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

Average Error: 5.8 → 5.1

Time: 23.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -4255057376079.22509765625:\\ \;\;\;\;\left(\left(y \cdot x\right) \cdot \left(z \cdot 18\right) - 4 \cdot a\right) \cdot t - \left(\left(k \cdot j\right) \cdot 27 + \left(\left(x \cdot i\right) \cdot 4 - b \cdot c\right)\right)\\ \mathbf{elif}\;t \le -2.88427376324024654932368842967755587761 \cdot 10^{-181}:\\ \;\;\;\;\left(\left(t \cdot x\right) \cdot \left(18 \cdot \left(z \cdot y\right)\right) - \left(4 \cdot t\right) \cdot a\right) - \left(\left(k \cdot j\right) \cdot 27 + \left(\left(x \cdot i\right) \cdot 4 - b \cdot c\right)\right)\\ \mathbf{elif}\;t \le -5.563520560047814269291659806298884197516 \cdot 10^{-270} \lor \neg \left(t \le 3.066741772996178315911197664139485865687 \cdot 10^{-209}\right):\\ \;\;\;\;\left(\left(y \cdot x\right) \cdot \left(z \cdot 18\right) - 4 \cdot a\right) \cdot t - \left(\left(k \cdot j\right) \cdot 27 + \left(\left(x \cdot i\right) \cdot 4 - b \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\left(k \cdot j\right) \cdot 27 + \left(\left(x \cdot i\right) \cdot 4 - b \cdot c\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r119471 = x;
        double r119472 = 18.0;
        double r119473 = r119471 * r119472;
        double r119474 = y;
        double r119475 = r119473 * r119474;
        double r119476 = z;
        double r119477 = r119475 * r119476;
        double r119478 = t;
        double r119479 = r119477 * r119478;
        double r119480 = a;
        double r119481 = 4.0;
        double r119482 = r119480 * r119481;
        double r119483 = r119482 * r119478;
        double r119484 = r119479 - r119483;
        double r119485 = b;
        double r119486 = c;
        double r119487 = r119485 * r119486;
        double r119488 = r119484 + r119487;
        double r119489 = r119471 * r119481;
        double r119490 = i;
        double r119491 = r119489 * r119490;
        double r119492 = r119488 - r119491;
        double r119493 = j;
        double r119494 = 27.0;
        double r119495 = r119493 * r119494;
        double r119496 = k;
        double r119497 = r119495 * r119496;
        double r119498 = r119492 - r119497;
        return r119498;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r119499 = t;
        double r119500 = -4255057376079.225;
        bool r119501 = r119499 <= r119500;
        double r119502 = y;
        double r119503 = x;
        double r119504 = r119502 * r119503;
        double r119505 = z;
        double r119506 = 18.0;
        double r119507 = r119505 * r119506;
        double r119508 = r119504 * r119507;
        double r119509 = 4.0;
        double r119510 = a;
        double r119511 = r119509 * r119510;
        double r119512 = r119508 - r119511;
        double r119513 = r119512 * r119499;
        double r119514 = k;
        double r119515 = j;
        double r119516 = r119514 * r119515;
        double r119517 = 27.0;
        double r119518 = r119516 * r119517;
        double r119519 = i;
        double r119520 = r119503 * r119519;
        double r119521 = r119520 * r119509;
        double r119522 = b;
        double r119523 = c;
        double r119524 = r119522 * r119523;
        double r119525 = r119521 - r119524;
        double r119526 = r119518 + r119525;
        double r119527 = r119513 - r119526;
        double r119528 = -2.8842737632402465e-181;
        bool r119529 = r119499 <= r119528;
        double r119530 = r119499 * r119503;
        double r119531 = r119505 * r119502;
        double r119532 = r119506 * r119531;
        double r119533 = r119530 * r119532;
        double r119534 = r119509 * r119499;
        double r119535 = r119534 * r119510;
        double r119536 = r119533 - r119535;
        double r119537 = r119536 - r119526;
        double r119538 = -5.563520560047814e-270;
        bool r119539 = r119499 <= r119538;
        double r119540 = 3.0667417729961783e-209;
        bool r119541 = r119499 <= r119540;
        double r119542 = !r119541;
        bool r119543 = r119539 || r119542;
        double r119544 = -r119526;
        double r119545 = r119543 ? r119527 : r119544;
        double r119546 = r119529 ? r119537 : r119545;
        double r119547 = r119501 ? r119527 : r119546;
        return r119547;
}

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 < -4255057376079.225 or -2.8842737632402465e-181 < t < -5.563520560047814e-270 or 3.0667417729961783e-209 < t

   1. Initial program 4.8

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

   2. Simplified 4.6

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

   4. Applied associate-*r* 4.5

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

   6. Applied pow1 4.5

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

   7. Applied pow1 4.5

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

   8. Applied pow-prod-down 4.5

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

   9. Applied pow1 4.5

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

   10. Applied pow-prod-down 4.5

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

   11. Simplified 4.4

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

   13. Applied pow1 4.4

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

   14. Applied pow-prod-down 4.4

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

   15. Simplified 4.7

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

   if -4255057376079.225 < t < -2.8842737632402465e-181

   1. Initial program 5.5

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

   2. Simplified 5.5

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

   4. Applied associate-*r* 5.4

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

   6. Applied pow1 5.4

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

   7. Applied pow1 5.4

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

   8. Applied pow-prod-down 5.4

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

   9. Applied pow1 5.4

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

   10. Applied pow-prod-down 5.4

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

   11. Simplified 5.3

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

   12. Taylor expanded around inf 6.1

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

   13. Simplified 5.3

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

   14. Taylor expanded around inf 6.1

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

   15. Simplified 4.3

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

   if -5.563520560047814e-270 < t < 3.0667417729961783e-209

   1. Initial program 10.8

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

   2. Simplified 10.7

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

   4. Applied associate-*r* 10.6

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

   5. Taylor expanded around 0 8.0

      \[\leadsto \color{blue}{0} - \left(\left(j \cdot k\right) \cdot 27 + \left(\left(x \cdot i\right) \cdot 4 - b \cdot c\right)\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4255057376079.22509765625:\\ \;\;\;\;\left(\left(y \cdot x\right) \cdot \left(z \cdot 18\right) - 4 \cdot a\right) \cdot t - \left(\left(k \cdot j\right) \cdot 27 + \left(\left(x \cdot i\right) \cdot 4 - b \cdot c\right)\right)\\ \mathbf{elif}\;t \le -2.88427376324024654932368842967755587761 \cdot 10^{-181}:\\ \;\;\;\;\left(\left(t \cdot x\right) \cdot \left(18 \cdot \left(z \cdot y\right)\right) - \left(4 \cdot t\right) \cdot a\right) - \left(\left(k \cdot j\right) \cdot 27 + \left(\left(x \cdot i\right) \cdot 4 - b \cdot c\right)\right)\\ \mathbf{elif}\;t \le -5.563520560047814269291659806298884197516 \cdot 10^{-270} \lor \neg \left(t \le 3.066741772996178315911197664139485865687 \cdot 10^{-209}\right):\\ \;\;\;\;\left(\left(y \cdot x\right) \cdot \left(z \cdot 18\right) - 4 \cdot a\right) \cdot t - \left(\left(k \cdot j\right) \cdot 27 + \left(\left(x \cdot i\right) \cdot 4 - b \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\left(k \cdot j\right) \cdot 27 + \left(\left(x \cdot i\right) \cdot 4 - b \cdot c\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))