Average Error: 5.6 → 1.8
Time: 31.5s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i = -\infty \lor \neg \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 1.191425524707375816183030560675713891259 \cdot 10^{284}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot \left(z \cdot x\right), 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b, c, t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right)\right) - 4 \cdot \left(i \cdot x\right)\right) + \left(x \cdot 4\right) \cdot \left(\left(-i\right) + i\right)\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]
\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 = -\infty \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 1.191425524707375816183030560675713891259 \cdot 10^{284}\right):\\
\;\;\;\;\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)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\mathsf{fma}\left(b, c, t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right)\right) - 4 \cdot \left(i \cdot x\right)\right) + \left(x \cdot 4\right) \cdot \left(\left(-i\right) + i\right)\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 r125398 = x;
        double r125399 = 18.0;
        double r125400 = r125398 * r125399;
        double r125401 = y;
        double r125402 = r125400 * r125401;
        double r125403 = z;
        double r125404 = r125402 * r125403;
        double r125405 = t;
        double r125406 = r125404 * r125405;
        double r125407 = a;
        double r125408 = 4.0;
        double r125409 = r125407 * r125408;
        double r125410 = r125409 * r125405;
        double r125411 = r125406 - r125410;
        double r125412 = b;
        double r125413 = c;
        double r125414 = r125412 * r125413;
        double r125415 = r125411 + r125414;
        double r125416 = r125398 * r125408;
        double r125417 = i;
        double r125418 = r125416 * r125417;
        double r125419 = r125415 - r125418;
        double r125420 = j;
        double r125421 = 27.0;
        double r125422 = r125420 * r125421;
        double r125423 = k;
        double r125424 = r125422 * r125423;
        double r125425 = r125419 - r125424;
        return r125425;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r125426 = x;
        double r125427 = 18.0;
        double r125428 = r125426 * r125427;
        double r125429 = y;
        double r125430 = r125428 * r125429;
        double r125431 = z;
        double r125432 = r125430 * r125431;
        double r125433 = t;
        double r125434 = r125432 * r125433;
        double r125435 = a;
        double r125436 = 4.0;
        double r125437 = r125435 * r125436;
        double r125438 = r125437 * r125433;
        double r125439 = r125434 - r125438;
        double r125440 = b;
        double r125441 = c;
        double r125442 = r125440 * r125441;
        double r125443 = r125439 + r125442;
        double r125444 = r125426 * r125436;
        double r125445 = i;
        double r125446 = r125444 * r125445;
        double r125447 = r125443 - r125446;
        double r125448 = -inf.0;
        bool r125449 = r125447 <= r125448;
        double r125450 = 1.1914255247073758e+284;
        bool r125451 = r125447 <= r125450;
        double r125452 = !r125451;
        bool r125453 = r125449 || r125452;
        double r125454 = r125433 * r125429;
        double r125455 = r125431 * r125426;
        double r125456 = r125454 * r125455;
        double r125457 = r125426 * r125445;
        double r125458 = fma(r125433, r125435, r125457);
        double r125459 = j;
        double r125460 = 27.0;
        double r125461 = r125459 * r125460;
        double r125462 = k;
        double r125463 = r125461 * r125462;
        double r125464 = fma(r125436, r125458, r125463);
        double r125465 = -r125464;
        double r125466 = fma(r125441, r125440, r125465);
        double r125467 = fma(r125456, r125427, r125466);
        double r125468 = r125432 - r125437;
        double r125469 = r125433 * r125468;
        double r125470 = fma(r125440, r125441, r125469);
        double r125471 = r125445 * r125426;
        double r125472 = r125436 * r125471;
        double r125473 = r125470 - r125472;
        double r125474 = -r125445;
        double r125475 = r125474 + r125445;
        double r125476 = r125444 * r125475;
        double r125477 = r125473 + r125476;
        double r125478 = r125477 - r125463;
        double r125479 = r125453 ? r125467 : r125478;
        return r125479;
}

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)) < -inf.0 or 1.1914255247073758e+284 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

    1. Initial program 46.8

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

    2. Simplified13.0

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

    if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 1.1914255247073758e+284

    1. Initial program 0.3

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

    3. Applied associate-*l*3.2

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

    5. Applied add-sqr-sqrt34.2

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

    6. Applied prod-diff34.2

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

    7. Simplified0.3

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

    8. Simplified0.3

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i = -\infty \lor \neg \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 1.191425524707375816183030560675713891259 \cdot 10^{284}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot \left(z \cdot x\right), 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(b, c, t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right)\right) - 4 \cdot \left(i \cdot x\right)\right) + \left(x \cdot 4\right) \cdot \left(\left(-i\right) + i\right)\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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)))