Average Error: 5.3 → 1.8
Time: 12.2s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.4288223988477578 \cdot 10^{-37} \lor \neg \left(t \le 2.64647452676486563 \cdot 10^{-55}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]
\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}\;t \le -6.4288223988477578 \cdot 10^{-37} \lor \neg \left(t \le 2.64647452676486563 \cdot 10^{-55}\right):\\
\;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\left(\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\\

\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 r112422 = x;
        double r112423 = 18.0;
        double r112424 = r112422 * r112423;
        double r112425 = y;
        double r112426 = r112424 * r112425;
        double r112427 = z;
        double r112428 = r112426 * r112427;
        double r112429 = t;
        double r112430 = r112428 * r112429;
        double r112431 = a;
        double r112432 = 4.0;
        double r112433 = r112431 * r112432;
        double r112434 = r112433 * r112429;
        double r112435 = r112430 - r112434;
        double r112436 = b;
        double r112437 = c;
        double r112438 = r112436 * r112437;
        double r112439 = r112435 + r112438;
        double r112440 = r112422 * r112432;
        double r112441 = i;
        double r112442 = r112440 * r112441;
        double r112443 = r112439 - r112442;
        double r112444 = j;
        double r112445 = 27.0;
        double r112446 = r112444 * r112445;
        double r112447 = k;
        double r112448 = r112446 * r112447;
        double r112449 = r112443 - r112448;
        return r112449;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r112450 = t;
        double r112451 = -6.428822398847758e-37;
        bool r112452 = r112450 <= r112451;
        double r112453 = 2.6464745267648656e-55;
        bool r112454 = r112450 <= r112453;
        double r112455 = !r112454;
        bool r112456 = r112452 || r112455;
        double r112457 = x;
        double r112458 = 18.0;
        double r112459 = r112457 * r112458;
        double r112460 = y;
        double r112461 = r112459 * r112460;
        double r112462 = z;
        double r112463 = r112461 * r112462;
        double r112464 = a;
        double r112465 = 4.0;
        double r112466 = r112464 * r112465;
        double r112467 = r112463 - r112466;
        double r112468 = b;
        double r112469 = c;
        double r112470 = r112468 * r112469;
        double r112471 = i;
        double r112472 = r112465 * r112471;
        double r112473 = j;
        double r112474 = 27.0;
        double r112475 = k;
        double r112476 = r112474 * r112475;
        double r112477 = r112473 * r112476;
        double r112478 = fma(r112457, r112472, r112477);
        double r112479 = r112470 - r112478;
        double r112480 = fma(r112450, r112467, r112479);
        double r112481 = r112462 * r112450;
        double r112482 = r112460 * r112481;
        double r112483 = r112459 * r112482;
        double r112484 = r112466 * r112450;
        double r112485 = r112483 - r112484;
        double r112486 = r112485 + r112470;
        double r112487 = r112457 * r112465;
        double r112488 = r112487 * r112471;
        double r112489 = r112486 - r112488;
        double r112490 = r112473 * r112474;
        double r112491 = r112490 * r112475;
        double r112492 = r112489 - r112491;
        double r112493 = r112456 ? r112480 : r112492;
        return r112493;
}

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 t < -6.428822398847758e-37 or 2.6464745267648656e-55 < t

    1. Initial program 2.1

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

    2. Simplified2.1

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

    4. Applied associate-*l*2.2

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

    if -6.428822398847758e-37 < t < 2.6464745267648656e-55

    1. Initial program 8.1

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

    3. Applied associate-*l*4.3

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

    5. Applied associate-*l*1.5

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.4288223988477578 \cdot 10^{-37} \lor \neg \left(t \le 2.64647452676486563 \cdot 10^{-55}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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)))