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

Average Error: 5.7 → 1.9

Time: 24.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -18869685775798424479823402359192282988540:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(k \cdot j\right)\\ \mathbf{elif}\;t \le 1.786777171204849913027233772322421856416 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(b \cdot c + \left(x \cdot \left(\left(18 \cdot y\right) \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(\sqrt[3]{27 \cdot \left(k \cdot j\right)} \cdot \sqrt[3]{27 \cdot \left(k \cdot j\right)}\right) \cdot \sqrt[3]{27 \cdot \left(k \cdot j\right)}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r97500 = x;
        double r97501 = 18.0;
        double r97502 = r97500 * r97501;
        double r97503 = y;
        double r97504 = r97502 * r97503;
        double r97505 = z;
        double r97506 = r97504 * r97505;
        double r97507 = t;
        double r97508 = r97506 * r97507;
        double r97509 = a;
        double r97510 = 4.0;
        double r97511 = r97509 * r97510;
        double r97512 = r97511 * r97507;
        double r97513 = r97508 - r97512;
        double r97514 = b;
        double r97515 = c;
        double r97516 = r97514 * r97515;
        double r97517 = r97513 + r97516;
        double r97518 = r97500 * r97510;
        double r97519 = i;
        double r97520 = r97518 * r97519;
        double r97521 = r97517 - r97520;
        double r97522 = j;
        double r97523 = 27.0;
        double r97524 = r97522 * r97523;
        double r97525 = k;
        double r97526 = r97524 * r97525;
        double r97527 = r97521 - r97526;
        return r97527;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r97528 = t;
        double r97529 = -1.8869685775798424e+40;
        bool r97530 = r97528 <= r97529;
        double r97531 = b;
        double r97532 = c;
        double r97533 = r97531 * r97532;
        double r97534 = 18.0;
        double r97535 = x;
        double r97536 = z;
        double r97537 = y;
        double r97538 = r97536 * r97537;
        double r97539 = r97535 * r97538;
        double r97540 = r97534 * r97539;
        double r97541 = r97528 * r97540;
        double r97542 = a;
        double r97543 = 4.0;
        double r97544 = r97542 * r97543;
        double r97545 = r97544 * r97528;
        double r97546 = r97541 - r97545;
        double r97547 = r97533 + r97546;
        double r97548 = r97535 * r97543;
        double r97549 = i;
        double r97550 = r97548 * r97549;
        double r97551 = r97547 - r97550;
        double r97552 = 27.0;
        double r97553 = k;
        double r97554 = j;
        double r97555 = r97553 * r97554;
        double r97556 = r97552 * r97555;
        double r97557 = r97551 - r97556;
        double r97558 = 1.78677717120485e-06;
        bool r97559 = r97528 <= r97558;
        double r97560 = r97534 * r97537;
        double r97561 = r97536 * r97528;
        double r97562 = r97560 * r97561;
        double r97563 = r97535 * r97562;
        double r97564 = r97563 - r97545;
        double r97565 = r97533 + r97564;
        double r97566 = r97565 - r97550;
        double r97567 = r97554 * r97552;
        double r97568 = r97567 * r97553;
        double r97569 = r97566 - r97568;
        double r97570 = r97535 * r97534;
        double r97571 = r97570 * r97537;
        double r97572 = r97571 * r97536;
        double r97573 = r97572 * r97528;
        double r97574 = r97573 - r97545;
        double r97575 = r97574 + r97533;
        double r97576 = r97575 - r97550;
        double r97577 = cbrt(r97556);
        double r97578 = r97577 * r97577;
        double r97579 = r97578 * r97577;
        double r97580 = r97576 - r97579;
        double r97581 = r97559 ? r97569 : r97580;
        double r97582 = r97530 ? r97557 : r97581;
        return r97582;
}

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 3 regimes

2. if t < -1.8869685775798424e+40

   1. Initial program 1.6

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

   3. Applied associate-*l* 1.7

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

   5. Applied *-un-lft-identity 1.7

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

   6. Applied associate-*l* 1.7

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

   7. Simplified 1.6

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

   9. Applied pow1 1.6

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

   10. Applied pow1 1.6

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

   11. Applied pow1 1.6

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

   12. Applied pow1 1.6

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

   13. Applied pow1 1.6

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

   14. Applied pow-prod-down 1.6

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

   15. Applied pow-prod-down 1.6

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

   16. Applied pow-prod-down 1.6

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

   17. Applied pow-prod-down 1.6

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

   18. Simplified 1.6

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

   if -1.8869685775798424e+40 < t < 1.78677717120485e-06

   1. Initial program 7.7

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

   3. Applied pow1 7.7

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

   4. Applied pow1 7.7

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

   5. Applied pow1 7.7

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

   6. Applied pow1 7.7

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

   7. Applied pow1 7.7

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

   8. Applied pow-prod-down 7.7

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

   9. Applied pow-prod-down 7.7

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

   10. Applied pow-prod-down 7.7

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

   11. Applied pow-prod-down 7.7

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

   12. Simplified 1.9

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

   if 1.78677717120485e-06 < t

   1. Initial program 1.9

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

   3. Applied associate-*l* 1.9

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

   5. Applied *-un-lft-identity 1.9

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

   6. Applied associate-*l* 1.9

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

   7. Simplified 1.8

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

   9. Applied add-cube-cbrt 2.0

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -18869685775798424479823402359192282988540:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(k \cdot j\right)\\ \mathbf{elif}\;t \le 1.786777171204849913027233772322421856416 \cdot 10^{-6}:\\ \;\;\;\;\left(\left(b \cdot c + \left(x \cdot \left(\left(18 \cdot y\right) \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(\sqrt[3]{27 \cdot \left(k \cdot j\right)} \cdot \sqrt[3]{27 \cdot \left(k \cdot j\right)}\right) \cdot \sqrt[3]{27 \cdot \left(k \cdot j\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(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)))