Average Error: 5.5 → 0.8
Time: 40.2s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k = -\infty \lor \neg \left(\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \le 2.291150336217132422785785995759037448958 \cdot 10^{305}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(t \cdot y\right) \cdot z\right) \cdot x, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(27 \cdot k\right) \cdot j\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]
\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(\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 = -\infty \lor \neg \left(\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 \le 2.291150336217132422785785995759037448958 \cdot 10^{305}\right):\\
\;\;\;\;\mathsf{fma}\left(\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(27 \cdot k\right) \cdot j\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 r103260 = x;
        double r103261 = 18.0;
        double r103262 = r103260 * r103261;
        double r103263 = y;
        double r103264 = r103262 * r103263;
        double r103265 = z;
        double r103266 = r103264 * r103265;
        double r103267 = t;
        double r103268 = r103266 * r103267;
        double r103269 = a;
        double r103270 = 4.0;
        double r103271 = r103269 * r103270;
        double r103272 = r103271 * r103267;
        double r103273 = r103268 - r103272;
        double r103274 = b;
        double r103275 = c;
        double r103276 = r103274 * r103275;
        double r103277 = r103273 + r103276;
        double r103278 = r103260 * r103270;
        double r103279 = i;
        double r103280 = r103278 * r103279;
        double r103281 = r103277 - r103280;
        double r103282 = j;
        double r103283 = 27.0;
        double r103284 = r103282 * r103283;
        double r103285 = k;
        double r103286 = r103284 * r103285;
        double r103287 = r103281 - r103286;
        return r103287;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r103288 = x;
        double r103289 = 18.0;
        double r103290 = r103288 * r103289;
        double r103291 = y;
        double r103292 = r103290 * r103291;
        double r103293 = z;
        double r103294 = r103292 * r103293;
        double r103295 = t;
        double r103296 = r103294 * r103295;
        double r103297 = a;
        double r103298 = 4.0;
        double r103299 = r103297 * r103298;
        double r103300 = r103299 * r103295;
        double r103301 = r103296 - r103300;
        double r103302 = b;
        double r103303 = c;
        double r103304 = r103302 * r103303;
        double r103305 = r103301 + r103304;
        double r103306 = r103288 * r103298;
        double r103307 = i;
        double r103308 = r103306 * r103307;
        double r103309 = r103305 - r103308;
        double r103310 = j;
        double r103311 = 27.0;
        double r103312 = r103310 * r103311;
        double r103313 = k;
        double r103314 = r103312 * r103313;
        double r103315 = r103309 - r103314;
        double r103316 = -inf.0;
        bool r103317 = r103315 <= r103316;
        double r103318 = 2.2911503362171324e+305;
        bool r103319 = r103315 <= r103318;
        double r103320 = !r103319;
        bool r103321 = r103317 || r103320;
        double r103322 = r103295 * r103291;
        double r103323 = r103322 * r103293;
        double r103324 = r103323 * r103288;
        double r103325 = r103288 * r103307;
        double r103326 = fma(r103295, r103297, r103325);
        double r103327 = r103311 * r103313;
        double r103328 = r103327 * r103310;
        double r103329 = fma(r103298, r103326, r103328);
        double r103330 = -r103329;
        double r103331 = fma(r103303, r103302, r103330);
        double r103332 = fma(r103324, r103289, r103331);
        double r103333 = r103321 ? r103332 : r103315;
        return r103333;
}

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)) (* (* j 27.0) k)) < -inf.0 or 2.2911503362171324e+305 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k))

    1. Initial program 59.9

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

    2. Simplified14.9

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

    4. Applied associate-*r*7.5

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

    5. Taylor expanded around 0 6.5

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

    7. Applied associate-*r*6.6

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

    if -inf.0 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)) < 2.2911503362171324e+305

    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 simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k = -\infty \lor \neg \left(\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \le 2.291150336217132422785785995759037448958 \cdot 10^{305}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(t \cdot y\right) \cdot z\right) \cdot x, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(27 \cdot k\right) \cdot j\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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)))