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

Average Error: 5.6 → 3.1

Time: 27.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot \left(z \cdot x\right), 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 3.790565547516040000125064508344150956225 \cdot 10^{245}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \sqrt{27} \cdot \left(\left(\sqrt{27} \cdot k\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(t \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(k \cdot j\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r565370 = x;
        double r565371 = 18.0;
        double r565372 = r565370 * r565371;
        double r565373 = y;
        double r565374 = r565372 * r565373;
        double r565375 = z;
        double r565376 = r565374 * r565375;
        double r565377 = t;
        double r565378 = r565376 * r565377;
        double r565379 = a;
        double r565380 = 4.0;
        double r565381 = r565379 * r565380;
        double r565382 = r565381 * r565377;
        double r565383 = r565378 - r565382;
        double r565384 = b;
        double r565385 = c;
        double r565386 = r565384 * r565385;
        double r565387 = r565383 + r565386;
        double r565388 = r565370 * r565380;
        double r565389 = i;
        double r565390 = r565388 * r565389;
        double r565391 = r565387 - r565390;
        double r565392 = j;
        double r565393 = 27.0;
        double r565394 = r565392 * r565393;
        double r565395 = k;
        double r565396 = r565394 * r565395;
        double r565397 = r565391 - r565396;
        return r565397;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r565398 = x;
        double r565399 = 18.0;
        double r565400 = r565398 * r565399;
        double r565401 = y;
        double r565402 = r565400 * r565401;
        double r565403 = z;
        double r565404 = r565402 * r565403;
        double r565405 = t;
        double r565406 = r565404 * r565405;
        double r565407 = a;
        double r565408 = 4.0;
        double r565409 = r565407 * r565408;
        double r565410 = r565409 * r565405;
        double r565411 = r565406 - r565410;
        double r565412 = b;
        double r565413 = c;
        double r565414 = r565412 * r565413;
        double r565415 = r565411 + r565414;
        double r565416 = r565398 * r565408;
        double r565417 = i;
        double r565418 = r565416 * r565417;
        double r565419 = r565415 - r565418;
        double r565420 = -inf.0;
        bool r565421 = r565419 <= r565420;
        double r565422 = r565405 * r565401;
        double r565423 = r565403 * r565398;
        double r565424 = r565422 * r565423;
        double r565425 = r565398 * r565417;
        double r565426 = fma(r565405, r565407, r565425);
        double r565427 = j;
        double r565428 = 27.0;
        double r565429 = r565427 * r565428;
        double r565430 = k;
        double r565431 = r565429 * r565430;
        double r565432 = fma(r565408, r565426, r565431);
        double r565433 = -r565432;
        double r565434 = fma(r565413, r565412, r565433);
        double r565435 = fma(r565424, r565399, r565434);
        double r565436 = 3.79056554751604e+245;
        bool r565437 = r565419 <= r565436;
        double r565438 = sqrt(r565428);
        double r565439 = r565438 * r565430;
        double r565440 = r565439 * r565427;
        double r565441 = r565438 * r565440;
        double r565442 = r565419 - r565441;
        double r565443 = r565405 * r565403;
        double r565444 = r565402 * r565443;
        double r565445 = r565444 - r565410;
        double r565446 = r565445 + r565414;
        double r565447 = r565446 - r565418;
        double r565448 = r565430 * r565427;
        double r565449 = r565428 * r565448;
        double r565450 = r565447 - r565449;
        double r565451 = r565437 ? r565442 : r565450;
        double r565452 = r565421 ? r565435 : r565451;
        return r565452;
}

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

Target

Original	5.6
Target	1.5
Herbie	3.1

\[\begin{array}{l} \mathbf{if}\;t \lt -1.62108153975413982700795070153457058168 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.6802794380522243500308832153677940369:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Simplified 14.1

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

   if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 3.79056554751604e+245

   1. Initial program 0.4

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

   3. Applied associate-*l* 0.4

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

   4. Taylor expanded around 0 0.3

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

   6. Applied add-sqr-sqrt 0.3

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

   7. Applied associate-*l* 0.3

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

   9. Applied associate-*r* 0.4

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

   if 3.79056554751604e+245 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 20.8

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

   3. Applied associate-*l* 20.8

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

   4. Taylor expanded around 0 20.8

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

   6. Applied associate-*l* 17.4

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

   7. Simplified 17.4

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot \left(z \cdot x\right), 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\right)\\ \mathbf{elif}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 3.790565547516040000125064508344150956225 \cdot 10^{245}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \sqrt{27} \cdot \left(\left(\sqrt{27} \cdot k\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(t \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(k \cdot j\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"
  :precision binary64

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b))) (if (< t 165.680279438052224) (+ (- (* (* 18 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4)) (- (* c b) (* 27 (* k j)))) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))