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

Average Error: 5.6 → 1.4

Time: 24.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i = -\infty:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(\left(t \cdot z\right) \cdot y\right) \cdot \left(x \cdot 18\right) - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 3.252609788198741362599457824359920825411 \cdot 10^{268}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot k\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(x \cdot \left(18 \cdot \left(\left(t \cdot z\right) \cdot y\right)\right) - \left(a \cdot 4\right) \cdot t\right)\right) - x \cdot \left(4 \cdot i\right)\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r37879742 = x;
        double r37879743 = 18.0;
        double r37879744 = r37879742 * r37879743;
        double r37879745 = y;
        double r37879746 = r37879744 * r37879745;
        double r37879747 = z;
        double r37879748 = r37879746 * r37879747;
        double r37879749 = t;
        double r37879750 = r37879748 * r37879749;
        double r37879751 = a;
        double r37879752 = 4.0;
        double r37879753 = r37879751 * r37879752;
        double r37879754 = r37879753 * r37879749;
        double r37879755 = r37879750 - r37879754;
        double r37879756 = b;
        double r37879757 = c;
        double r37879758 = r37879756 * r37879757;
        double r37879759 = r37879755 + r37879758;
        double r37879760 = r37879742 * r37879752;
        double r37879761 = i;
        double r37879762 = r37879760 * r37879761;
        double r37879763 = r37879759 - r37879762;
        double r37879764 = j;
        double r37879765 = 27.0;
        double r37879766 = r37879764 * r37879765;
        double r37879767 = k;
        double r37879768 = r37879766 * r37879767;
        double r37879769 = r37879763 - r37879768;
        return r37879769;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r37879770 = t;
        double r37879771 = x;
        double r37879772 = 18.0;
        double r37879773 = r37879771 * r37879772;
        double r37879774 = y;
        double r37879775 = r37879773 * r37879774;
        double r37879776 = z;
        double r37879777 = r37879775 * r37879776;
        double r37879778 = r37879770 * r37879777;
        double r37879779 = a;
        double r37879780 = 4.0;
        double r37879781 = r37879779 * r37879780;
        double r37879782 = r37879781 * r37879770;
        double r37879783 = r37879778 - r37879782;
        double r37879784 = c;
        double r37879785 = b;
        double r37879786 = r37879784 * r37879785;
        double r37879787 = r37879783 + r37879786;
        double r37879788 = r37879771 * r37879780;
        double r37879789 = i;
        double r37879790 = r37879788 * r37879789;
        double r37879791 = r37879787 - r37879790;
        double r37879792 = -inf.0;
        bool r37879793 = r37879791 <= r37879792;
        double r37879794 = r37879770 * r37879776;
        double r37879795 = r37879794 * r37879774;
        double r37879796 = r37879795 * r37879773;
        double r37879797 = r37879796 - r37879782;
        double r37879798 = r37879786 + r37879797;
        double r37879799 = r37879798 - r37879790;
        double r37879800 = j;
        double r37879801 = 27.0;
        double r37879802 = r37879800 * r37879801;
        double r37879803 = k;
        double r37879804 = r37879802 * r37879803;
        double r37879805 = r37879799 - r37879804;
        double r37879806 = 3.2526097881987414e+268;
        bool r37879807 = r37879791 <= r37879806;
        double r37879808 = r37879800 * r37879803;
        double r37879809 = r37879808 * r37879801;
        double r37879810 = r37879791 - r37879809;
        double r37879811 = r37879772 * r37879795;
        double r37879812 = r37879771 * r37879811;
        double r37879813 = r37879812 - r37879782;
        double r37879814 = r37879786 + r37879813;
        double r37879815 = r37879780 * r37879789;
        double r37879816 = r37879771 * r37879815;
        double r37879817 = r37879814 - r37879816;
        double r37879818 = r37879817 - r37879804;
        double r37879819 = r37879807 ? r37879810 : r37879818;
        double r37879820 = r37879793 ? r37879805 : r37879819;
        return r37879820;
}

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	5.6
Target	1.6
Herbie	1.4

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

Derivation

1. Split input into 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 64.0

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

   3. Applied associate-*l* 36.0

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

   5. Applied associate-*l* 6.9

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

   if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 3.2526097881987414e+268

   1. Initial program 0.3

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

   2. Taylor expanded around 0 0.2

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

   if 3.2526097881987414e+268 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 26.8

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

   3. Applied associate-*l* 18.4

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

   5. Applied associate-*l* 9.9

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

   7. Applied associate-*l* 9.6

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

   9. Applied associate-*l* 9.3

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i = -\infty:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(\left(t \cdot z\right) \cdot y\right) \cdot \left(x \cdot 18\right) - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{elif}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 3.252609788198741362599457824359920825411 \cdot 10^{268}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot k\right) \cdot 27\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(x \cdot \left(18 \cdot \left(\left(t \cdot z\right) \cdot y\right)\right) - \left(a \cdot 4\right) \cdot t\right)\right) - x \cdot \left(4 \cdot i\right)\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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)))