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

Average Error: 5.8 → 4.1

Time: 25.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -6.163897221999321950368730647359960400126 \cdot 10^{68}:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(\left(18 \cdot \left(x \cdot y\right)\right) \cdot z\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;z \le 3.513749760257365308486787390070311830711 \cdot 10^{54}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(t \cdot \left(\left(y \cdot z\right) \cdot x\right)\right) \cdot 18 - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;z \le 9.457178616359135458824395063988188906067 \cdot 10^{237}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(18 \cdot \left(x \cdot y\right)\right) \cdot \left(t \cdot z\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - k \cdot \left(j \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(\left(18 \cdot \left(x \cdot y\right)\right) \cdot z\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r37798686 = x;
        double r37798687 = 18.0;
        double r37798688 = r37798686 * r37798687;
        double r37798689 = y;
        double r37798690 = r37798688 * r37798689;
        double r37798691 = z;
        double r37798692 = r37798690 * r37798691;
        double r37798693 = t;
        double r37798694 = r37798692 * r37798693;
        double r37798695 = a;
        double r37798696 = 4.0;
        double r37798697 = r37798695 * r37798696;
        double r37798698 = r37798697 * r37798693;
        double r37798699 = r37798694 - r37798698;
        double r37798700 = b;
        double r37798701 = c;
        double r37798702 = r37798700 * r37798701;
        double r37798703 = r37798699 + r37798702;
        double r37798704 = r37798686 * r37798696;
        double r37798705 = i;
        double r37798706 = r37798704 * r37798705;
        double r37798707 = r37798703 - r37798706;
        double r37798708 = j;
        double r37798709 = 27.0;
        double r37798710 = r37798708 * r37798709;
        double r37798711 = k;
        double r37798712 = r37798710 * r37798711;
        double r37798713 = r37798707 - r37798712;
        return r37798713;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r37798714 = z;
        double r37798715 = -6.163897221999322e+68;
        bool r37798716 = r37798714 <= r37798715;
        double r37798717 = b;
        double r37798718 = c;
        double r37798719 = r37798717 * r37798718;
        double r37798720 = t;
        double r37798721 = 18.0;
        double r37798722 = x;
        double r37798723 = y;
        double r37798724 = r37798722 * r37798723;
        double r37798725 = r37798721 * r37798724;
        double r37798726 = r37798725 * r37798714;
        double r37798727 = r37798720 * r37798726;
        double r37798728 = a;
        double r37798729 = 4.0;
        double r37798730 = r37798728 * r37798729;
        double r37798731 = r37798720 * r37798730;
        double r37798732 = r37798727 - r37798731;
        double r37798733 = r37798719 + r37798732;
        double r37798734 = r37798722 * r37798729;
        double r37798735 = i;
        double r37798736 = r37798734 * r37798735;
        double r37798737 = r37798733 - r37798736;
        double r37798738 = j;
        double r37798739 = 27.0;
        double r37798740 = k;
        double r37798741 = r37798739 * r37798740;
        double r37798742 = r37798738 * r37798741;
        double r37798743 = r37798737 - r37798742;
        double r37798744 = 3.513749760257365e+54;
        bool r37798745 = r37798714 <= r37798744;
        double r37798746 = r37798723 * r37798714;
        double r37798747 = r37798746 * r37798722;
        double r37798748 = r37798720 * r37798747;
        double r37798749 = r37798748 * r37798721;
        double r37798750 = r37798749 - r37798731;
        double r37798751 = r37798719 + r37798750;
        double r37798752 = r37798751 - r37798736;
        double r37798753 = r37798752 - r37798742;
        double r37798754 = 9.457178616359135e+237;
        bool r37798755 = r37798714 <= r37798754;
        double r37798756 = r37798720 * r37798714;
        double r37798757 = r37798725 * r37798756;
        double r37798758 = r37798757 - r37798731;
        double r37798759 = r37798719 + r37798758;
        double r37798760 = r37798759 - r37798736;
        double r37798761 = r37798738 * r37798739;
        double r37798762 = r37798740 * r37798761;
        double r37798763 = r37798760 - r37798762;
        double r37798764 = r37798755 ? r37798763 : r37798743;
        double r37798765 = r37798745 ? r37798753 : r37798764;
        double r37798766 = r37798716 ? r37798743 : r37798765;
        return r37798766;
}

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	1.7
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 3 regimes

2. if z < -6.163897221999322e+68 or 9.457178616359135e+237 < z

   1. Initial program 9.1

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

   2. Taylor expanded around 0 9.1

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

   4. Applied associate-*l* 9.1

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

   if -6.163897221999322e+68 < z < 3.513749760257365e+54

   1. Initial program 4.7

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

   2. Taylor expanded around 0 4.6

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

   4. Applied associate-*l* 4.7

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

   5. Taylor expanded around inf 1.7

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

   if 3.513749760257365e+54 < z < 9.457178616359135e+237

   1. Initial program 6.6

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

   2. Taylor expanded around 0 6.6

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

   4. Applied associate-*l* 8.5

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

4. Final simplification 4.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.163897221999321950368730647359960400126 \cdot 10^{68}:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(\left(18 \cdot \left(x \cdot y\right)\right) \cdot z\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;z \le 3.513749760257365308486787390070311830711 \cdot 10^{54}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(t \cdot \left(\left(y \cdot z\right) \cdot x\right)\right) \cdot 18 - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;z \le 9.457178616359135458824395063988188906067 \cdot 10^{237}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(18 \cdot \left(x \cdot y\right)\right) \cdot \left(t \cdot z\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - k \cdot \left(j \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(\left(18 \cdot \left(x \cdot y\right)\right) \cdot z\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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)))