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

Average Error: 5.7 → 4.2

Time: 29.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.339915709177500803043603435967818880006 \cdot 10^{130}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot \left(y \cdot 18\right)\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(k \cdot j\right)\right)\right)\\ \mathbf{elif}\;z \le 4.983042576879064654649325286831293190518 \cdot 10^{-72}:\\ \;\;\;\;\left(t \cdot \left(x \cdot \left(\left(y \cdot 18\right) \cdot z\right) - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r498726 = x;
        double r498727 = 18.0;
        double r498728 = r498726 * r498727;
        double r498729 = y;
        double r498730 = r498728 * r498729;
        double r498731 = z;
        double r498732 = r498730 * r498731;
        double r498733 = t;
        double r498734 = r498732 * r498733;
        double r498735 = a;
        double r498736 = 4.0;
        double r498737 = r498735 * r498736;
        double r498738 = r498737 * r498733;
        double r498739 = r498734 - r498738;
        double r498740 = b;
        double r498741 = c;
        double r498742 = r498740 * r498741;
        double r498743 = r498739 + r498742;
        double r498744 = r498726 * r498736;
        double r498745 = i;
        double r498746 = r498744 * r498745;
        double r498747 = r498743 - r498746;
        double r498748 = j;
        double r498749 = 27.0;
        double r498750 = r498748 * r498749;
        double r498751 = k;
        double r498752 = r498750 * r498751;
        double r498753 = r498747 - r498752;
        return r498753;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r498754 = z;
        double r498755 = -1.3399157091775008e+130;
        bool r498756 = r498754 <= r498755;
        double r498757 = t;
        double r498758 = x;
        double r498759 = y;
        double r498760 = 18.0;
        double r498761 = r498759 * r498760;
        double r498762 = r498758 * r498761;
        double r498763 = r498762 * r498754;
        double r498764 = a;
        double r498765 = 4.0;
        double r498766 = r498764 * r498765;
        double r498767 = r498763 - r498766;
        double r498768 = r498757 * r498767;
        double r498769 = b;
        double r498770 = c;
        double r498771 = r498769 * r498770;
        double r498772 = r498768 + r498771;
        double r498773 = r498758 * r498765;
        double r498774 = i;
        double r498775 = r498773 * r498774;
        double r498776 = 27.0;
        double r498777 = sqrt(r498776);
        double r498778 = k;
        double r498779 = j;
        double r498780 = r498778 * r498779;
        double r498781 = r498777 * r498780;
        double r498782 = r498777 * r498781;
        double r498783 = r498775 + r498782;
        double r498784 = r498772 - r498783;
        double r498785 = 4.983042576879065e-72;
        bool r498786 = r498754 <= r498785;
        double r498787 = r498761 * r498754;
        double r498788 = r498758 * r498787;
        double r498789 = r498788 - r498766;
        double r498790 = r498757 * r498789;
        double r498791 = r498790 + r498771;
        double r498792 = r498779 * r498776;
        double r498793 = r498792 * r498778;
        double r498794 = r498775 + r498793;
        double r498795 = r498791 - r498794;
        double r498796 = r498758 * r498760;
        double r498797 = r498796 * r498759;
        double r498798 = r498797 * r498754;
        double r498799 = r498798 - r498766;
        double r498800 = r498757 * r498799;
        double r498801 = r498800 + r498771;
        double r498802 = r498776 * r498778;
        double r498803 = r498779 * r498802;
        double r498804 = r498775 + r498803;
        double r498805 = r498801 - r498804;
        double r498806 = r498786 ? r498795 : r498805;
        double r498807 = r498756 ? r498784 : r498806;
        return r498807;
}

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	5.7
Target	1.6
Herbie	4.2

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

Derivation

1. Split input into 3 regimes

2. if z < -1.3399157091775008e+130

   1. Initial program 7.9

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

   2. Simplified 7.9

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

   4. Applied associate-*l* 7.9

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

   5. Simplified 7.9

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

   6. Taylor expanded around 0 7.8

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

   8. Applied add-sqr-sqrt 7.8

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

   9. Applied associate-*l* 7.8

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

   if -1.3399157091775008e+130 < z < 4.983042576879065e-72

   1. Initial program 5.1

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

   2. Simplified 5.1

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

   4. Applied associate-*l* 5.1

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

   5. Simplified 5.1

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

   7. Applied associate-*l* 2.5

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

   if 4.983042576879065e-72 < z

   1. Initial program 6.2

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

   2. Simplified 6.2

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

   4. Applied associate-*l* 6.3

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

4. Final simplification 4.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.339915709177500803043603435967818880006 \cdot 10^{130}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot \left(y \cdot 18\right)\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(k \cdot j\right)\right)\right)\\ \mathbf{elif}\;z \le 4.983042576879064654649325286831293190518 \cdot 10^{-72}:\\ \;\;\;\;\left(t \cdot \left(x \cdot \left(\left(y \cdot 18\right) \cdot z\right) - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))