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

Average Error: 5.6 → 5.3

Time: 20.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.814761326386550777279758521488119181836 \cdot 10^{156}:\\ \;\;\;\;b \cdot c - \left(x \cdot \left(4 \cdot i\right) + j \cdot \left(27 \cdot k\right)\right)\\ \mathbf{elif}\;x \le 2.490620401820025423975805319912306702311 \cdot 10^{-84}:\\ \;\;\;\;\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\sqrt[3]{t} \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right)\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) + t \cdot \left(-a \cdot 4\right)\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r547348 = x;
        double r547349 = 18.0;
        double r547350 = r547348 * r547349;
        double r547351 = y;
        double r547352 = r547350 * r547351;
        double r547353 = z;
        double r547354 = r547352 * r547353;
        double r547355 = t;
        double r547356 = r547354 * r547355;
        double r547357 = a;
        double r547358 = 4.0;
        double r547359 = r547357 * r547358;
        double r547360 = r547359 * r547355;
        double r547361 = r547356 - r547360;
        double r547362 = b;
        double r547363 = c;
        double r547364 = r547362 * r547363;
        double r547365 = r547361 + r547364;
        double r547366 = r547348 * r547358;
        double r547367 = i;
        double r547368 = r547366 * r547367;
        double r547369 = r547365 - r547368;
        double r547370 = j;
        double r547371 = 27.0;
        double r547372 = r547370 * r547371;
        double r547373 = k;
        double r547374 = r547372 * r547373;
        double r547375 = r547369 - r547374;
        return r547375;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r547376 = x;
        double r547377 = -2.8147613263865508e+156;
        bool r547378 = r547376 <= r547377;
        double r547379 = b;
        double r547380 = c;
        double r547381 = r547379 * r547380;
        double r547382 = 4.0;
        double r547383 = i;
        double r547384 = r547382 * r547383;
        double r547385 = r547376 * r547384;
        double r547386 = j;
        double r547387 = 27.0;
        double r547388 = k;
        double r547389 = r547387 * r547388;
        double r547390 = r547386 * r547389;
        double r547391 = r547385 + r547390;
        double r547392 = r547381 - r547391;
        double r547393 = 2.4906204018200254e-84;
        bool r547394 = r547376 <= r547393;
        double r547395 = t;
        double r547396 = cbrt(r547395);
        double r547397 = r547396 * r547396;
        double r547398 = 18.0;
        double r547399 = r547376 * r547398;
        double r547400 = y;
        double r547401 = r547399 * r547400;
        double r547402 = z;
        double r547403 = r547401 * r547402;
        double r547404 = a;
        double r547405 = r547404 * r547382;
        double r547406 = r547403 - r547405;
        double r547407 = r547396 * r547406;
        double r547408 = r547397 * r547407;
        double r547409 = r547408 + r547381;
        double r547410 = r547376 * r547382;
        double r547411 = r547410 * r547383;
        double r547412 = r547386 * r547387;
        double r547413 = r547412 * r547388;
        double r547414 = r547411 + r547413;
        double r547415 = r547409 - r547414;
        double r547416 = r547402 * r547400;
        double r547417 = r547376 * r547416;
        double r547418 = r547395 * r547417;
        double r547419 = r547398 * r547418;
        double r547420 = -r547405;
        double r547421 = r547395 * r547420;
        double r547422 = r547419 + r547421;
        double r547423 = r547422 + r547381;
        double r547424 = r547423 - r547414;
        double r547425 = r547394 ? r547415 : r547424;
        double r547426 = r547378 ? r547392 : r547425;
        return r547426;
}

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	5.3

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

Derivation

1. Split input into 3 regimes

2. if x < -2.8147613263865508e+156

   1. Initial program 19.9

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

   2. Simplified 19.9

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

   4. Applied associate-*l* 19.9

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

   6. Applied associate-*l* 19.8

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

   8. Applied associate-*l* 19.7

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

   9. Taylor expanded around 0 22.6

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

   if -2.8147613263865508e+156 < x < 2.4906204018200254e-84

   1. Initial program 3.0

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

   2. Simplified 3.0

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

   4. Applied add-cube-cbrt 3.4

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

   5. Applied associate-*l* 3.4

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

   if 2.4906204018200254e-84 < x

   1. Initial program 8.8

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

   2. Simplified 8.8

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

   4. Applied sub-neg 8.8

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

   5. Applied distribute-lft-in 8.8

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

   6. Simplified 6.0

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

4. Final simplification 5.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.814761326386550777279758521488119181836 \cdot 10^{156}:\\ \;\;\;\;b \cdot c - \left(x \cdot \left(4 \cdot i\right) + j \cdot \left(27 \cdot k\right)\right)\\ \mathbf{elif}\;x \le 2.490620401820025423975805319912306702311 \cdot 10^{-84}:\\ \;\;\;\;\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\sqrt[3]{t} \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right)\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) + t \cdot \left(-a \cdot 4\right)\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(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)))