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

Average Error: 5.1 → 4.4

Time: 29.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 5.612796806482376315037030586204465630066 \cdot 10^{68}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right)\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right)\right) \cdot t\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r508533 = x;
        double r508534 = 18.0;
        double r508535 = r508533 * r508534;
        double r508536 = y;
        double r508537 = r508535 * r508536;
        double r508538 = z;
        double r508539 = r508537 * r508538;
        double r508540 = t;
        double r508541 = r508539 * r508540;
        double r508542 = a;
        double r508543 = 4.0;
        double r508544 = r508542 * r508543;
        double r508545 = r508544 * r508540;
        double r508546 = r508541 - r508545;
        double r508547 = b;
        double r508548 = c;
        double r508549 = r508547 * r508548;
        double r508550 = r508546 + r508549;
        double r508551 = r508533 * r508543;
        double r508552 = i;
        double r508553 = r508551 * r508552;
        double r508554 = r508550 - r508553;
        double r508555 = j;
        double r508556 = 27.0;
        double r508557 = r508555 * r508556;
        double r508558 = k;
        double r508559 = r508557 * r508558;
        double r508560 = r508554 - r508559;
        return r508560;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r508561 = t;
        double r508562 = 5.612796806482376e+68;
        bool r508563 = r508561 <= r508562;
        double r508564 = c;
        double r508565 = b;
        double r508566 = x;
        double r508567 = 18.0;
        double r508568 = r508566 * r508567;
        double r508569 = y;
        double r508570 = r508568 * r508569;
        double r508571 = z;
        double r508572 = r508571 * r508561;
        double r508573 = r508570 * r508572;
        double r508574 = fma(r508564, r508565, r508573);
        double r508575 = 4.0;
        double r508576 = a;
        double r508577 = i;
        double r508578 = r508566 * r508577;
        double r508579 = fma(r508561, r508576, r508578);
        double r508580 = 27.0;
        double r508581 = k;
        double r508582 = j;
        double r508583 = r508581 * r508582;
        double r508584 = r508580 * r508583;
        double r508585 = fma(r508575, r508579, r508584);
        double r508586 = r508574 - r508585;
        double r508587 = r508569 * r508571;
        double r508588 = r508568 * r508587;
        double r508589 = r508588 * r508561;
        double r508590 = fma(r508564, r508565, r508589);
        double r508591 = r508590 - r508585;
        double r508592 = r508563 ? r508586 : r508591;
        return r508592;
}

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	5.1
Target	1.6
Herbie	4.4

\[\begin{array}{l} \mathbf{if}\;t \lt -1.62108153975413982700795070153457058168 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.6802794380522243500308832153677940369:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if t < 5.612796806482376e+68

   1. Initial program 5.6

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

   2. Simplified 5.6

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

   4. Applied associate-*l* 5.6

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

   6. Applied *-un-lft-identity 5.6

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

   7. Applied associate-*l* 5.6

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

   8. Simplified 5.5

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

   10. Applied associate-*l* 4.8

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

   if 5.612796806482376e+68 < t

   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

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

   4. Applied associate-*l* 1.3

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

   6. Applied *-un-lft-identity 1.3

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

   7. Applied associate-*l* 1.3

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

   8. Simplified 1.2

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

   10. Applied associate-*l* 1.6

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

4. Final simplification 4.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 5.612796806482376315037030586204465630066 \cdot 10^{68}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right)\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, b, \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right)\right) \cdot t\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), 27 \cdot \left(k \cdot j\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +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 165.680279438052224) (+ (- (* (* 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)))