Average Error: 5.4 → 3.7
Time: 15.3s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.15846637414394352 \cdot 10^{131} \lor \neg \left(z \le 3.14022389051985691 \cdot 10^{43}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(z \cdot y\right)\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]
\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 -4.15846637414394352 \cdot 10^{131} \lor \neg \left(z \le 3.14022389051985691 \cdot 10^{43}\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}:\\
\;\;\;\;\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(z \cdot y\right)\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\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 r135297 = x;
        double r135298 = 18.0;
        double r135299 = r135297 * r135298;
        double r135300 = y;
        double r135301 = r135299 * r135300;
        double r135302 = z;
        double r135303 = r135301 * r135302;
        double r135304 = t;
        double r135305 = r135303 * r135304;
        double r135306 = a;
        double r135307 = 4.0;
        double r135308 = r135306 * r135307;
        double r135309 = r135308 * r135304;
        double r135310 = r135305 - r135309;
        double r135311 = b;
        double r135312 = c;
        double r135313 = r135311 * r135312;
        double r135314 = r135310 + r135313;
        double r135315 = r135297 * r135307;
        double r135316 = i;
        double r135317 = r135315 * r135316;
        double r135318 = r135314 - r135317;
        double r135319 = j;
        double r135320 = 27.0;
        double r135321 = r135319 * r135320;
        double r135322 = k;
        double r135323 = r135321 * r135322;
        double r135324 = r135318 - r135323;
        return r135324;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r135325 = z;
        double r135326 = -4.1584663741439435e+131;
        bool r135327 = r135325 <= r135326;
        double r135328 = 3.140223890519857e+43;
        bool r135329 = r135325 <= r135328;
        double r135330 = !r135329;
        bool r135331 = r135327 || r135330;
        double r135332 = t;
        double r135333 = x;
        double r135334 = 18.0;
        double r135335 = r135333 * r135334;
        double r135336 = y;
        double r135337 = r135335 * r135336;
        double r135338 = r135337 * r135325;
        double r135339 = a;
        double r135340 = 4.0;
        double r135341 = r135339 * r135340;
        double r135342 = r135338 - r135341;
        double r135343 = b;
        double r135344 = c;
        double r135345 = r135343 * r135344;
        double r135346 = i;
        double r135347 = r135340 * r135346;
        double r135348 = j;
        double r135349 = 27.0;
        double r135350 = k;
        double r135351 = r135349 * r135350;
        double r135352 = r135348 * r135351;
        double r135353 = fma(r135333, r135347, r135352);
        double r135354 = r135345 - r135353;
        double r135355 = fma(r135332, r135342, r135354);
        double r135356 = r135325 * r135336;
        double r135357 = r135333 * r135356;
        double r135358 = r135334 * r135357;
        double r135359 = r135358 - r135341;
        double r135360 = r135348 * r135349;
        double r135361 = r135360 * r135350;
        double r135362 = fma(r135333, r135347, r135361);
        double r135363 = r135345 - r135362;
        double r135364 = fma(r135332, r135359, r135363);
        double r135365 = r135331 ? r135355 : r135364;
        return r135365;
}

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 z < -4.1584663741439435e+131 or 3.140223890519857e+43 < z

    1. Initial program 7.9

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

    2. Simplified7.9

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

    4. Applied associate-*l*8.0

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

    if -4.1584663741439435e+131 < z < 3.140223890519857e+43

    1. Initial program 4.4

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

    2. Simplified4.4

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

    3. Taylor expanded around inf 1.9

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

  4. Final simplification3.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.15846637414394352 \cdot 10^{131} \lor \neg \left(z \le 3.14022389051985691 \cdot 10^{43}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(z \cdot y\right)\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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)))