Average Error: 5.2 → 4.9
Time: 37.1s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 2.446814143058209 \cdot 10^{-97}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(t \cdot z\right) \cdot \left(\left(x \cdot 18.0\right) \cdot y\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot 18.0\right) \cdot \left(y \cdot z\right) - a \cdot 4.0, t, b \cdot c - \mathsf{fma}\left(27.0, j \cdot k, \left(4.0 \cdot x\right) \cdot i\right)\right)\\ \end{array}\]
\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k
↓
\begin{array}{l}
\mathbf{if}\;t \le 2.446814143058209 \cdot 10^{-97}:\\
\;\;\;\;\left(\left(b \cdot c + \left(\left(t \cdot z\right) \cdot \left(\left(x \cdot 18.0\right) \cdot y\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\left(x \cdot 18.0\right) \cdot \left(y \cdot z\right) - a \cdot 4.0, t, b \cdot c - \mathsf{fma}\left(27.0, j \cdot k, \left(4.0 \cdot x\right) \cdot i\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 r4605482 = x;
        double r4605483 = 18.0;
        double r4605484 = r4605482 * r4605483;
        double r4605485 = y;
        double r4605486 = r4605484 * r4605485;
        double r4605487 = z;
        double r4605488 = r4605486 * r4605487;
        double r4605489 = t;
        double r4605490 = r4605488 * r4605489;
        double r4605491 = a;
        double r4605492 = 4.0;
        double r4605493 = r4605491 * r4605492;
        double r4605494 = r4605493 * r4605489;
        double r4605495 = r4605490 - r4605494;
        double r4605496 = b;
        double r4605497 = c;
        double r4605498 = r4605496 * r4605497;
        double r4605499 = r4605495 + r4605498;
        double r4605500 = r4605482 * r4605492;
        double r4605501 = i;
        double r4605502 = r4605500 * r4605501;
        double r4605503 = r4605499 - r4605502;
        double r4605504 = j;
        double r4605505 = 27.0;
        double r4605506 = r4605504 * r4605505;
        double r4605507 = k;
        double r4605508 = r4605506 * r4605507;
        double r4605509 = r4605503 - r4605508;
        return r4605509;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r4605510 = t;
        double r4605511 = 2.446814143058209e-97;
        bool r4605512 = r4605510 <= r4605511;
        double r4605513 = b;
        double r4605514 = c;
        double r4605515 = r4605513 * r4605514;
        double r4605516 = z;
        double r4605517 = r4605510 * r4605516;
        double r4605518 = x;
        double r4605519 = 18.0;
        double r4605520 = r4605518 * r4605519;
        double r4605521 = y;
        double r4605522 = r4605520 * r4605521;
        double r4605523 = r4605517 * r4605522;
        double r4605524 = a;
        double r4605525 = 4.0;
        double r4605526 = r4605524 * r4605525;
        double r4605527 = r4605526 * r4605510;
        double r4605528 = r4605523 - r4605527;
        double r4605529 = r4605515 + r4605528;
        double r4605530 = r4605525 * r4605518;
        double r4605531 = i;
        double r4605532 = r4605530 * r4605531;
        double r4605533 = r4605529 - r4605532;
        double r4605534 = k;
        double r4605535 = j;
        double r4605536 = 27.0;
        double r4605537 = r4605535 * r4605536;
        double r4605538 = r4605534 * r4605537;
        double r4605539 = r4605533 - r4605538;
        double r4605540 = r4605521 * r4605516;
        double r4605541 = r4605520 * r4605540;
        double r4605542 = r4605541 - r4605526;
        double r4605543 = r4605535 * r4605534;
        double r4605544 = fma(r4605536, r4605543, r4605532);
        double r4605545 = r4605515 - r4605544;
        double r4605546 = fma(r4605542, r4605510, r4605545);
        double r4605547 = r4605512 ? r4605539 : r4605546;
        return r4605547;
}

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 < 2.446814143058209e-97

    1. Initial program 6.5

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    2. Using strategy rm

    3. Applied associate-*l*5.7

      \[\leadsto \left(\left(\left(\color{blue}{\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot \left(z \cdot t\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    if 2.446814143058209e-97 < t

    1. Initial program 1.9

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    2. Simplified2.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot z\right) \cdot \left(x \cdot 18.0\right) - a \cdot 4.0, t, c \cdot b - \mathsf{fma}\left(27.0, k \cdot j, \left(x \cdot 4.0\right) \cdot i\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 2.446814143058209 \cdot 10^{-97}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(t \cdot z\right) \cdot \left(\left(x \cdot 18.0\right) \cdot y\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot 18.0\right) \cdot \left(y \cdot z\right) - a \cdot 4.0, t, b \cdot c - \mathsf{fma}\left(27.0, j \cdot k, \left(4.0 \cdot x\right) \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +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)))