Average Error: 5.9 → 1.5
Time: 28.5s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -18589615164.687671661376953125:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(\left(\left(18 \cdot x\right) \cdot t\right) \cdot y\right) \cdot z - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), 27 \cdot \left(j \cdot k\right)\right)\right)\\ \mathbf{elif}\;z \le 2.168478244128475081119441153156288006138 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(t \cdot \left(\left(z \cdot y\right) \cdot x\right)\right) \cdot 18 - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), k \cdot \left(27 \cdot j\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, z \cdot \left(18 \cdot \left(t \cdot \left(y \cdot x\right)\right)\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), 27 \cdot \left(j \cdot k\right)\right)\right)\\ \end{array}\]
\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}\;z \le -18589615164.687671661376953125:\\
\;\;\;\;\mathsf{fma}\left(b, c, \left(\left(\left(18 \cdot x\right) \cdot t\right) \cdot y\right) \cdot z - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), 27 \cdot \left(j \cdot k\right)\right)\right)\\

\mathbf{elif}\;z \le 2.168478244128475081119441153156288006138 \cdot 10^{-38}:\\
\;\;\;\;\mathsf{fma}\left(b, c, \left(t \cdot \left(\left(z \cdot y\right) \cdot x\right)\right) \cdot 18 - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), k \cdot \left(27 \cdot j\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(b, c, z \cdot \left(18 \cdot \left(t \cdot \left(y \cdot x\right)\right)\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), 27 \cdot \left(j \cdot k\right)\right)\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 r4843388 = x;
        double r4843389 = 18.0;
        double r4843390 = r4843388 * r4843389;
        double r4843391 = y;
        double r4843392 = r4843390 * r4843391;
        double r4843393 = z;
        double r4843394 = r4843392 * r4843393;
        double r4843395 = t;
        double r4843396 = r4843394 * r4843395;
        double r4843397 = a;
        double r4843398 = 4.0;
        double r4843399 = r4843397 * r4843398;
        double r4843400 = r4843399 * r4843395;
        double r4843401 = r4843396 - r4843400;
        double r4843402 = b;
        double r4843403 = c;
        double r4843404 = r4843402 * r4843403;
        double r4843405 = r4843401 + r4843404;
        double r4843406 = r4843388 * r4843398;
        double r4843407 = i;
        double r4843408 = r4843406 * r4843407;
        double r4843409 = r4843405 - r4843408;
        double r4843410 = j;
        double r4843411 = 27.0;
        double r4843412 = r4843410 * r4843411;
        double r4843413 = k;
        double r4843414 = r4843412 * r4843413;
        double r4843415 = r4843409 - r4843414;
        return r4843415;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r4843416 = z;
        double r4843417 = -18589615164.68767;
        bool r4843418 = r4843416 <= r4843417;
        double r4843419 = b;
        double r4843420 = c;
        double r4843421 = 18.0;
        double r4843422 = x;
        double r4843423 = r4843421 * r4843422;
        double r4843424 = t;
        double r4843425 = r4843423 * r4843424;
        double r4843426 = y;
        double r4843427 = r4843425 * r4843426;
        double r4843428 = r4843427 * r4843416;
        double r4843429 = 4.0;
        double r4843430 = a;
        double r4843431 = i;
        double r4843432 = r4843431 * r4843422;
        double r4843433 = fma(r4843424, r4843430, r4843432);
        double r4843434 = 27.0;
        double r4843435 = j;
        double r4843436 = k;
        double r4843437 = r4843435 * r4843436;
        double r4843438 = r4843434 * r4843437;
        double r4843439 = fma(r4843429, r4843433, r4843438);
        double r4843440 = r4843428 - r4843439;
        double r4843441 = fma(r4843419, r4843420, r4843440);
        double r4843442 = 2.168478244128475e-38;
        bool r4843443 = r4843416 <= r4843442;
        double r4843444 = r4843416 * r4843426;
        double r4843445 = r4843444 * r4843422;
        double r4843446 = r4843424 * r4843445;
        double r4843447 = r4843446 * r4843421;
        double r4843448 = r4843434 * r4843435;
        double r4843449 = r4843436 * r4843448;
        double r4843450 = fma(r4843429, r4843433, r4843449);
        double r4843451 = r4843447 - r4843450;
        double r4843452 = fma(r4843419, r4843420, r4843451);
        double r4843453 = r4843426 * r4843422;
        double r4843454 = r4843424 * r4843453;
        double r4843455 = r4843421 * r4843454;
        double r4843456 = r4843416 * r4843455;
        double r4843457 = r4843456 - r4843439;
        double r4843458 = fma(r4843419, r4843420, r4843457);
        double r4843459 = r4843443 ? r4843452 : r4843458;
        double r4843460 = r4843418 ? r4843441 : r4843459;
        return r4843460;
}

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 3 regimes

  2. if z < -18589615164.68767

    1. Initial program 7.2

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

    2. Simplified7.2

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

    4. Applied associate-*r*1.5

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

    6. Applied associate-*r*1.8

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

    8. Applied associate-*l*1.8

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

    if -18589615164.68767 < z < 2.168478244128475e-38

    1. Initial program 5.2

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

    2. Simplified5.2

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

    3. Taylor expanded around inf 1.4

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

    if 2.168478244128475e-38 < z

    1. Initial program 6.2

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

    2. Simplified6.2

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

    4. Applied associate-*r*1.6

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

    6. Applied associate-*r*2.0

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

    8. Applied associate-*l*1.9

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

    9. Taylor expanded around inf 1.6

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -18589615164.687671661376953125:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(\left(\left(18 \cdot x\right) \cdot t\right) \cdot y\right) \cdot z - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), 27 \cdot \left(j \cdot k\right)\right)\right)\\ \mathbf{elif}\;z \le 2.168478244128475081119441153156288006138 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(t \cdot \left(\left(z \cdot y\right) \cdot x\right)\right) \cdot 18 - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), k \cdot \left(27 \cdot j\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, z \cdot \left(18 \cdot \left(t \cdot \left(y \cdot x\right)\right)\right) - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), 27 \cdot \left(j \cdot k\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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)))