Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1

Average Error: 5.5 → 0.5

Time: 1.0m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k = -\infty:\\ \;\;\;\;\mathsf{fma}\left(b, c, 18.0 \cdot \left(y \cdot \left(x \cdot \left(t \cdot z\right)\right)\right) - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot 27.0\right) \cdot j\right)\right)\\ \mathbf{elif}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k \le 2.761847535126534 \cdot 10^{+307}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, 18.0 \cdot \left(y \cdot \left(x \cdot \left(t \cdot z\right)\right)\right) - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot 27.0\right) \cdot j\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r4191296 = x;
        double r4191297 = 18.0;
        double r4191298 = r4191296 * r4191297;
        double r4191299 = y;
        double r4191300 = r4191298 * r4191299;
        double r4191301 = z;
        double r4191302 = r4191300 * r4191301;
        double r4191303 = t;
        double r4191304 = r4191302 * r4191303;
        double r4191305 = a;
        double r4191306 = 4.0;
        double r4191307 = r4191305 * r4191306;
        double r4191308 = r4191307 * r4191303;
        double r4191309 = r4191304 - r4191308;
        double r4191310 = b;
        double r4191311 = c;
        double r4191312 = r4191310 * r4191311;
        double r4191313 = r4191309 + r4191312;
        double r4191314 = r4191296 * r4191306;
        double r4191315 = i;
        double r4191316 = r4191314 * r4191315;
        double r4191317 = r4191313 - r4191316;
        double r4191318 = j;
        double r4191319 = 27.0;
        double r4191320 = r4191318 * r4191319;
        double r4191321 = k;
        double r4191322 = r4191320 * r4191321;
        double r4191323 = r4191317 - r4191322;
        return r4191323;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r4191324 = t;
        double r4191325 = x;
        double r4191326 = 18.0;
        double r4191327 = r4191325 * r4191326;
        double r4191328 = y;
        double r4191329 = r4191327 * r4191328;
        double r4191330 = z;
        double r4191331 = r4191329 * r4191330;
        double r4191332 = r4191324 * r4191331;
        double r4191333 = a;
        double r4191334 = 4.0;
        double r4191335 = r4191333 * r4191334;
        double r4191336 = r4191335 * r4191324;
        double r4191337 = r4191332 - r4191336;
        double r4191338 = c;
        double r4191339 = b;
        double r4191340 = r4191338 * r4191339;
        double r4191341 = r4191337 + r4191340;
        double r4191342 = r4191325 * r4191334;
        double r4191343 = i;
        double r4191344 = r4191342 * r4191343;
        double r4191345 = r4191341 - r4191344;
        double r4191346 = 27.0;
        double r4191347 = j;
        double r4191348 = r4191346 * r4191347;
        double r4191349 = k;
        double r4191350 = r4191348 * r4191349;
        double r4191351 = r4191345 - r4191350;
        double r4191352 = -inf.0;
        bool r4191353 = r4191351 <= r4191352;
        double r4191354 = r4191324 * r4191330;
        double r4191355 = r4191325 * r4191354;
        double r4191356 = r4191328 * r4191355;
        double r4191357 = r4191326 * r4191356;
        double r4191358 = r4191325 * r4191343;
        double r4191359 = fma(r4191324, r4191333, r4191358);
        double r4191360 = r4191349 * r4191346;
        double r4191361 = r4191360 * r4191347;
        double r4191362 = fma(r4191334, r4191359, r4191361);
        double r4191363 = r4191357 - r4191362;
        double r4191364 = fma(r4191339, r4191338, r4191363);
        double r4191365 = 2.761847535126534e+307;
        bool r4191366 = r4191351 <= r4191365;
        double r4191367 = r4191366 ? r4191351 : r4191364;
        double r4191368 = r4191353 ? r4191364 : r4191367;
        return r4191368;
}

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 2.761847535126534e+307 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k))

   1. Initial program 62.7

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

   2. Simplified 13.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, c, z \cdot \left(\left(t \cdot x\right) \cdot \left(y \cdot 18.0\right)\right) - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), 27.0 \cdot \left(k \cdot j\right)\right)\right)}\]
   3. Using strategy rm

   4. Applied associate-*r* 5.9

      \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{\left(z \cdot \left(t \cdot x\right)\right) \cdot \left(y \cdot 18.0\right)} - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), 27.0 \cdot \left(k \cdot j\right)\right)\right)\]
   5. Using strategy rm

   6. Applied associate-*r* 3.4

      \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{\left(\left(z \cdot t\right) \cdot x\right)} \cdot \left(y \cdot 18.0\right) - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), 27.0 \cdot \left(k \cdot j\right)\right)\right)\]
   7. Using strategy rm

   8. Applied associate-*r* 3.3

      \[\leadsto \mathsf{fma}\left(b, c, \color{blue}{\left(\left(\left(z \cdot t\right) \cdot x\right) \cdot y\right) \cdot 18.0} - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), 27.0 \cdot \left(k \cdot j\right)\right)\right)\]
   9. Using strategy rm

   10. Applied associate-*r* 3.3

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

   if -inf.0 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)) < 2.761847535126534e+307

   1. Initial program 0.3

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k = -\infty:\\ \;\;\;\;\mathsf{fma}\left(b, c, 18.0 \cdot \left(y \cdot \left(x \cdot \left(t \cdot z\right)\right)\right) - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot 27.0\right) \cdot j\right)\right)\\ \mathbf{elif}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k \le 2.761847535126534 \cdot 10^{+307}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, 18.0 \cdot \left(y \cdot \left(x \cdot \left(t \cdot z\right)\right)\right) - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot 27.0\right) \cdot j\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +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)))