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

Average Error: 5.5 → 2.3

Time: 18.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.231030079622034677757047713069532532969 \cdot 10^{-84} \lor \neg \left(x \le 3.987060137731610333464591129755943173549 \cdot 10^{-19}\right):\\ \;\;\;\;\left(\left(\left(x \cdot \left(18 \cdot \left(\left(y \cdot z\right) \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r971765 = x;
        double r971766 = 18.0;
        double r971767 = r971765 * r971766;
        double r971768 = y;
        double r971769 = r971767 * r971768;
        double r971770 = z;
        double r971771 = r971769 * r971770;
        double r971772 = t;
        double r971773 = r971771 * r971772;
        double r971774 = a;
        double r971775 = 4.0;
        double r971776 = r971774 * r971775;
        double r971777 = r971776 * r971772;
        double r971778 = r971773 - r971777;
        double r971779 = b;
        double r971780 = c;
        double r971781 = r971779 * r971780;
        double r971782 = r971778 + r971781;
        double r971783 = r971765 * r971775;
        double r971784 = i;
        double r971785 = r971783 * r971784;
        double r971786 = r971782 - r971785;
        double r971787 = j;
        double r971788 = 27.0;
        double r971789 = r971787 * r971788;
        double r971790 = k;
        double r971791 = r971789 * r971790;
        double r971792 = r971786 - r971791;
        return r971792;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r971793 = x;
        double r971794 = -4.231030079622035e-84;
        bool r971795 = r971793 <= r971794;
        double r971796 = 3.9870601377316103e-19;
        bool r971797 = r971793 <= r971796;
        double r971798 = !r971797;
        bool r971799 = r971795 || r971798;
        double r971800 = 18.0;
        double r971801 = y;
        double r971802 = z;
        double r971803 = r971801 * r971802;
        double r971804 = t;
        double r971805 = r971803 * r971804;
        double r971806 = r971800 * r971805;
        double r971807 = r971793 * r971806;
        double r971808 = a;
        double r971809 = 4.0;
        double r971810 = r971808 * r971809;
        double r971811 = r971810 * r971804;
        double r971812 = r971807 - r971811;
        double r971813 = b;
        double r971814 = c;
        double r971815 = r971813 * r971814;
        double r971816 = r971812 + r971815;
        double r971817 = r971793 * r971809;
        double r971818 = i;
        double r971819 = r971817 * r971818;
        double r971820 = r971816 - r971819;
        double r971821 = j;
        double r971822 = 27.0;
        double r971823 = r971821 * r971822;
        double r971824 = k;
        double r971825 = r971823 * r971824;
        double r971826 = r971820 - r971825;
        double r971827 = r971793 * r971800;
        double r971828 = r971827 * r971801;
        double r971829 = r971828 * r971802;
        double r971830 = r971829 * r971804;
        double r971831 = r971830 - r971811;
        double r971832 = r971831 + r971815;
        double r971833 = r971832 - r971819;
        double r971834 = r971822 * r971824;
        double r971835 = r971821 * r971834;
        double r971836 = r971833 - r971835;
        double r971837 = r971799 ? r971826 : r971836;
        return r971837;
}

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	1.5
Herbie	2.3

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

Derivation

1. Split input into 2 regimes

2. if x < -4.231030079622035e-84 or 3.9870601377316103e-19 < x

   1. Initial program 9.9

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

   3. Applied associate-*l* 6.6

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

   5. Applied associate-*l* 3.1

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

   7. Applied associate-*l* 3.0

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

   if -4.231030079622035e-84 < x < 3.9870601377316103e-19

   1. Initial program 1.6

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

   3. Applied associate-*l* 1.6

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

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.231030079622034677757047713069532532969 \cdot 10^{-84} \lor \neg \left(x \le 3.987060137731610333464591129755943173549 \cdot 10^{-19}\right):\\ \;\;\;\;\left(\left(\left(x \cdot \left(18 \cdot \left(\left(y \cdot z\right) \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"
  :precision binary64

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4)) (- (* c b) (* 27 (* k j)))) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))