Average Error: 5.8 → 1.0
Time: 37.2s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le -1.591976256934889034187455180953636735584 \cdot 10^{302} \lor \neg \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 2.618961756211518564383950228017988163092 \cdot 10^{294}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(z \cdot t\right) \cdot y\right) \cdot x, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]
\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 r98349 = x;
        double r98350 = 18.0;
        double r98351 = r98349 * r98350;
        double r98352 = y;
        double r98353 = r98351 * r98352;
        double r98354 = z;
        double r98355 = r98353 * r98354;
        double r98356 = t;
        double r98357 = r98355 * r98356;
        double r98358 = a;
        double r98359 = 4.0;
        double r98360 = r98358 * r98359;
        double r98361 = r98360 * r98356;
        double r98362 = r98357 - r98361;
        double r98363 = b;
        double r98364 = c;
        double r98365 = r98363 * r98364;
        double r98366 = r98362 + r98365;
        double r98367 = r98349 * r98359;
        double r98368 = i;
        double r98369 = r98367 * r98368;
        double r98370 = r98366 - r98369;
        double r98371 = j;
        double r98372 = 27.0;
        double r98373 = r98371 * r98372;
        double r98374 = k;
        double r98375 = r98373 * r98374;
        double r98376 = r98370 - r98375;
        return r98376;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r98377 = x;
        double r98378 = 18.0;
        double r98379 = r98377 * r98378;
        double r98380 = y;
        double r98381 = r98379 * r98380;
        double r98382 = z;
        double r98383 = r98381 * r98382;
        double r98384 = t;
        double r98385 = r98383 * r98384;
        double r98386 = a;
        double r98387 = 4.0;
        double r98388 = r98386 * r98387;
        double r98389 = r98388 * r98384;
        double r98390 = r98385 - r98389;
        double r98391 = b;
        double r98392 = c;
        double r98393 = r98391 * r98392;
        double r98394 = r98390 + r98393;
        double r98395 = r98377 * r98387;
        double r98396 = i;
        double r98397 = r98395 * r98396;
        double r98398 = r98394 - r98397;
        double r98399 = -1.591976256934889e+302;
        bool r98400 = r98398 <= r98399;
        double r98401 = 2.6189617562115186e+294;
        bool r98402 = r98398 <= r98401;
        double r98403 = !r98402;
        bool r98404 = r98400 || r98403;
        double r98405 = r98382 * r98384;
        double r98406 = r98405 * r98380;
        double r98407 = r98406 * r98377;
        double r98408 = r98377 * r98396;
        double r98409 = fma(r98384, r98386, r98408);
        double r98410 = j;
        double r98411 = 27.0;
        double r98412 = r98410 * r98411;
        double r98413 = k;
        double r98414 = r98412 * r98413;
        double r98415 = fma(r98387, r98409, r98414);
        double r98416 = -r98415;
        double r98417 = fma(r98392, r98391, r98416);
        double r98418 = fma(r98407, r98378, r98417);
        double r98419 = r98398 - r98414;
        double r98420 = r98404 ? r98418 : r98419;
        return r98420;
}

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)))