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

Average Error: 5.7 → 1.9

Time: 19.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.09206231284064533924647487158246602957 \cdot 10^{-45} \lor \neg \left(t \le 7.712463495126096893546357295759619116123 \cdot 10^{-24}\right):\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot \left(y \cdot 18\right)\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot \left(\left(\left(t \cdot x\right) \cdot z\right) \cdot y\right) + a \cdot \left(\left(-4\right) \cdot t\right)\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r848545 = x;
        double r848546 = 18.0;
        double r848547 = r848545 * r848546;
        double r848548 = y;
        double r848549 = r848547 * r848548;
        double r848550 = z;
        double r848551 = r848549 * r848550;
        double r848552 = t;
        double r848553 = r848551 * r848552;
        double r848554 = a;
        double r848555 = 4.0;
        double r848556 = r848554 * r848555;
        double r848557 = r848556 * r848552;
        double r848558 = r848553 - r848557;
        double r848559 = b;
        double r848560 = c;
        double r848561 = r848559 * r848560;
        double r848562 = r848558 + r848561;
        double r848563 = r848545 * r848555;
        double r848564 = i;
        double r848565 = r848563 * r848564;
        double r848566 = r848562 - r848565;
        double r848567 = j;
        double r848568 = 27.0;
        double r848569 = r848567 * r848568;
        double r848570 = k;
        double r848571 = r848569 * r848570;
        double r848572 = r848566 - r848571;
        return r848572;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r848573 = t;
        double r848574 = -2.0920623128406453e-45;
        bool r848575 = r848573 <= r848574;
        double r848576 = 7.712463495126097e-24;
        bool r848577 = r848573 <= r848576;
        double r848578 = !r848577;
        bool r848579 = r848575 || r848578;
        double r848580 = x;
        double r848581 = y;
        double r848582 = 18.0;
        double r848583 = r848581 * r848582;
        double r848584 = r848580 * r848583;
        double r848585 = z;
        double r848586 = r848584 * r848585;
        double r848587 = a;
        double r848588 = 4.0;
        double r848589 = r848587 * r848588;
        double r848590 = r848586 - r848589;
        double r848591 = r848573 * r848590;
        double r848592 = b;
        double r848593 = c;
        double r848594 = r848592 * r848593;
        double r848595 = r848591 + r848594;
        double r848596 = r848580 * r848588;
        double r848597 = i;
        double r848598 = r848596 * r848597;
        double r848599 = j;
        double r848600 = 27.0;
        double r848601 = r848599 * r848600;
        double r848602 = k;
        double r848603 = r848601 * r848602;
        double r848604 = r848598 + r848603;
        double r848605 = r848595 - r848604;
        double r848606 = r848573 * r848580;
        double r848607 = r848606 * r848585;
        double r848608 = r848607 * r848581;
        double r848609 = r848582 * r848608;
        double r848610 = -r848588;
        double r848611 = r848610 * r848573;
        double r848612 = r848587 * r848611;
        double r848613 = r848609 + r848612;
        double r848614 = r848613 + r848594;
        double r848615 = r848614 - r848604;
        double r848616 = r848579 ? r848605 : r848615;
        return r848616;
}

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	5.7
Target	1.6
Herbie	1.9

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

Derivation

1. Split input into 2 regimes

2. if t < -2.0920623128406453e-45 or 7.712463495126097e-24 < t

   1. Initial program 2.0

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

   2. Simplified 2.0

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

   4. Applied associate-*l* 2.0

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

   5. Simplified 2.0

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

   if -2.0920623128406453e-45 < t < 7.712463495126097e-24

   1. Initial program 8.6

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

   2. Simplified 8.6

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

   4. Applied sub-neg 8.6

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

   5. Applied distribute-lft-in 8.6

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

   6. Simplified 9.0

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

   7. Simplified 9.0

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

   9. Applied associate-*r* 5.9

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

   11. Applied associate-*r* 1.9

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

   13. Applied distribute-rgt-neg-in 1.9

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

   14. Applied associate-*l* 1.8

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.09206231284064533924647487158246602957 \cdot 10^{-45} \lor \neg \left(t \le 7.712463495126096893546357295759619116123 \cdot 10^{-24}\right):\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot \left(y \cdot 18\right)\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot \left(\left(\left(t \cdot x\right) \cdot z\right) \cdot y\right) + a \cdot \left(\left(-4\right) \cdot t\right)\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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)))