Average Error: 5.8 → 1.0
Time: 40.6s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le -1.591976256934889034187455180953636735584 \cdot 10^{302} \lor \neg \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 2.618961756211518564383950228017988163092 \cdot 10^{294}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(z \cdot t\right) \cdot y\right) \cdot x, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]
\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}\;\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 \le -1.591976256934889034187455180953636735584 \cdot 10^{302} \lor \neg \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 \le 2.618961756211518564383950228017988163092 \cdot 10^{294}\right):\\
\;\;\;\;\mathsf{fma}\left(\left(\left(z \cdot t\right) \cdot y\right) \cdot x, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\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\\

\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 r100530 = x;
        double r100531 = 18.0;
        double r100532 = r100530 * r100531;
        double r100533 = y;
        double r100534 = r100532 * r100533;
        double r100535 = z;
        double r100536 = r100534 * r100535;
        double r100537 = t;
        double r100538 = r100536 * r100537;
        double r100539 = a;
        double r100540 = 4.0;
        double r100541 = r100539 * r100540;
        double r100542 = r100541 * r100537;
        double r100543 = r100538 - r100542;
        double r100544 = b;
        double r100545 = c;
        double r100546 = r100544 * r100545;
        double r100547 = r100543 + r100546;
        double r100548 = r100530 * r100540;
        double r100549 = i;
        double r100550 = r100548 * r100549;
        double r100551 = r100547 - r100550;
        double r100552 = j;
        double r100553 = 27.0;
        double r100554 = r100552 * r100553;
        double r100555 = k;
        double r100556 = r100554 * r100555;
        double r100557 = r100551 - r100556;
        return r100557;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r100558 = x;
        double r100559 = 18.0;
        double r100560 = r100558 * r100559;
        double r100561 = y;
        double r100562 = r100560 * r100561;
        double r100563 = z;
        double r100564 = r100562 * r100563;
        double r100565 = t;
        double r100566 = r100564 * r100565;
        double r100567 = a;
        double r100568 = 4.0;
        double r100569 = r100567 * r100568;
        double r100570 = r100569 * r100565;
        double r100571 = r100566 - r100570;
        double r100572 = b;
        double r100573 = c;
        double r100574 = r100572 * r100573;
        double r100575 = r100571 + r100574;
        double r100576 = r100558 * r100568;
        double r100577 = i;
        double r100578 = r100576 * r100577;
        double r100579 = r100575 - r100578;
        double r100580 = -1.591976256934889e+302;
        bool r100581 = r100579 <= r100580;
        double r100582 = 2.6189617562115186e+294;
        bool r100583 = r100579 <= r100582;
        double r100584 = !r100583;
        bool r100585 = r100581 || r100584;
        double r100586 = r100563 * r100565;
        double r100587 = r100586 * r100561;
        double r100588 = r100587 * r100558;
        double r100589 = r100558 * r100577;
        double r100590 = fma(r100565, r100567, r100589);
        double r100591 = j;
        double r100592 = 27.0;
        double r100593 = r100591 * r100592;
        double r100594 = k;
        double r100595 = r100593 * r100594;
        double r100596 = fma(r100568, r100590, r100595);
        double r100597 = -r100596;
        double r100598 = fma(r100573, r100572, r100597);
        double r100599 = fma(r100588, r100559, r100598);
        double r100600 = r100579 - r100595;
        double r100601 = r100585 ? r100599 : r100600;
        return r100601;
}

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 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < -1.591976256934889e+302 or 2.6189617562115186e+294 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

    1. Initial program 49.0

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

    2. Simplified13.5

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

    4. Applied associate-*r*7.4

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

    6. Applied associate-*l*16.5

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

    7. Simplified16.5

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

    9. Applied associate-*r*6.0

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

    10. Simplified6.0

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

    if -1.591976256934889e+302 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 2.6189617562115186e+294

    1. Initial program 0.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le -1.591976256934889034187455180953636735584 \cdot 10^{302} \lor \neg \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 2.618961756211518564383950228017988163092 \cdot 10^{294}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(z \cdot t\right) \cdot y\right) \cdot x, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  :precision binary64
  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))