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

Average Error: 3.8 → 1.4

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -6.4129760095180493 \cdot 10^{-217}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\ \mathbf{elif}\;t \le 2.4602073916559479 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - {\left(\left(9 \cdot \left(t \cdot y\right)\right) \cdot z\right)}^{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r847720 = x;
        double r847721 = 2.0;
        double r847722 = r847720 * r847721;
        double r847723 = y;
        double r847724 = 9.0;
        double r847725 = r847723 * r847724;
        double r847726 = z;
        double r847727 = r847725 * r847726;
        double r847728 = t;
        double r847729 = r847727 * r847728;
        double r847730 = r847722 - r847729;
        double r847731 = a;
        double r847732 = 27.0;
        double r847733 = r847731 * r847732;
        double r847734 = b;
        double r847735 = r847733 * r847734;
        double r847736 = r847730 + r847735;
        return r847736;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r847737 = t;
        double r847738 = -6.412976009518049e-217;
        bool r847739 = r847737 <= r847738;
        double r847740 = 2.0;
        double r847741 = x;
        double r847742 = 27.0;
        double r847743 = a;
        double r847744 = b;
        double r847745 = r847743 * r847744;
        double r847746 = r847742 * r847745;
        double r847747 = 9.0;
        double r847748 = z;
        double r847749 = y;
        double r847750 = r847748 * r847749;
        double r847751 = r847737 * r847750;
        double r847752 = r847747 * r847751;
        double r847753 = r847746 - r847752;
        double r847754 = fma(r847740, r847741, r847753);
        double r847755 = 2.460207391655948e-15;
        bool r847756 = r847737 <= r847755;
        double r847757 = r847737 * r847749;
        double r847758 = r847747 * r847757;
        double r847759 = r847758 * r847748;
        double r847760 = 1.0;
        double r847761 = pow(r847759, r847760);
        double r847762 = r847746 - r847761;
        double r847763 = fma(r847740, r847741, r847762);
        double r847764 = cbrt(r847747);
        double r847765 = r847764 * r847764;
        double r847766 = r847764 * r847751;
        double r847767 = r847765 * r847766;
        double r847768 = r847746 - r847767;
        double r847769 = fma(r847740, r847741, r847768);
        double r847770 = r847756 ? r847763 : r847769;
        double r847771 = r847739 ? r847754 : r847770;
        return r847771;
}

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	3.8
Target	2.5
Herbie	1.4

\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if t < -6.412976009518049e-217

   1. Initial program 2.8

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

   2. Simplified 2.8

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

   3. Taylor expanded around inf 2.7

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

   4. Simplified 2.7

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

   if -6.412976009518049e-217 < t < 2.460207391655948e-15

   1. Initial program 6.7

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

   2. Simplified 6.6

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

   3. Taylor expanded around inf 6.5

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

   4. Simplified 6.5

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

   6. Applied add-cube-cbrt 6.5

      \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - \color{blue}{\left(\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \sqrt[3]{9}\right)} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\]

   7. Applied associate-*l* 6.5

      \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - \color{blue}{\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)}\right)\]
   8. Using strategy rm

   9. Applied pow1 6.5

      \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot \color{blue}{{y}^{1}}\right)\right)\right)\right)\]

   10. Applied pow1 6.5

       \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(t \cdot \left(\color{blue}{{z}^{1}} \cdot {y}^{1}\right)\right)\right)\right)\]

   11. Applied pow-prod-down 6.5

       \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(t \cdot \color{blue}{{\left(z \cdot y\right)}^{1}}\right)\right)\right)\]

   12. Applied pow1 6.5

       \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(\color{blue}{{t}^{1}} \cdot {\left(z \cdot y\right)}^{1}\right)\right)\right)\]

   13. Applied pow-prod-down 6.5

       \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \color{blue}{{\left(t \cdot \left(z \cdot y\right)\right)}^{1}}\right)\right)\]

   14. Applied pow1 6.5

       \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{9}\right)}^{1}} \cdot {\left(t \cdot \left(z \cdot y\right)\right)}^{1}\right)\right)\]

   15. Applied pow-prod-down 6.5

       \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \color{blue}{{\left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)}^{1}}\right)\]

   16. Applied pow1 6.5

       \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - \left(\sqrt[3]{9} \cdot \color{blue}{{\left(\sqrt[3]{9}\right)}^{1}}\right) \cdot {\left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)}^{1}\right)\]

   17. Applied pow1 6.5

       \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - \left(\color{blue}{{\left(\sqrt[3]{9}\right)}^{1}} \cdot {\left(\sqrt[3]{9}\right)}^{1}\right) \cdot {\left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)}^{1}\right)\]

   18. Applied pow-prod-down 6.5

       \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - \color{blue}{{\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right)}^{1}} \cdot {\left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)}^{1}\right)\]

   19. Applied pow-prod-down 6.5

       \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - \color{blue}{{\left(\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\right)}^{1}}\right)\]

   20. Simplified 0.4

       \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - {\color{blue}{\left(\left(9 \cdot \left(t \cdot y\right)\right) \cdot z\right)}}^{1}\right)\]

   if 2.460207391655948e-15 < t

   1. Initial program 0.7

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

   2. Simplified 0.8

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

   3. Taylor expanded around inf 0.6

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

   4. Simplified 0.6

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

   6. Applied add-cube-cbrt 0.6

      \[\leadsto \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - \color{blue}{\left(\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \sqrt[3]{9}\right)} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\]

   7. Applied associate-*l* 0.6

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.4129760095180493 \cdot 10^{-217}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\ \mathbf{elif}\;t \le 2.4602073916559479 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - {\left(\left(9 \cdot \left(t \cdot y\right)\right) \cdot z\right)}^{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right) - \left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2) (* (* (* y 9) z) t)) (* a (* 27 b))) (+ (- (* x 2) (* 9 (* y (* t z)))) (* (* a 27) b)))

  (+ (- (* x 2) (* (* (* y 9) z) t)) (* (* a 27) b)))