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

Average Error: 5.6 → 4.7

Time: 22.0s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -3.1141079314986784 \cdot 10^{-44}:\\ \;\;\;\;\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(\sqrt[3]{\left(j \cdot 27\right) \cdot k} \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\right) \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\right)\\ \mathbf{elif}\;t \le 1.1312232233938015 \cdot 10^{-68}:\\ \;\;\;\;\left(t \cdot \left(-a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r866590 = x;
        double r866591 = 18.0;
        double r866592 = r866590 * r866591;
        double r866593 = y;
        double r866594 = r866592 * r866593;
        double r866595 = z;
        double r866596 = r866594 * r866595;
        double r866597 = t;
        double r866598 = r866596 * r866597;
        double r866599 = a;
        double r866600 = 4.0;
        double r866601 = r866599 * r866600;
        double r866602 = r866601 * r866597;
        double r866603 = r866598 - r866602;
        double r866604 = b;
        double r866605 = c;
        double r866606 = r866604 * r866605;
        double r866607 = r866603 + r866606;
        double r866608 = r866590 * r866600;
        double r866609 = i;
        double r866610 = r866608 * r866609;
        double r866611 = r866607 - r866610;
        double r866612 = j;
        double r866613 = 27.0;
        double r866614 = r866612 * r866613;
        double r866615 = k;
        double r866616 = r866614 * r866615;
        double r866617 = r866611 - r866616;
        return r866617;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r866618 = t;
        double r866619 = -3.1141079314986784e-44;
        bool r866620 = r866618 <= r866619;
        double r866621 = x;
        double r866622 = 18.0;
        double r866623 = r866621 * r866622;
        double r866624 = y;
        double r866625 = r866623 * r866624;
        double r866626 = z;
        double r866627 = r866625 * r866626;
        double r866628 = a;
        double r866629 = 4.0;
        double r866630 = r866628 * r866629;
        double r866631 = r866627 - r866630;
        double r866632 = r866618 * r866631;
        double r866633 = b;
        double r866634 = c;
        double r866635 = r866633 * r866634;
        double r866636 = r866632 + r866635;
        double r866637 = r866621 * r866629;
        double r866638 = i;
        double r866639 = r866637 * r866638;
        double r866640 = j;
        double r866641 = 27.0;
        double r866642 = r866640 * r866641;
        double r866643 = k;
        double r866644 = r866642 * r866643;
        double r866645 = cbrt(r866644);
        double r866646 = r866645 * r866645;
        double r866647 = r866646 * r866645;
        double r866648 = r866639 + r866647;
        double r866649 = r866636 - r866648;
        double r866650 = 1.1312232233938015e-68;
        bool r866651 = r866618 <= r866650;
        double r866652 = -r866630;
        double r866653 = r866618 * r866652;
        double r866654 = r866653 + r866635;
        double r866655 = r866639 + r866644;
        double r866656 = r866654 - r866655;
        double r866657 = r866622 * r866624;
        double r866658 = r866621 * r866657;
        double r866659 = r866658 * r866626;
        double r866660 = r866659 - r866630;
        double r866661 = r866618 * r866660;
        double r866662 = r866661 + r866635;
        double r866663 = r866641 * r866643;
        double r866664 = r866640 * r866663;
        double r866665 = r866639 + r866664;
        double r866666 = r866662 - r866665;
        double r866667 = r866651 ? r866656 : r866666;
        double r866668 = r866620 ? r866649 : r866667;
        return r866668;
}

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 < -3.1141079314986784e-44

   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 add-cube-cbrt 2.2

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

   if -3.1141079314986784e-44 < t < 1.1312232233938015e-68

   1. Initial program 8.7

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

   2. Simplified 8.7

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

   3. Taylor expanded around 0 6.8

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

   if 1.1312232233938015e-68 < t

   1. Initial program 2.4

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

   2. Simplified 2.4

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

   4. Applied associate-*l* 2.4

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

   6. Applied associate-*l* 2.4

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

4. Final simplification 4.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.1141079314986784 \cdot 10^{-44}:\\ \;\;\;\;\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(\sqrt[3]{\left(j \cdot 27\right) \cdot k} \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\right) \cdot \sqrt[3]{\left(j \cdot 27\right) \cdot k}\right)\\ \mathbf{elif}\;t \le 1.1312232233938015 \cdot 10^{-68}:\\ \;\;\;\;\left(t \cdot \left(-a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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)))