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

Average Error: 3.7 → 0.5

Time: 17.7s

Precision: 64

Math

\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t = -\infty:\\ \;\;\;\;\mathsf{fma}\left(-9, z \cdot \left(t \cdot y\right), a \cdot \left(27 \cdot b\right)\right) + x \cdot 2\\ \mathbf{elif}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \le 4.296785106576348656228515128564970887864 \cdot 10^{306}:\\ \;\;\;\;\left(x \cdot 2 - t \cdot \left(y \cdot \left(z \cdot 9\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 27\right) \cdot b + \left(x \cdot 2 - \left(z \cdot \left(t \cdot 9\right)\right) \cdot y\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r626708 = x;
        double r626709 = 2.0;
        double r626710 = r626708 * r626709;
        double r626711 = y;
        double r626712 = 9.0;
        double r626713 = r626711 * r626712;
        double r626714 = z;
        double r626715 = r626713 * r626714;
        double r626716 = t;
        double r626717 = r626715 * r626716;
        double r626718 = r626710 - r626717;
        double r626719 = a;
        double r626720 = 27.0;
        double r626721 = r626719 * r626720;
        double r626722 = b;
        double r626723 = r626721 * r626722;
        double r626724 = r626718 + r626723;
        return r626724;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r626725 = y;
        double r626726 = 9.0;
        double r626727 = r626725 * r626726;
        double r626728 = z;
        double r626729 = r626727 * r626728;
        double r626730 = t;
        double r626731 = r626729 * r626730;
        double r626732 = -inf.0;
        bool r626733 = r626731 <= r626732;
        double r626734 = -r626726;
        double r626735 = r626730 * r626725;
        double r626736 = r626728 * r626735;
        double r626737 = a;
        double r626738 = 27.0;
        double r626739 = b;
        double r626740 = r626738 * r626739;
        double r626741 = r626737 * r626740;
        double r626742 = fma(r626734, r626736, r626741);
        double r626743 = x;
        double r626744 = 2.0;
        double r626745 = r626743 * r626744;
        double r626746 = r626742 + r626745;
        double r626747 = 4.2967851065763487e+306;
        bool r626748 = r626731 <= r626747;
        double r626749 = r626728 * r626726;
        double r626750 = r626725 * r626749;
        double r626751 = r626730 * r626750;
        double r626752 = r626745 - r626751;
        double r626753 = r626737 * r626738;
        double r626754 = r626753 * r626739;
        double r626755 = r626752 + r626754;
        double r626756 = r626730 * r626726;
        double r626757 = r626728 * r626756;
        double r626758 = r626757 * r626725;
        double r626759 = r626745 - r626758;
        double r626760 = r626754 + r626759;
        double r626761 = r626748 ? r626755 : r626760;
        double r626762 = r626733 ? r626746 : r626761;
        return r626762;
}

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

Target

Original	3.7
Target	2.6
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811188954625810696587370427881 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* (* (* y 9.0) z) t) < -inf.0

   1. Initial program 64.0

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   2. Using strategy rm

   3. Applied sub-neg 64.0

      \[\leadsto \color{blue}{\left(x \cdot 2 + \left(-\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right)} + \left(a \cdot 27\right) \cdot b\]

   4. Applied associate-+l+ 64.0

      \[\leadsto \color{blue}{x \cdot 2 + \left(\left(-\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\right)}\]

   5. Simplified 64.0

      \[\leadsto x \cdot 2 + \color{blue}{\mathsf{fma}\left(b \cdot a, 27, \left(-t\right) \cdot \left(z \cdot \left(9 \cdot y\right)\right)\right)}\]

   6. Taylor expanded around inf 62.7

      \[\leadsto x \cdot 2 + \color{blue}{\left(27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)}\]

   7. Simplified 0.6

      \[\leadsto x \cdot 2 + \color{blue}{\mathsf{fma}\left(-9, z \cdot \left(y \cdot t\right), \left(27 \cdot b\right) \cdot a\right)}\]

   if -inf.0 < (* (* (* y 9.0) z) t) < 4.2967851065763487e+306

   1. Initial program 0.5

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   2. Using strategy rm

   3. Applied associate-*l* 0.5

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(y \cdot \left(9 \cdot z\right)\right)} \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

   4. Simplified 0.5

      \[\leadsto \left(x \cdot 2 - \left(y \cdot \color{blue}{\left(z \cdot 9\right)}\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

   if 4.2967851065763487e+306 < (* (* (* y 9.0) z) t)

   1. Initial program 62.7

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   2. Using strategy rm

   3. Applied associate-*l* 62.1

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(y \cdot \left(9 \cdot z\right)\right)} \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

   4. Simplified 62.1

      \[\leadsto \left(x \cdot 2 - \left(y \cdot \color{blue}{\left(z \cdot 9\right)}\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   5. Using strategy rm

   6. Applied associate-*l* 0.9

      \[\leadsto \left(x \cdot 2 - \color{blue}{y \cdot \left(\left(z \cdot 9\right) \cdot t\right)}\right) + \left(a \cdot 27\right) \cdot b\]

   7. Simplified 0.3

      \[\leadsto \left(x \cdot 2 - y \cdot \color{blue}{\left(z \cdot \left(9 \cdot t\right)\right)}\right) + \left(a \cdot 27\right) \cdot b\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t = -\infty:\\ \;\;\;\;\mathsf{fma}\left(-9, z \cdot \left(t \cdot y\right), a \cdot \left(27 \cdot b\right)\right) + x \cdot 2\\ \mathbf{elif}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \le 4.296785106576348656228515128564970887864 \cdot 10^{306}:\\ \;\;\;\;\left(x \cdot 2 - t \cdot \left(y \cdot \left(z \cdot 9\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 27\right) \cdot b + \left(x \cdot 2 - \left(z \cdot \left(t \cdot 9\right)\right) \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b))) (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b)))

  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))