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

Average Error: 5.3 → 6.2

Time: 7.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le 7.0669981561857078 \cdot 10^{-218} \lor \neg \left(k \le 4.00595738363528446 \cdot 10^{-73}\right):\\ \;\;\;\;t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(0 - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r796563 = x;
        double r796564 = 18.0;
        double r796565 = r796563 * r796564;
        double r796566 = y;
        double r796567 = r796565 * r796566;
        double r796568 = z;
        double r796569 = r796567 * r796568;
        double r796570 = t;
        double r796571 = r796569 * r796570;
        double r796572 = a;
        double r796573 = 4.0;
        double r796574 = r796572 * r796573;
        double r796575 = r796574 * r796570;
        double r796576 = r796571 - r796575;
        double r796577 = b;
        double r796578 = c;
        double r796579 = r796577 * r796578;
        double r796580 = r796576 + r796579;
        double r796581 = r796563 * r796573;
        double r796582 = i;
        double r796583 = r796581 * r796582;
        double r796584 = r796580 - r796583;
        double r796585 = j;
        double r796586 = 27.0;
        double r796587 = r796585 * r796586;
        double r796588 = k;
        double r796589 = r796587 * r796588;
        double r796590 = r796584 - r796589;
        return r796590;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r796591 = k;
        double r796592 = 7.066998156185708e-218;
        bool r796593 = r796591 <= r796592;
        double r796594 = 4.0059573836352845e-73;
        bool r796595 = r796591 <= r796594;
        double r796596 = !r796595;
        bool r796597 = r796593 || r796596;
        double r796598 = t;
        double r796599 = x;
        double r796600 = 18.0;
        double r796601 = r796599 * r796600;
        double r796602 = y;
        double r796603 = r796601 * r796602;
        double r796604 = z;
        double r796605 = r796603 * r796604;
        double r796606 = a;
        double r796607 = 4.0;
        double r796608 = r796606 * r796607;
        double r796609 = r796605 - r796608;
        double r796610 = r796598 * r796609;
        double r796611 = b;
        double r796612 = c;
        double r796613 = r796611 * r796612;
        double r796614 = r796599 * r796607;
        double r796615 = i;
        double r796616 = r796614 * r796615;
        double r796617 = j;
        double r796618 = 27.0;
        double r796619 = r796618 * r796591;
        double r796620 = r796617 * r796619;
        double r796621 = r796616 + r796620;
        double r796622 = r796613 - r796621;
        double r796623 = r796610 + r796622;
        double r796624 = 0.0;
        double r796625 = r796624 - r796608;
        double r796626 = r796598 * r796625;
        double r796627 = r796617 * r796618;
        double r796628 = r796627 * r796591;
        double r796629 = r796616 + r796628;
        double r796630 = r796613 - r796629;
        double r796631 = r796626 + r796630;
        double r796632 = r796597 ? r796623 : r796631;
        return r796632;
}

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	5.3
Target	1.5
Herbie	6.2

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

Derivation

1. Split input into 2 regimes

2. if k < 7.066998156185708e-218 or 4.0059573836352845e-73 < k

   1. Initial program 5.4

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

   2. Simplified 5.4

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

   4. Applied associate-*l* 5.4

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

   if 7.066998156185708e-218 < k < 4.0059573836352845e-73

   1. Initial program 5.0

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

   2. Simplified 5.0

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

   3. Taylor expanded around 0 11.1

      \[\leadsto t \cdot \left(\color{blue}{0} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 7.0669981561857078 \cdot 10^{-218} \lor \neg \left(k \le 4.00595738363528446 \cdot 10^{-73}\right):\\ \;\;\;\;t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(0 - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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)))