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

Average Error: 5.7 → 1.6

Time: 27.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(t \cdot y\right) \cdot z\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x}, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), j \cdot \left(27 \cdot k\right)\right)\right)\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 1.583419617970655285712273062224011067798 \cdot 10^{284}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot \left(z \cdot y\right)\right) \cdot x, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), j \cdot \left(27 \cdot k\right)\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r648752 = x;
        double r648753 = 18.0;
        double r648754 = r648752 * r648753;
        double r648755 = y;
        double r648756 = r648754 * r648755;
        double r648757 = z;
        double r648758 = r648756 * r648757;
        double r648759 = t;
        double r648760 = r648758 * r648759;
        double r648761 = a;
        double r648762 = 4.0;
        double r648763 = r648761 * r648762;
        double r648764 = r648763 * r648759;
        double r648765 = r648760 - r648764;
        double r648766 = b;
        double r648767 = c;
        double r648768 = r648766 * r648767;
        double r648769 = r648765 + r648768;
        double r648770 = r648752 * r648762;
        double r648771 = i;
        double r648772 = r648770 * r648771;
        double r648773 = r648769 - r648772;
        double r648774 = j;
        double r648775 = 27.0;
        double r648776 = r648774 * r648775;
        double r648777 = k;
        double r648778 = r648776 * r648777;
        double r648779 = r648773 - r648778;
        return r648779;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r648780 = x;
        double r648781 = 18.0;
        double r648782 = r648780 * r648781;
        double r648783 = y;
        double r648784 = r648782 * r648783;
        double r648785 = z;
        double r648786 = r648784 * r648785;
        double r648787 = t;
        double r648788 = r648786 * r648787;
        double r648789 = a;
        double r648790 = 4.0;
        double r648791 = r648789 * r648790;
        double r648792 = r648791 * r648787;
        double r648793 = r648788 - r648792;
        double r648794 = b;
        double r648795 = c;
        double r648796 = r648794 * r648795;
        double r648797 = r648793 + r648796;
        double r648798 = r648780 * r648790;
        double r648799 = i;
        double r648800 = r648798 * r648799;
        double r648801 = r648797 - r648800;
        double r648802 = -inf.0;
        bool r648803 = r648801 <= r648802;
        double r648804 = r648787 * r648783;
        double r648805 = r648804 * r648785;
        double r648806 = cbrt(r648780);
        double r648807 = r648806 * r648806;
        double r648808 = r648805 * r648807;
        double r648809 = r648808 * r648806;
        double r648810 = r648780 * r648799;
        double r648811 = fma(r648787, r648789, r648810);
        double r648812 = j;
        double r648813 = 27.0;
        double r648814 = k;
        double r648815 = r648813 * r648814;
        double r648816 = r648812 * r648815;
        double r648817 = fma(r648790, r648811, r648816);
        double r648818 = -r648817;
        double r648819 = fma(r648795, r648794, r648818);
        double r648820 = fma(r648809, r648781, r648819);
        double r648821 = 1.5834196179706553e+284;
        bool r648822 = r648801 <= r648821;
        double r648823 = r648812 * r648813;
        double r648824 = r648823 * r648814;
        double r648825 = r648801 - r648824;
        double r648826 = r648785 * r648783;
        double r648827 = r648787 * r648826;
        double r648828 = r648827 * r648780;
        double r648829 = fma(r648828, r648781, r648819);
        double r648830 = r648822 ? r648825 : r648829;
        double r648831 = r648803 ? r648820 : r648830;
        return r648831;
}

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	1.6

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

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. Simplified 14.4

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

   4. Applied associate-*r* 5.3

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

   6. Applied associate-*l* 5.5

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

   8. Applied add-cube-cbrt 5.9

      \[\leadsto \mathsf{fma}\left(\left(\left(t \cdot y\right) \cdot z\right) \cdot \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), j \cdot \left(27 \cdot k\right)\right)\right)\right)\]

   9. Applied associate-*r* 5.9

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\left(t \cdot y\right) \cdot z\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x}}, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), j \cdot \left(27 \cdot k\right)\right)\right)\right)\]

   if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 1.5834196179706553e+284

   1. Initial program 0.4

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

   if 1.5834196179706553e+284 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 37.4

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

   2. Simplified 12.5

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

   4. Applied associate-*r* 7.6

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

   6. Applied associate-*l* 7.6

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

   8. Applied *-un-lft-identity 7.6

      \[\leadsto \mathsf{fma}\left(\left(\left(t \cdot y\right) \cdot z\right) \cdot \color{blue}{\left(1 \cdot x\right)}, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), j \cdot \left(27 \cdot k\right)\right)\right)\right)\]

   9. Applied associate-*r* 7.6

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\left(t \cdot y\right) \cdot z\right) \cdot 1\right) \cdot x}, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), j \cdot \left(27 \cdot k\right)\right)\right)\right)\]

   10. Simplified 13.6

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(t \cdot y\right) \cdot z\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x}, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), j \cdot \left(27 \cdot k\right)\right)\right)\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 1.583419617970655285712273062224011067798 \cdot 10^{284}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot \left(z \cdot y\right)\right) \cdot x, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), j \cdot \left(27 \cdot k\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +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)))