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

Average Error: 5.5 → 3.9

Time: 25.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -4.817835946533626758757686769633005516507 \cdot 10^{-98}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(y \cdot \left(t \cdot z\right)\right) \cdot \left(x \cdot 18\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(4 \cdot x\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;y \le 1.919804578831060539244494100787191703918 \cdot 10^{-79}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(y \cdot z\right) \cdot t\right) \cdot \left(x \cdot 18\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(4 \cdot x\right) \cdot i\right) - 27 \cdot \left(j \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(y \cdot \left(t \cdot z\right)\right) \cdot \left(x \cdot 18\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(4 \cdot x\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r35241740 = x;
        double r35241741 = 18.0;
        double r35241742 = r35241740 * r35241741;
        double r35241743 = y;
        double r35241744 = r35241742 * r35241743;
        double r35241745 = z;
        double r35241746 = r35241744 * r35241745;
        double r35241747 = t;
        double r35241748 = r35241746 * r35241747;
        double r35241749 = a;
        double r35241750 = 4.0;
        double r35241751 = r35241749 * r35241750;
        double r35241752 = r35241751 * r35241747;
        double r35241753 = r35241748 - r35241752;
        double r35241754 = b;
        double r35241755 = c;
        double r35241756 = r35241754 * r35241755;
        double r35241757 = r35241753 + r35241756;
        double r35241758 = r35241740 * r35241750;
        double r35241759 = i;
        double r35241760 = r35241758 * r35241759;
        double r35241761 = r35241757 - r35241760;
        double r35241762 = j;
        double r35241763 = 27.0;
        double r35241764 = r35241762 * r35241763;
        double r35241765 = k;
        double r35241766 = r35241764 * r35241765;
        double r35241767 = r35241761 - r35241766;
        return r35241767;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r35241768 = y;
        double r35241769 = -4.817835946533627e-98;
        bool r35241770 = r35241768 <= r35241769;
        double r35241771 = b;
        double r35241772 = c;
        double r35241773 = r35241771 * r35241772;
        double r35241774 = t;
        double r35241775 = z;
        double r35241776 = r35241774 * r35241775;
        double r35241777 = r35241768 * r35241776;
        double r35241778 = x;
        double r35241779 = 18.0;
        double r35241780 = r35241778 * r35241779;
        double r35241781 = r35241777 * r35241780;
        double r35241782 = a;
        double r35241783 = 4.0;
        double r35241784 = r35241782 * r35241783;
        double r35241785 = r35241774 * r35241784;
        double r35241786 = r35241781 - r35241785;
        double r35241787 = r35241773 + r35241786;
        double r35241788 = r35241783 * r35241778;
        double r35241789 = i;
        double r35241790 = r35241788 * r35241789;
        double r35241791 = r35241787 - r35241790;
        double r35241792 = j;
        double r35241793 = 27.0;
        double r35241794 = k;
        double r35241795 = r35241793 * r35241794;
        double r35241796 = r35241792 * r35241795;
        double r35241797 = r35241791 - r35241796;
        double r35241798 = 1.9198045788310605e-79;
        bool r35241799 = r35241768 <= r35241798;
        double r35241800 = r35241768 * r35241775;
        double r35241801 = r35241800 * r35241774;
        double r35241802 = r35241801 * r35241780;
        double r35241803 = r35241802 - r35241785;
        double r35241804 = r35241773 + r35241803;
        double r35241805 = r35241804 - r35241790;
        double r35241806 = r35241792 * r35241794;
        double r35241807 = r35241793 * r35241806;
        double r35241808 = r35241805 - r35241807;
        double r35241809 = r35241799 ? r35241808 : r35241797;
        double r35241810 = r35241770 ? r35241797 : r35241809;
        return r35241810;
}

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	5.5
Target	1.4
Herbie	3.9

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

Derivation

1. Split input into 2 regimes

2. if y < -4.817835946533627e-98 or 1.9198045788310605e-79 < y

   1. Initial program 8.5

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

   3. Applied associate-*l* 10.5

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

   5. Applied associate-*l* 10.6

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

   7. Applied associate-*l* 10.6

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

   9. Applied associate-*l* 6.2

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

   if -4.817835946533627e-98 < y < 1.9198045788310605e-79

   1. Initial program 0.9

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

   3. Applied associate-*l* 0.7

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

   5. Applied associate-*l* 0.7

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

   6. Taylor expanded around 0 0.6

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

4. Final simplification 3.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.817835946533626758757686769633005516507 \cdot 10^{-98}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(y \cdot \left(t \cdot z\right)\right) \cdot \left(x \cdot 18\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(4 \cdot x\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;y \le 1.919804578831060539244494100787191703918 \cdot 10^{-79}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(y \cdot z\right) \cdot t\right) \cdot \left(x \cdot 18\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(4 \cdot x\right) \cdot i\right) - 27 \cdot \left(j \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(y \cdot \left(t \cdot z\right)\right) \cdot \left(x \cdot 18\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(4 \cdot x\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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)))