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

Average Error: 5.2 → 4.7

Time: 24.2s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.6023650830861039 \cdot 10^{-202}:\\ \;\;\;\;\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(\left(j \cdot 27\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}\right)\\ \mathbf{elif}\;t \le 1.24034573106726284 \cdot 10^{-70}:\\ \;\;\;\;\left(t \cdot \left(-a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r1331303 = x;
        double r1331304 = 18.0;
        double r1331305 = r1331303 * r1331304;
        double r1331306 = y;
        double r1331307 = r1331305 * r1331306;
        double r1331308 = z;
        double r1331309 = r1331307 * r1331308;
        double r1331310 = t;
        double r1331311 = r1331309 * r1331310;
        double r1331312 = a;
        double r1331313 = 4.0;
        double r1331314 = r1331312 * r1331313;
        double r1331315 = r1331314 * r1331310;
        double r1331316 = r1331311 - r1331315;
        double r1331317 = b;
        double r1331318 = c;
        double r1331319 = r1331317 * r1331318;
        double r1331320 = r1331316 + r1331319;
        double r1331321 = r1331303 * r1331313;
        double r1331322 = i;
        double r1331323 = r1331321 * r1331322;
        double r1331324 = r1331320 - r1331323;
        double r1331325 = j;
        double r1331326 = 27.0;
        double r1331327 = r1331325 * r1331326;
        double r1331328 = k;
        double r1331329 = r1331327 * r1331328;
        double r1331330 = r1331324 - r1331329;
        return r1331330;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r1331331 = t;
        double r1331332 = -2.602365083086104e-202;
        bool r1331333 = r1331331 <= r1331332;
        double r1331334 = x;
        double r1331335 = 18.0;
        double r1331336 = r1331334 * r1331335;
        double r1331337 = y;
        double r1331338 = r1331336 * r1331337;
        double r1331339 = z;
        double r1331340 = r1331338 * r1331339;
        double r1331341 = a;
        double r1331342 = 4.0;
        double r1331343 = r1331341 * r1331342;
        double r1331344 = r1331340 - r1331343;
        double r1331345 = r1331331 * r1331344;
        double r1331346 = b;
        double r1331347 = c;
        double r1331348 = r1331346 * r1331347;
        double r1331349 = r1331345 + r1331348;
        double r1331350 = r1331334 * r1331342;
        double r1331351 = i;
        double r1331352 = r1331350 * r1331351;
        double r1331353 = j;
        double r1331354 = 27.0;
        double r1331355 = r1331353 * r1331354;
        double r1331356 = k;
        double r1331357 = cbrt(r1331356);
        double r1331358 = r1331357 * r1331357;
        double r1331359 = r1331355 * r1331358;
        double r1331360 = r1331359 * r1331357;
        double r1331361 = r1331352 + r1331360;
        double r1331362 = r1331349 - r1331361;
        double r1331363 = 1.2403457310672628e-70;
        bool r1331364 = r1331331 <= r1331363;
        double r1331365 = -r1331343;
        double r1331366 = r1331331 * r1331365;
        double r1331367 = r1331366 + r1331348;
        double r1331368 = r1331354 * r1331356;
        double r1331369 = r1331353 * r1331368;
        double r1331370 = r1331352 + r1331369;
        double r1331371 = r1331367 - r1331370;
        double r1331372 = r1331337 * r1331339;
        double r1331373 = r1331336 * r1331372;
        double r1331374 = r1331373 - r1331343;
        double r1331375 = r1331331 * r1331374;
        double r1331376 = r1331375 + r1331348;
        double r1331377 = r1331376 - r1331370;
        double r1331378 = r1331364 ? r1331371 : r1331377;
        double r1331379 = r1331333 ? r1331362 : r1331378;
        return r1331379;
}

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	5.2
Target	1.5
Herbie	4.7

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

Derivation

1. Split input into 3 regimes

2. if t < -2.602365083086104e-202

   1. Initial program 4.2

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

   2. Simplified 4.2

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

   4. Applied add-cube-cbrt 4.4

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

   5. Applied associate-*r* 4.4

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

   if -2.602365083086104e-202 < t < 1.2403457310672628e-70

   1. Initial program 8.3

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

   2. Simplified 8.3

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

   4. Applied associate-*l* 8.4

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

   5. Taylor expanded around 0 6.0

      \[\leadsto \left(t \cdot \left(\color{blue}{0} - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\]

   if 1.2403457310672628e-70 < t

   1. Initial program 2.3

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

   2. Simplified 2.3

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

   4. Applied associate-*l* 2.3

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

   6. Applied associate-*l* 3.5

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

4. Final simplification 4.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.6023650830861039 \cdot 10^{-202}:\\ \;\;\;\;\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(\left(j \cdot 27\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}\right)\\ \mathbf{elif}\;t \le 1.24034573106726284 \cdot 10^{-70}:\\ \;\;\;\;\left(t \cdot \left(-a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))