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

Average Error: 3.7 → 0.8

Time: 19.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -87076559.25117374956607818603515625 \lor \neg \left(t \le 2.019127621004269860664729859013204689276 \cdot 10^{71}\right):\\ \;\;\;\;x \cdot 2 + \left(a \cdot \left(27 \cdot b\right) - 9 \cdot \left(\left(z \cdot y\right) \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 + \left(\left(\left(-9\right) \cdot y\right) \cdot t\right) \cdot z\right) + b \cdot \left(a \cdot 27\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r474618 = x;
        double r474619 = 2.0;
        double r474620 = r474618 * r474619;
        double r474621 = y;
        double r474622 = 9.0;
        double r474623 = r474621 * r474622;
        double r474624 = z;
        double r474625 = r474623 * r474624;
        double r474626 = t;
        double r474627 = r474625 * r474626;
        double r474628 = r474620 - r474627;
        double r474629 = a;
        double r474630 = 27.0;
        double r474631 = r474629 * r474630;
        double r474632 = b;
        double r474633 = r474631 * r474632;
        double r474634 = r474628 + r474633;
        return r474634;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r474635 = t;
        double r474636 = -87076559.25117375;
        bool r474637 = r474635 <= r474636;
        double r474638 = 2.0191276210042699e+71;
        bool r474639 = r474635 <= r474638;
        double r474640 = !r474639;
        bool r474641 = r474637 || r474640;
        double r474642 = x;
        double r474643 = 2.0;
        double r474644 = r474642 * r474643;
        double r474645 = a;
        double r474646 = 27.0;
        double r474647 = b;
        double r474648 = r474646 * r474647;
        double r474649 = r474645 * r474648;
        double r474650 = 9.0;
        double r474651 = z;
        double r474652 = y;
        double r474653 = r474651 * r474652;
        double r474654 = r474653 * r474635;
        double r474655 = r474650 * r474654;
        double r474656 = r474649 - r474655;
        double r474657 = r474644 + r474656;
        double r474658 = -r474650;
        double r474659 = r474658 * r474652;
        double r474660 = r474659 * r474635;
        double r474661 = r474660 * r474651;
        double r474662 = r474644 + r474661;
        double r474663 = r474645 * r474646;
        double r474664 = r474647 * r474663;
        double r474665 = r474662 + r474664;
        double r474666 = r474641 ? r474657 : r474665;
        return r474666;
}

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	2.7
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if t < -87076559.25117375 or 2.0191276210042699e+71 < 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. Using strategy rm

   3. Applied pow1 0.7

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

   4. Applied pow1 0.7

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

   5. Applied pow1 0.7

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

   6. Applied pow1 0.7

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

   7. Applied pow-prod-down 0.7

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

   8. Applied pow-prod-down 0.7

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

   9. Applied pow-prod-down 0.7

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

   10. Simplified 8.3

       \[\leadsto \left(x \cdot 2 - {\color{blue}{\left(\left(t \cdot z\right) \cdot \left(9 \cdot y\right)\right)}}^{1}\right) + \left(a \cdot 27\right) \cdot b\]
   11. Using strategy rm

   12. Applied sub-neg 8.3

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

   13. Applied associate-+l+ 8.3

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

   14. Simplified 0.8

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

   if -87076559.25117375 < t < 2.0191276210042699e+71

   1. Initial program 5.2

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

   3. Applied pow1 5.2

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

   4. Applied pow1 5.2

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

   5. Applied pow1 5.2

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

   6. Applied pow1 5.2

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

   7. Applied pow-prod-down 5.2

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

   8. Applied pow-prod-down 5.2

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

   9. Applied pow-prod-down 5.2

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

   10. Simplified 0.8

       \[\leadsto \left(x \cdot 2 - {\color{blue}{\left(\left(t \cdot z\right) \cdot \left(9 \cdot y\right)\right)}}^{1}\right) + \left(a \cdot 27\right) \cdot b\]
   11. Using strategy rm

   12. Applied sub-neg 0.8

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

   13. Simplified 0.8

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -87076559.25117374956607818603515625 \lor \neg \left(t \le 2.019127621004269860664729859013204689276 \cdot 10^{71}\right):\\ \;\;\;\;x \cdot 2 + \left(a \cdot \left(27 \cdot b\right) - 9 \cdot \left(\left(z \cdot y\right) \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 + \left(\left(\left(-9\right) \cdot y\right) \cdot t\right) \cdot z\right) + b \cdot \left(a \cdot 27\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(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)))