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

Average Error: 5.6 → 4.7

Time: 21.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.35973034078404779 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(t, {\left(\left(z \cdot y\right) \cdot \left(x \cdot 18\right)\right)}^{1} - a \cdot 4, \mathsf{fma}\left(b, c, -\mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\right)\\ \mathbf{elif}\;t \le 6.2161625191812012 \cdot 10^{-206}:\\ \;\;\;\;\mathsf{fma}\left(t, {0}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r801521 = x;
        double r801522 = 18.0;
        double r801523 = r801521 * r801522;
        double r801524 = y;
        double r801525 = r801523 * r801524;
        double r801526 = z;
        double r801527 = r801525 * r801526;
        double r801528 = t;
        double r801529 = r801527 * r801528;
        double r801530 = a;
        double r801531 = 4.0;
        double r801532 = r801530 * r801531;
        double r801533 = r801532 * r801528;
        double r801534 = r801529 - r801533;
        double r801535 = b;
        double r801536 = c;
        double r801537 = r801535 * r801536;
        double r801538 = r801534 + r801537;
        double r801539 = r801521 * r801531;
        double r801540 = i;
        double r801541 = r801539 * r801540;
        double r801542 = r801538 - r801541;
        double r801543 = j;
        double r801544 = 27.0;
        double r801545 = r801543 * r801544;
        double r801546 = k;
        double r801547 = r801545 * r801546;
        double r801548 = r801542 - r801547;
        return r801548;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r801549 = t;
        double r801550 = -2.359730340784048e-109;
        bool r801551 = r801549 <= r801550;
        double r801552 = z;
        double r801553 = y;
        double r801554 = r801552 * r801553;
        double r801555 = x;
        double r801556 = 18.0;
        double r801557 = r801555 * r801556;
        double r801558 = r801554 * r801557;
        double r801559 = 1.0;
        double r801560 = pow(r801558, r801559);
        double r801561 = a;
        double r801562 = 4.0;
        double r801563 = r801561 * r801562;
        double r801564 = r801560 - r801563;
        double r801565 = b;
        double r801566 = c;
        double r801567 = i;
        double r801568 = r801562 * r801567;
        double r801569 = j;
        double r801570 = 27.0;
        double r801571 = k;
        double r801572 = r801570 * r801571;
        double r801573 = r801569 * r801572;
        double r801574 = fma(r801555, r801568, r801573);
        double r801575 = -r801574;
        double r801576 = fma(r801565, r801566, r801575);
        double r801577 = fma(r801549, r801564, r801576);
        double r801578 = 6.216162519181201e-206;
        bool r801579 = r801549 <= r801578;
        double r801580 = 0.0;
        double r801581 = pow(r801580, r801559);
        double r801582 = r801581 - r801563;
        double r801583 = r801565 * r801566;
        double r801584 = r801583 - r801574;
        double r801585 = fma(r801549, r801582, r801584);
        double r801586 = cbrt(r801553);
        double r801587 = r801586 * r801586;
        double r801588 = r801557 * r801587;
        double r801589 = r801588 * r801586;
        double r801590 = r801589 * r801552;
        double r801591 = r801590 - r801563;
        double r801592 = fma(r801549, r801591, r801584);
        double r801593 = r801579 ? r801585 : r801592;
        double r801594 = r801551 ? r801577 : r801593;
        return r801594;
}

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

Target

Original	5.6
Target	1.5
Herbie	4.7

\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.680279438052224:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if t < -2.359730340784048e-109

   1. Initial program 3.2

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

   2. Simplified 3.3

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

   4. Applied associate-*l* 3.4

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

   6. Applied add-cube-cbrt 3.5

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)}\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   7. Applied associate-*r* 3.5

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

   9. Applied pow1 3.5

      \[\leadsto \mathsf{fma}\left(t, \left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}\right) \cdot \color{blue}{{z}^{1}} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   10. Applied pow1 3.5

       \[\leadsto \mathsf{fma}\left(t, \left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{1}}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   11. Applied pow1 3.5

       \[\leadsto \mathsf{fma}\left(t, \left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{1}}\right)\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   12. Applied pow1 3.5

       \[\leadsto \mathsf{fma}\left(t, \left(\left(\left(x \cdot 18\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{1}} \cdot {\left(\sqrt[3]{y}\right)}^{1}\right)\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   13. Applied pow-prod-down 3.5

       \[\leadsto \mathsf{fma}\left(t, \left(\left(\left(x \cdot 18\right) \cdot \color{blue}{{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{1}}\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   14. Applied pow1 3.5

       \[\leadsto \mathsf{fma}\left(t, \left(\left(\left(x \cdot \color{blue}{{18}^{1}}\right) \cdot {\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{1}\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   15. Applied pow1 3.5

       \[\leadsto \mathsf{fma}\left(t, \left(\left(\left(\color{blue}{{x}^{1}} \cdot {18}^{1}\right) \cdot {\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{1}\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   16. Applied pow-prod-down 3.5

       \[\leadsto \mathsf{fma}\left(t, \left(\left(\color{blue}{{\left(x \cdot 18\right)}^{1}} \cdot {\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{1}\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   17. Applied pow-prod-down 3.5

       \[\leadsto \mathsf{fma}\left(t, \left(\color{blue}{{\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)}^{1}} \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   18. Applied pow-prod-down 3.5

       \[\leadsto \mathsf{fma}\left(t, \color{blue}{{\left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}\right)}^{1}} \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   19. Applied pow-prod-down 3.5

       \[\leadsto \mathsf{fma}\left(t, \color{blue}{{\left(\left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}\right) \cdot z\right)}^{1}} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   20. Simplified 3.4

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

   22. Applied fma-neg 3.4

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

   if -2.359730340784048e-109 < t < 6.216162519181201e-206

   1. Initial program 9.6

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

   2. Simplified 9.6

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

   4. Applied associate-*l* 9.7

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

   6. Applied add-cube-cbrt 9.7

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)}\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   7. Applied associate-*r* 9.7

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

   9. Applied pow1 9.7

      \[\leadsto \mathsf{fma}\left(t, \left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}\right) \cdot \color{blue}{{z}^{1}} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   10. Applied pow1 9.7

       \[\leadsto \mathsf{fma}\left(t, \left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{1}}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   11. Applied pow1 9.7

       \[\leadsto \mathsf{fma}\left(t, \left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{1}}\right)\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   12. Applied pow1 9.7

       \[\leadsto \mathsf{fma}\left(t, \left(\left(\left(x \cdot 18\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{1}} \cdot {\left(\sqrt[3]{y}\right)}^{1}\right)\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   13. Applied pow-prod-down 9.7

       \[\leadsto \mathsf{fma}\left(t, \left(\left(\left(x \cdot 18\right) \cdot \color{blue}{{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{1}}\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   14. Applied pow1 9.7

       \[\leadsto \mathsf{fma}\left(t, \left(\left(\left(x \cdot \color{blue}{{18}^{1}}\right) \cdot {\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{1}\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   15. Applied pow1 9.7

       \[\leadsto \mathsf{fma}\left(t, \left(\left(\left(\color{blue}{{x}^{1}} \cdot {18}^{1}\right) \cdot {\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{1}\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   16. Applied pow-prod-down 9.7

       \[\leadsto \mathsf{fma}\left(t, \left(\left(\color{blue}{{\left(x \cdot 18\right)}^{1}} \cdot {\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{1}\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   17. Applied pow-prod-down 9.7

       \[\leadsto \mathsf{fma}\left(t, \left(\color{blue}{{\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)}^{1}} \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   18. Applied pow-prod-down 9.7

       \[\leadsto \mathsf{fma}\left(t, \color{blue}{{\left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}\right)}^{1}} \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   19. Applied pow-prod-down 9.7

       \[\leadsto \mathsf{fma}\left(t, \color{blue}{{\left(\left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}\right) \cdot z\right)}^{1}} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   20. Simplified 10.2

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

   21. Taylor expanded around 0 6.2

       \[\leadsto \mathsf{fma}\left(t, {\color{blue}{0}}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   if 6.216162519181201e-206 < t

   1. Initial program 4.2

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

   2. Simplified 4.2

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

   4. Applied associate-*l* 4.3

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

   6. Applied add-cube-cbrt 4.4

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)}\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   7. Applied associate-*r* 4.4

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

4. Final simplification 4.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.35973034078404779 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(t, {\left(\left(z \cdot y\right) \cdot \left(x \cdot 18\right)\right)}^{1} - a \cdot 4, \mathsf{fma}\left(b, c, -\mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\right)\\ \mathbf{elif}\;t \le 6.2161625191812012 \cdot 10^{-206}:\\ \;\;\;\;\mathsf{fma}\left(t, {0}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(\left(x \cdot 18\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"
  :precision binary64

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4)) (- (* c b) (* 27 (* k j)))) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))