cdot y\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i = -\infty:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot \left(t \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i \le 1.736218430949862 \cdot 10^{+238}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - j \cdot \left(27.0 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(18.0 \cdot \left(\left(t \cdot \left(z \cdot x\right)\right) \cdot y\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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)))