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

Average Error: 5.4 → 1.4

Time: 8.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -138023816.617849797 \lor \neg \left(y \le 4.51721422379651299 \cdot 10^{34}\right):\\ \;\;\;\;\left(\left(t \cdot \left(18 \cdot \left(x \cdot z\right)\right)\right) \cdot y + t \cdot \left(-a \cdot 4\right)\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left({\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r697479 = x;
        double r697480 = 18.0;
        double r697481 = r697479 * r697480;
        double r697482 = y;
        double r697483 = r697481 * r697482;
        double r697484 = z;
        double r697485 = r697483 * r697484;
        double r697486 = t;
        double r697487 = r697485 * r697486;
        double r697488 = a;
        double r697489 = 4.0;
        double r697490 = r697488 * r697489;
        double r697491 = r697490 * r697486;
        double r697492 = r697487 - r697491;
        double r697493 = b;
        double r697494 = c;
        double r697495 = r697493 * r697494;
        double r697496 = r697492 + r697495;
        double r697497 = r697479 * r697489;
        double r697498 = i;
        double r697499 = r697497 * r697498;
        double r697500 = r697496 - r697499;
        double r697501 = j;
        double r697502 = 27.0;
        double r697503 = r697501 * r697502;
        double r697504 = k;
        double r697505 = r697503 * r697504;
        double r697506 = r697500 - r697505;
        return r697506;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r697507 = y;
        double r697508 = -138023816.6178498;
        bool r697509 = r697507 <= r697508;
        double r697510 = 4.517214223796513e+34;
        bool r697511 = r697507 <= r697510;
        double r697512 = !r697511;
        bool r697513 = r697509 || r697512;
        double r697514 = t;
        double r697515 = 18.0;
        double r697516 = x;
        double r697517 = z;
        double r697518 = r697516 * r697517;
        double r697519 = r697515 * r697518;
        double r697520 = r697514 * r697519;
        double r697521 = r697520 * r697507;
        double r697522 = a;
        double r697523 = 4.0;
        double r697524 = r697522 * r697523;
        double r697525 = -r697524;
        double r697526 = r697514 * r697525;
        double r697527 = r697521 + r697526;
        double r697528 = b;
        double r697529 = c;
        double r697530 = r697528 * r697529;
        double r697531 = r697516 * r697523;
        double r697532 = i;
        double r697533 = r697531 * r697532;
        double r697534 = j;
        double r697535 = 27.0;
        double r697536 = r697534 * r697535;
        double r697537 = k;
        double r697538 = r697536 * r697537;
        double r697539 = r697533 + r697538;
        double r697540 = r697530 - r697539;
        double r697541 = r697527 + r697540;
        double r697542 = r697517 * r697507;
        double r697543 = r697516 * r697542;
        double r697544 = r697515 * r697543;
        double r697545 = 1.0;
        double r697546 = pow(r697544, r697545);
        double r697547 = r697546 - r697524;
        double r697548 = r697514 * r697547;
        double r697549 = r697535 * r697537;
        double r697550 = r697534 * r697549;
        double r697551 = r697533 + r697550;
        double r697552 = r697530 - r697551;
        double r697553 = r697548 + r697552;
        double r697554 = r697513 ? r697541 : r697553;
        return r697554;
}

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	5.4
Target	1.6
Herbie	1.4

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

Derivation

1. Split input into 2 regimes

2. if y < -138023816.6178498 or 4.517214223796513e+34 < y

   1. Initial program 11.2

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

   2. Simplified 11.2

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

   4. Applied pow1 11.2

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

   5. Applied pow1 11.2

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

   6. Applied pow1 11.2

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

   7. Applied pow1 11.2

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

   8. Applied pow-prod-down 11.2

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

   9. Applied pow-prod-down 11.2

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

   10. Applied pow-prod-down 11.2

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

   11. Simplified 13.4

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

   13. Applied associate-*r* 6.9

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

   15. Applied associate-*r* 7.0

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

   17. Applied sub-neg 7.0

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

   18. Applied distribute-lft-in 7.0

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

   19. Simplified 1.5

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

   if -138023816.6178498 < y < 4.517214223796513e+34

   1. Initial program 1.4

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

   2. Simplified 1.4

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

   4. Applied pow1 1.4

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

   5. Applied pow1 1.4

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

   6. Applied pow1 1.4

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

   7. Applied pow1 1.4

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

   8. Applied pow-prod-down 1.4

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

   9. Applied pow-prod-down 1.4

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

   10. Applied pow-prod-down 1.4

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

   11. Simplified 1.4

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

   13. Applied associate-*l* 1.3

       \[\leadsto t \cdot \left({\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \color{blue}{j \cdot \left(27 \cdot k\right)}\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -138023816.617849797 \lor \neg \left(y \le 4.51721422379651299 \cdot 10^{34}\right):\\ \;\;\;\;\left(\left(t \cdot \left(18 \cdot \left(x \cdot z\right)\right)\right) \cdot y + t \cdot \left(-a \cdot 4\right)\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left({\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}^{1} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(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)))