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

Average Error: 5.6 → 1.1

Time: 13.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i = -\infty \lor \neg \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 2.13292152496892302 \cdot 10^{291}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot \left(z \cdot 18\right), x, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(k \cdot j\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r766415 = x;
        double r766416 = 18.0;
        double r766417 = r766415 * r766416;
        double r766418 = y;
        double r766419 = r766417 * r766418;
        double r766420 = z;
        double r766421 = r766419 * r766420;
        double r766422 = t;
        double r766423 = r766421 * r766422;
        double r766424 = a;
        double r766425 = 4.0;
        double r766426 = r766424 * r766425;
        double r766427 = r766426 * r766422;
        double r766428 = r766423 - r766427;
        double r766429 = b;
        double r766430 = c;
        double r766431 = r766429 * r766430;
        double r766432 = r766428 + r766431;
        double r766433 = r766415 * r766425;
        double r766434 = i;
        double r766435 = r766433 * r766434;
        double r766436 = r766432 - r766435;
        double r766437 = j;
        double r766438 = 27.0;
        double r766439 = r766437 * r766438;
        double r766440 = k;
        double r766441 = r766439 * r766440;
        double r766442 = r766436 - r766441;
        return r766442;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r766443 = x;
        double r766444 = 18.0;
        double r766445 = r766443 * r766444;
        double r766446 = y;
        double r766447 = r766445 * r766446;
        double r766448 = z;
        double r766449 = r766447 * r766448;
        double r766450 = t;
        double r766451 = r766449 * r766450;
        double r766452 = a;
        double r766453 = 4.0;
        double r766454 = r766452 * r766453;
        double r766455 = r766454 * r766450;
        double r766456 = r766451 - r766455;
        double r766457 = b;
        double r766458 = c;
        double r766459 = r766457 * r766458;
        double r766460 = r766456 + r766459;
        double r766461 = r766443 * r766453;
        double r766462 = i;
        double r766463 = r766461 * r766462;
        double r766464 = r766460 - r766463;
        double r766465 = -inf.0;
        bool r766466 = r766464 <= r766465;
        double r766467 = 2.132921524968923e+291;
        bool r766468 = r766464 <= r766467;
        double r766469 = !r766468;
        bool r766470 = r766466 || r766469;
        double r766471 = r766450 * r766446;
        double r766472 = r766448 * r766444;
        double r766473 = r766471 * r766472;
        double r766474 = r766443 * r766462;
        double r766475 = fma(r766450, r766452, r766474);
        double r766476 = j;
        double r766477 = 27.0;
        double r766478 = r766476 * r766477;
        double r766479 = k;
        double r766480 = r766478 * r766479;
        double r766481 = fma(r766453, r766475, r766480);
        double r766482 = r766459 - r766481;
        double r766483 = fma(r766473, r766443, r766482);
        double r766484 = sqrt(r766477);
        double r766485 = r766479 * r766476;
        double r766486 = r766484 * r766485;
        double r766487 = r766484 * r766486;
        double r766488 = r766464 - r766487;
        double r766489 = r766470 ? r766483 : r766488;
        return r766489;
}

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	1.6
Herbie	1.1

\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.680279438052224:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < -inf.0 or 2.132921524968923e+291 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 50.3

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

   2. Simplified 7.8

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

   if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 2.132921524968923e+291

   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. Taylor expanded around 0 0.2

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

   4. Applied add-sqr-sqrt 0.2

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

   5. Applied associate-*l* 0.3

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \color{blue}{\sqrt{27} \cdot \left(\sqrt{27} \cdot \left(k \cdot j\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i = -\infty \lor \neg \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i \le 2.13292152496892302 \cdot 10^{291}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot y\right) \cdot \left(z \cdot 18\right), x, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(k \cdot j\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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 165.68027943805222) (+ (- (* (* 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)))