Average Error: 5.5 → 1.8
Time: 1.0m
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.59691870065664 \cdot 10^{+26}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(18.0 \cdot x\right) \cdot \left(\left(t \cdot z\right) \cdot y\right) - t \cdot \left(a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;x \le 5.303827427874542 \cdot 10^{-20}:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(\left(\left(18.0 \cdot x\right) \cdot y\right) \cdot z\right) - t \cdot \left(a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - \left(k \cdot 27.0\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(18.0 \cdot x\right) \cdot \left(\left(t \cdot z\right) \cdot y\right) - t \cdot \left(a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \end{array}\]
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r16038625 = x;
        double r16038626 = 18.0;
        double r16038627 = r16038625 * r16038626;
        double r16038628 = y;
        double r16038629 = r16038627 * r16038628;
        double r16038630 = z;
        double r16038631 = r16038629 * r16038630;
        double r16038632 = t;
        double r16038633 = r16038631 * r16038632;
        double r16038634 = a;
        double r16038635 = 4.0;
        double r16038636 = r16038634 * r16038635;
        double r16038637 = r16038636 * r16038632;
        double r16038638 = r16038633 - r16038637;
        double r16038639 = b;
        double r16038640 = c;
        double r16038641 = r16038639 * r16038640;
        double r16038642 = r16038638 + r16038641;
        double r16038643 = r16038625 * r16038635;
        double r16038644 = i;
        double r16038645 = r16038643 * r16038644;
        double r16038646 = r16038642 - r16038645;
        double r16038647 = j;
        double r16038648 = 27.0;
        double r16038649 = r16038647 * r16038648;
        double r16038650 = k;
        double r16038651 = r16038649 * r16038650;
        double r16038652 = r16038646 - r16038651;
        return r16038652;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r16038653 = x;
        double r16038654 = -1.59691870065664e+26;
        bool r16038655 = r16038653 <= r16038654;
        double r16038656 = b;
        double r16038657 = c;
        double r16038658 = r16038656 * r16038657;
        double r16038659 = 18.0;
        double r16038660 = r16038659 * r16038653;
        double r16038661 = t;
        double r16038662 = z;
        double r16038663 = r16038661 * r16038662;
        double r16038664 = y;
        double r16038665 = r16038663 * r16038664;
        double r16038666 = r16038660 * r16038665;
        double r16038667 = a;
        double r16038668 = 4.0;
        double r16038669 = r16038667 * r16038668;
        double r16038670 = r16038661 * r16038669;
        double r16038671 = r16038666 - r16038670;
        double r16038672 = r16038658 + r16038671;
        double r16038673 = r16038668 * r16038653;
        double r16038674 = i;
        double r16038675 = r16038673 * r16038674;
        double r16038676 = r16038672 - r16038675;
        double r16038677 = k;
        double r16038678 = j;
        double r16038679 = 27.0;
        double r16038680 = r16038678 * r16038679;
        double r16038681 = r16038677 * r16038680;
        double r16038682 = r16038676 - r16038681;
        double r16038683 = 5.303827427874542e-20;
        bool r16038684 = r16038653 <= r16038683;
        double r16038685 = r16038660 * r16038664;
        double r16038686 = r16038685 * r16038662;
        double r16038687 = r16038661 * r16038686;
        double r16038688 = r16038687 - r16038670;
        double r16038689 = r16038658 + r16038688;
        double r16038690 = r16038689 - r16038675;
        double r16038691 = r16038677 * r16038679;
        double r16038692 = r16038691 * r16038678;
        double r16038693 = r16038690 - r16038692;
        double r16038694 = r16038684 ? r16038693 : r16038682;
        double r16038695 = r16038655 ? r16038682 : r16038694;
        return r16038695;
}


\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.59691870065664 \cdot 10^{+26}:\\
\;\;\;\;\left(\left(b \cdot c + \left(\left(18.0 \cdot x\right) \cdot \left(\left(t \cdot z\right) \cdot y\right) - t \cdot \left(a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\

\mathbf{elif}\;x \le 5.303827427874542 \cdot 10^{-20}:\\
\;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(\left(\left(18.0 \cdot x\right) \cdot y\right) \cdot z\right) - t \cdot \left(a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - \left(k \cdot 27.0\right) \cdot j\\

\mathbf{else}:\\
\;\;\;\;\left(\left(b \cdot c + \left(\left(18.0 \cdot x\right) \cdot \left(\left(t \cdot z\right) \cdot y\right) - t \cdot \left(a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\

\end{array}

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

Derivation

  1. Split input into 2 regimes

  2. if x < -1.59691870065664e+26 or 5.303827427874542e-20 < x

    1. Initial program 12.6

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    2. Using strategy rm

    3. Applied associate-*l*9.7

      \[\leadsto \left(\left(\left(\color{blue}{\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot \left(z \cdot t\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    4. Using strategy rm

    5. Applied associate-*l*1.8

      \[\leadsto \left(\left(\left(\color{blue}{\left(x \cdot 18.0\right) \cdot \left(y \cdot \left(z \cdot t\right)\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    if -1.59691870065664e+26 < x < 5.303827427874542e-20

    1. Initial program 1.8

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    2. Using strategy rm

    3. Applied associate-*l*1.8

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \color{blue}{j \cdot \left(27.0 \cdot k\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.59691870065664 \cdot 10^{+26}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(18.0 \cdot x\right) \cdot \left(\left(t \cdot z\right) \cdot y\right) - t \cdot \left(a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;x \le 5.303827427874542 \cdot 10^{-20}:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(\left(\left(18.0 \cdot x\right) \cdot y\right) \cdot z\right) - t \cdot \left(a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - \left(k \cdot 27.0\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(18.0 \cdot x\right) \cdot \left(\left(t \cdot z\right) \cdot y\right) - t \cdot \left(a \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))