Average Error: 5.6 → 4.5
Time: 16.3s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.388095490955714183817090599266464810179 \cdot 10^{-146} \lor \neg \left(t \le 1.04728534392942456179731135613637724381 \cdot 10^{-109}\right):\\ \;\;\;\;\mathsf{fma}\left(t, x \cdot \left(\left(18 \cdot y\right) \cdot z\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]
\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}\;t \le -3.388095490955714183817090599266464810179 \cdot 10^{-146} \lor \neg \left(t \le 1.04728534392942456179731135613637724381 \cdot 10^{-109}\right):\\
\;\;\;\;\mathsf{fma}\left(t, x \cdot \left(\left(18 \cdot y\right) \cdot z\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\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 r182171 = x;
        double r182172 = 18.0;
        double r182173 = r182171 * r182172;
        double r182174 = y;
        double r182175 = r182173 * r182174;
        double r182176 = z;
        double r182177 = r182175 * r182176;
        double r182178 = t;
        double r182179 = r182177 * r182178;
        double r182180 = a;
        double r182181 = 4.0;
        double r182182 = r182180 * r182181;
        double r182183 = r182182 * r182178;
        double r182184 = r182179 - r182183;
        double r182185 = b;
        double r182186 = c;
        double r182187 = r182185 * r182186;
        double r182188 = r182184 + r182187;
        double r182189 = r182171 * r182181;
        double r182190 = i;
        double r182191 = r182189 * r182190;
        double r182192 = r182188 - r182191;
        double r182193 = j;
        double r182194 = 27.0;
        double r182195 = r182193 * r182194;
        double r182196 = k;
        double r182197 = r182195 * r182196;
        double r182198 = r182192 - r182197;
        return r182198;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r182199 = t;
        double r182200 = -3.388095490955714e-146;
        bool r182201 = r182199 <= r182200;
        double r182202 = 1.0472853439294246e-109;
        bool r182203 = r182199 <= r182202;
        double r182204 = !r182203;
        bool r182205 = r182201 || r182204;
        double r182206 = x;
        double r182207 = 18.0;
        double r182208 = y;
        double r182209 = r182207 * r182208;
        double r182210 = z;
        double r182211 = r182209 * r182210;
        double r182212 = r182206 * r182211;
        double r182213 = a;
        double r182214 = 4.0;
        double r182215 = r182213 * r182214;
        double r182216 = r182212 - r182215;
        double r182217 = b;
        double r182218 = c;
        double r182219 = r182217 * r182218;
        double r182220 = i;
        double r182221 = r182214 * r182220;
        double r182222 = j;
        double r182223 = 27.0;
        double r182224 = r182222 * r182223;
        double r182225 = k;
        double r182226 = r182224 * r182225;
        double r182227 = fma(r182206, r182221, r182226);
        double r182228 = r182219 - r182227;
        double r182229 = fma(r182199, r182216, r182228);
        double r182230 = 0.0;
        double r182231 = r182230 - r182215;
        double r182232 = r182223 * r182225;
        double r182233 = r182222 * r182232;
        double r182234 = fma(r182206, r182221, r182233);
        double r182235 = r182219 - r182234;
        double r182236 = fma(r182199, r182231, r182235);
        double r182237 = r182205 ? r182229 : r182236;
        return r182237;
}

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 < -3.388095490955714e-146 or 1.0472853439294246e-109 < t

    1. Initial program 3.5

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

    2. Simplified3.5

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

    4. Applied associate-*l*3.6

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

    6. Applied associate-*l*3.7

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

    if -3.388095490955714e-146 < t < 1.0472853439294246e-109

    1. Initial program 9.0

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

    2. Simplified9.0

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

    4. Applied associate-*l*9.0

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

    5. Taylor expanded around 0 5.6

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{0} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.388095490955714183817090599266464810179 \cdot 10^{-146} \lor \neg \left(t \le 1.04728534392942456179731135613637724381 \cdot 10^{-109}\right):\\ \;\;\;\;\mathsf{fma}\left(t, x \cdot \left(\left(18 \cdot y\right) \cdot z\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 +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)))