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

Average Error: 5.6 → 1.9

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -6877626944398136512086016:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(z \cdot \left(\left(y \cdot x\right) \cdot 18\right)\right) \cdot t - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(k \cdot j\right)\\ \mathbf{elif}\;t \le 56517438608387023208118747136:\\ \;\;\;\;\left(\left(\left(\left(y \cdot \left(t \cdot z\right)\right) \cdot \left(x \cdot 18\right) - t \cdot \left(a \cdot 4\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(z \cdot \left(\left(y \cdot x\right) \cdot 18\right)\right) \cdot t - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(k \cdot j\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r38518650 = x;
        double r38518651 = 18.0;
        double r38518652 = r38518650 * r38518651;
        double r38518653 = y;
        double r38518654 = r38518652 * r38518653;
        double r38518655 = z;
        double r38518656 = r38518654 * r38518655;
        double r38518657 = t;
        double r38518658 = r38518656 * r38518657;
        double r38518659 = a;
        double r38518660 = 4.0;
        double r38518661 = r38518659 * r38518660;
        double r38518662 = r38518661 * r38518657;
        double r38518663 = r38518658 - r38518662;
        double r38518664 = b;
        double r38518665 = c;
        double r38518666 = r38518664 * r38518665;
        double r38518667 = r38518663 + r38518666;
        double r38518668 = r38518650 * r38518660;
        double r38518669 = i;
        double r38518670 = r38518668 * r38518669;
        double r38518671 = r38518667 - r38518670;
        double r38518672 = j;
        double r38518673 = 27.0;
        double r38518674 = r38518672 * r38518673;
        double r38518675 = k;
        double r38518676 = r38518674 * r38518675;
        double r38518677 = r38518671 - r38518676;
        return r38518677;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r38518678 = t;
        double r38518679 = -6.877626944398137e+24;
        bool r38518680 = r38518678 <= r38518679;
        double r38518681 = b;
        double r38518682 = c;
        double r38518683 = r38518681 * r38518682;
        double r38518684 = z;
        double r38518685 = y;
        double r38518686 = x;
        double r38518687 = r38518685 * r38518686;
        double r38518688 = 18.0;
        double r38518689 = r38518687 * r38518688;
        double r38518690 = r38518684 * r38518689;
        double r38518691 = r38518690 * r38518678;
        double r38518692 = a;
        double r38518693 = 4.0;
        double r38518694 = r38518692 * r38518693;
        double r38518695 = r38518678 * r38518694;
        double r38518696 = r38518691 - r38518695;
        double r38518697 = r38518683 + r38518696;
        double r38518698 = r38518686 * r38518693;
        double r38518699 = i;
        double r38518700 = r38518698 * r38518699;
        double r38518701 = r38518697 - r38518700;
        double r38518702 = 27.0;
        double r38518703 = k;
        double r38518704 = j;
        double r38518705 = r38518703 * r38518704;
        double r38518706 = r38518702 * r38518705;
        double r38518707 = r38518701 - r38518706;
        double r38518708 = 5.651743860838702e+28;
        bool r38518709 = r38518678 <= r38518708;
        double r38518710 = r38518678 * r38518684;
        double r38518711 = r38518685 * r38518710;
        double r38518712 = r38518686 * r38518688;
        double r38518713 = r38518711 * r38518712;
        double r38518714 = r38518713 - r38518695;
        double r38518715 = r38518714 + r38518683;
        double r38518716 = r38518715 - r38518700;
        double r38518717 = r38518702 * r38518704;
        double r38518718 = r38518717 * r38518703;
        double r38518719 = r38518716 - r38518718;
        double r38518720 = r38518709 ? r38518719 : r38518707;
        double r38518721 = r38518680 ? r38518707 : r38518720;
        return r38518721;
}

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	1.7
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 < -6.877626944398137e+24 or 5.651743860838702e+28 < t

   1. Initial program 1.7

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

   2. Taylor expanded around 0 1.7

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

   3. Taylor expanded around 0 1.7

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

   if -6.877626944398137e+24 < t < 5.651743860838702e+28

   1. Initial program 7.5

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

   3. Applied associate-*l* 4.2

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

   5. Applied associate-*l* 2.0

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6877626944398136512086016:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(z \cdot \left(\left(y \cdot x\right) \cdot 18\right)\right) \cdot t - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(k \cdot j\right)\\ \mathbf{elif}\;t \le 56517438608387023208118747136:\\ \;\;\;\;\left(\left(\left(\left(y \cdot \left(t \cdot z\right)\right) \cdot \left(x \cdot 18\right) - t \cdot \left(a \cdot 4\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(z \cdot \left(\left(y \cdot x\right) \cdot 18\right)\right) \cdot t - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(k \cdot j\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18.0 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4.0)) (- (* c b) (* 27.0 (* k j)))) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))