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

Average Error: 3.5 → 2.2

Time: 14.2s

Precision: 64

Math

\[\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.4982446069550864 \cdot 10^{+80}:\\ \;\;\;\;x \cdot 2.0 + \left(b \cdot \left(a \cdot 27.0\right) - \sqrt{9.0} \cdot \left(\sqrt{9.0} \cdot \left(\left(y \cdot z\right) \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2.0 - \left(y \cdot 9.0\right) \cdot \left(z \cdot t\right)\right) + b \cdot \left(a \cdot 27.0\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r13810681 = x;
        double r13810682 = 2.0;
        double r13810683 = r13810681 * r13810682;
        double r13810684 = y;
        double r13810685 = 9.0;
        double r13810686 = r13810684 * r13810685;
        double r13810687 = z;
        double r13810688 = r13810686 * r13810687;
        double r13810689 = t;
        double r13810690 = r13810688 * r13810689;
        double r13810691 = r13810683 - r13810690;
        double r13810692 = a;
        double r13810693 = 27.0;
        double r13810694 = r13810692 * r13810693;
        double r13810695 = b;
        double r13810696 = r13810694 * r13810695;
        double r13810697 = r13810691 + r13810696;
        return r13810697;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r13810698 = t;
        double r13810699 = -1.4982446069550864e+80;
        bool r13810700 = r13810698 <= r13810699;
        double r13810701 = x;
        double r13810702 = 2.0;
        double r13810703 = r13810701 * r13810702;
        double r13810704 = b;
        double r13810705 = a;
        double r13810706 = 27.0;
        double r13810707 = r13810705 * r13810706;
        double r13810708 = r13810704 * r13810707;
        double r13810709 = 9.0;
        double r13810710 = sqrt(r13810709);
        double r13810711 = y;
        double r13810712 = z;
        double r13810713 = r13810711 * r13810712;
        double r13810714 = r13810713 * r13810698;
        double r13810715 = r13810710 * r13810714;
        double r13810716 = r13810710 * r13810715;
        double r13810717 = r13810708 - r13810716;
        double r13810718 = r13810703 + r13810717;
        double r13810719 = r13810711 * r13810709;
        double r13810720 = r13810712 * r13810698;
        double r13810721 = r13810719 * r13810720;
        double r13810722 = r13810703 - r13810721;
        double r13810723 = r13810722 + r13810708;
        double r13810724 = r13810700 ? r13810718 : r13810723;
        return r13810724;
}

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	3.5
Target	2.5
Herbie	2.2

\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + a \cdot \left(27.0 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2.0 - 9.0 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27.0\right) \cdot b\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if t < -1.4982446069550864e+80

   1. Initial program 0.8

      \[\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\]
   2. Using strategy rm

   3. Applied sub-neg 0.8

      \[\leadsto \color{blue}{\left(x \cdot 2.0 + \left(-\left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right)\right)} + \left(a \cdot 27.0\right) \cdot b\]

   4. Applied associate-+l+ 0.8

      \[\leadsto \color{blue}{x \cdot 2.0 + \left(\left(-\left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\right)}\]

   5. Simplified 9.3

      \[\leadsto x \cdot 2.0 + \color{blue}{\left(\left(27.0 \cdot a\right) \cdot b - y \cdot \left(\left(z \cdot 9.0\right) \cdot t\right)\right)}\]

   6. Taylor expanded around inf 0.8

      \[\leadsto x \cdot 2.0 + \left(\left(27.0 \cdot a\right) \cdot b - \color{blue}{9.0 \cdot \left(t \cdot \left(z \cdot y\right)\right)}\right)\]
   7. Using strategy rm

   8. Applied add-sqr-sqrt 0.8

      \[\leadsto x \cdot 2.0 + \left(\left(27.0 \cdot a\right) \cdot b - \color{blue}{\left(\sqrt{9.0} \cdot \sqrt{9.0}\right)} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\]

   9. Applied associate-*l* 0.9

      \[\leadsto x \cdot 2.0 + \left(\left(27.0 \cdot a\right) \cdot b - \color{blue}{\sqrt{9.0} \cdot \left(\sqrt{9.0} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)}\right)\]

   if -1.4982446069550864e+80 < t

   1. Initial program 4.0

      \[\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\]
   2. Using strategy rm

   3. Applied associate-*l* 2.4

      \[\leadsto \left(x \cdot 2.0 - \color{blue}{\left(y \cdot 9.0\right) \cdot \left(z \cdot t\right)}\right) + \left(a \cdot 27.0\right) \cdot b\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.4982446069550864 \cdot 10^{+80}:\\ \;\;\;\;x \cdot 2.0 + \left(b \cdot \left(a \cdot 27.0\right) - \sqrt{9.0} \cdot \left(\sqrt{9.0} \cdot \left(\left(y \cdot z\right) \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2.0 - \left(y \cdot 9.0\right) \cdot \left(z \cdot t\right)\right) + b \cdot \left(a \cdot 27.0\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +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)))