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

Average Error: 5.4 → 1.4

Time: 8.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -138023816.617849797 \lor \neg \left(y \le 4.51721422379651299 \cdot 10^{34}\right):\\ \;\;\;\;\left(\left(t \cdot \left(18 \cdot \left(x \cdot z\right)\right)\right) \cdot y + t \cdot \left(-a \cdot 4\right)\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left({\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r129441 = x;
        double r129442 = 18.0;
        double r129443 = r129441 * r129442;
        double r129444 = y;
        double r129445 = r129443 * r129444;
        double r129446 = z;
        double r129447 = r129445 * r129446;
        double r129448 = t;
        double r129449 = r129447 * r129448;
        double r129450 = a;
        double r129451 = 4.0;
        double r129452 = r129450 * r129451;
        double r129453 = r129452 * r129448;
        double r129454 = r129449 - r129453;
        double r129455 = b;
        double r129456 = c;
        double r129457 = r129455 * r129456;
        double r129458 = r129454 + r129457;
        double r129459 = r129441 * r129451;
        double r129460 = i;
        double r129461 = r129459 * r129460;
        double r129462 = r129458 - r129461;
        double r129463 = j;
        double r129464 = 27.0;
        double r129465 = r129463 * r129464;
        double r129466 = k;
        double r129467 = r129465 * r129466;
        double r129468 = r129462 - r129467;
        return r129468;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r129469 = y;
        double r129470 = -138023816.6178498;
        bool r129471 = r129469 <= r129470;
        double r129472 = 4.517214223796513e+34;
        bool r129473 = r129469 <= r129472;
        double r129474 = !r129473;
        bool r129475 = r129471 || r129474;
        double r129476 = t;
        double r129477 = 18.0;
        double r129478 = x;
        double r129479 = z;
        double r129480 = r129478 * r129479;
        double r129481 = r129477 * r129480;
        double r129482 = r129476 * r129481;
        double r129483 = r129482 * r129469;
        double r129484 = a;
        double r129485 = 4.0;
        double r129486 = r129484 * r129485;
        double r129487 = -r129486;
        double r129488 = r129476 * r129487;
        double r129489 = r129483 + r129488;
        double r129490 = b;
        double r129491 = c;
        double r129492 = r129490 * r129491;
        double r129493 = r129478 * r129485;
        double r129494 = i;
        double r129495 = r129493 * r129494;
        double r129496 = j;
        double r129497 = 27.0;
        double r129498 = r129496 * r129497;
        double r129499 = k;
        double r129500 = r129498 * r129499;
        double r129501 = r129495 + r129500;
        double r129502 = r129492 - r129501;
        double r129503 = r129489 + r129502;
        double r129504 = r129479 * r129469;
        double r129505 = r129478 * r129504;
        double r129506 = r129477 * r129505;
        double r129507 = 1.0;
        double r129508 = pow(r129506, r129507);
        double r129509 = r129508 - r129486;
        double r129510 = r129476 * r129509;
        double r129511 = r129497 * r129499;
        double r129512 = r129496 * r129511;
        double r129513 = r129495 + r129512;
        double r129514 = r129492 - r129513;
        double r129515 = r129510 + r129514;
        double r129516 = r129475 ? r129503 : r129515;
        return r129516;
}

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

Derivation

1. Split input into 2 regimes

2. if y < -138023816.6178498 or 4.517214223796513e+34 < y

   1. Initial program 11.2

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

   2. Simplified 11.2

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

   4. Applied pow1 11.2

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

   5. Applied pow1 11.2

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

   6. Applied pow1 11.2

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

   7. Applied pow1 11.2

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

   8. Applied pow-prod-down 11.2

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

   9. Applied pow-prod-down 11.2

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

   10. Applied pow-prod-down 11.2

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

   11. Simplified 13.4

       \[\leadsto t \cdot \left({\color{blue}{\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]
   12. Using strategy rm

   13. Applied associate-*r* 6.9

       \[\leadsto t \cdot \left({\left(18 \cdot \color{blue}{\left(\left(x \cdot z\right) \cdot y\right)}\right)}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]
   14. Using strategy rm

   15. Applied associate-*r* 7.0

       \[\leadsto t \cdot \left({\color{blue}{\left(\left(18 \cdot \left(x \cdot z\right)\right) \cdot y\right)}}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]
   16. Using strategy rm

   17. Applied sub-neg 7.0

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

   18. Applied distribute-lft-in 7.0

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

   19. Simplified 1.5

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

   if -138023816.6178498 < y < 4.517214223796513e+34

   1. Initial program 1.4

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

   2. Simplified 1.4

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

   4. Applied pow1 1.4

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

   5. Applied pow1 1.4

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

   6. Applied pow1 1.4

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

   7. Applied pow1 1.4

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

   8. Applied pow-prod-down 1.4

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

   9. Applied pow-prod-down 1.4

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

   10. Applied pow-prod-down 1.4

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

   11. Simplified 1.4

       \[\leadsto t \cdot \left({\color{blue}{\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]
   12. Using strategy rm

   13. Applied associate-*l* 1.3

       \[\leadsto t \cdot \left({\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \color{blue}{j \cdot \left(27 \cdot k\right)}\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -138023816.617849797 \lor \neg \left(y \le 4.51721422379651299 \cdot 10^{34}\right):\\ \;\;\;\;\left(\left(t \cdot \left(18 \cdot \left(x \cdot z\right)\right)\right) \cdot y + t \cdot \left(-a \cdot 4\right)\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left({\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  :precision binary64
  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))