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

Average Error: 5.6 → 4.3

Time: 19.0s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.2028576290406759 \cdot 10^{164} \lor \neg \left(t \le 3.8698264799029388 \cdot 10^{-82}\right):\\ \;\;\;\;\left(\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot \left(y \cdot 18\right)\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r509406 = x;
        double r509407 = 18.0;
        double r509408 = r509406 * r509407;
        double r509409 = y;
        double r509410 = r509408 * r509409;
        double r509411 = z;
        double r509412 = r509410 * r509411;
        double r509413 = t;
        double r509414 = r509412 * r509413;
        double r509415 = a;
        double r509416 = 4.0;
        double r509417 = r509415 * r509416;
        double r509418 = r509417 * r509413;
        double r509419 = r509414 - r509418;
        double r509420 = b;
        double r509421 = c;
        double r509422 = r509420 * r509421;
        double r509423 = r509419 + r509422;
        double r509424 = r509406 * r509416;
        double r509425 = i;
        double r509426 = r509424 * r509425;
        double r509427 = r509423 - r509426;
        double r509428 = j;
        double r509429 = 27.0;
        double r509430 = r509428 * r509429;
        double r509431 = k;
        double r509432 = r509430 * r509431;
        double r509433 = r509427 - r509432;
        return r509433;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r509434 = t;
        double r509435 = -1.202857629040676e+164;
        bool r509436 = r509434 <= r509435;
        double r509437 = 3.869826479902939e-82;
        bool r509438 = r509434 <= r509437;
        double r509439 = !r509438;
        bool r509440 = r509436 || r509439;
        double r509441 = b;
        double r509442 = c;
        double r509443 = r509441 * r509442;
        double r509444 = 18.0;
        double r509445 = x;
        double r509446 = z;
        double r509447 = y;
        double r509448 = r509446 * r509447;
        double r509449 = r509445 * r509448;
        double r509450 = r509434 * r509449;
        double r509451 = r509444 * r509450;
        double r509452 = a;
        double r509453 = 4.0;
        double r509454 = r509452 * r509453;
        double r509455 = r509454 * r509434;
        double r509456 = r509451 - r509455;
        double r509457 = r509443 + r509456;
        double r509458 = r509445 * r509453;
        double r509459 = i;
        double r509460 = r509458 * r509459;
        double r509461 = r509457 - r509460;
        double r509462 = j;
        double r509463 = 27.0;
        double r509464 = k;
        double r509465 = r509463 * r509464;
        double r509466 = r509462 * r509465;
        double r509467 = r509461 - r509466;
        double r509468 = r509447 * r509444;
        double r509469 = r509445 * r509468;
        double r509470 = r509446 * r509434;
        double r509471 = r509469 * r509470;
        double r509472 = r509471 - r509455;
        double r509473 = r509472 + r509443;
        double r509474 = r509473 - r509460;
        double r509475 = r509474 - r509466;
        double r509476 = r509440 ? r509467 : r509475;
        return r509476;
}

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	4.3

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

Derivation

1. Split input into 2 regimes

2. if t < -1.202857629040676e+164 or 3.869826479902939e-82 < 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. Using strategy rm

   3. Applied associate-*l* 2.4

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

   5. Applied pow1 2.4

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot \color{blue}{{t}^{1}} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]

   6. Applied pow1 2.4

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \color{blue}{{z}^{1}}\right) \cdot {t}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]

   7. Applied pow1 2.4

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot \color{blue}{{y}^{1}}\right) \cdot {z}^{1}\right) \cdot {t}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]

   8. Applied pow1 2.4

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot \color{blue}{{18}^{1}}\right) \cdot {y}^{1}\right) \cdot {z}^{1}\right) \cdot {t}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]

   9. Applied pow1 2.4

      \[\leadsto \left(\left(\left(\left(\left(\left(\color{blue}{{x}^{1}} \cdot {18}^{1}\right) \cdot {y}^{1}\right) \cdot {z}^{1}\right) \cdot {t}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]

   10. Applied pow-prod-down 2.4

       \[\leadsto \left(\left(\left(\left(\left(\color{blue}{{\left(x \cdot 18\right)}^{1}} \cdot {y}^{1}\right) \cdot {z}^{1}\right) \cdot {t}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]

   11. Applied pow-prod-down 2.4

       \[\leadsto \left(\left(\left(\left(\color{blue}{{\left(\left(x \cdot 18\right) \cdot y\right)}^{1}} \cdot {z}^{1}\right) \cdot {t}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]

   12. Applied pow-prod-down 2.4

       \[\leadsto \left(\left(\left(\color{blue}{{\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right)}^{1}} \cdot {t}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]

   13. Applied pow-prod-down 2.4

       \[\leadsto \left(\left(\left(\color{blue}{{\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t\right)}^{1}} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]

   14. Simplified 2.8

       \[\leadsto \left(\left(\left({\color{blue}{\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)\right)}}^{1} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]

   if -1.202857629040676e+164 < t < 3.869826479902939e-82

   1. Initial program 7.1

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

   3. Applied associate-*l* 7.2

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

   5. Applied associate-*l* 7.2

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

   6. Simplified 7.2

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

   8. Applied associate-*l* 4.9

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

4. Final simplification 4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.2028576290406759 \cdot 10^{164} \lor \neg \left(t \le 3.8698264799029388 \cdot 10^{-82}\right):\\ \;\;\;\;\left(\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot \left(y \cdot 18\right)\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +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)))