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

Average Error: 5.8 → 4.7

Time: 7.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -6.7829313514797449 \cdot 10^{-133} \lor \neg \left(t \le 4.444074167892446 \cdot 10^{-95}\right):\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(0 - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r1278486 = x;
        double r1278487 = 18.0;
        double r1278488 = r1278486 * r1278487;
        double r1278489 = y;
        double r1278490 = r1278488 * r1278489;
        double r1278491 = z;
        double r1278492 = r1278490 * r1278491;
        double r1278493 = t;
        double r1278494 = r1278492 * r1278493;
        double r1278495 = a;
        double r1278496 = 4.0;
        double r1278497 = r1278495 * r1278496;
        double r1278498 = r1278497 * r1278493;
        double r1278499 = r1278494 - r1278498;
        double r1278500 = b;
        double r1278501 = c;
        double r1278502 = r1278500 * r1278501;
        double r1278503 = r1278499 + r1278502;
        double r1278504 = r1278486 * r1278496;
        double r1278505 = i;
        double r1278506 = r1278504 * r1278505;
        double r1278507 = r1278503 - r1278506;
        double r1278508 = j;
        double r1278509 = 27.0;
        double r1278510 = r1278508 * r1278509;
        double r1278511 = k;
        double r1278512 = r1278510 * r1278511;
        double r1278513 = r1278507 - r1278512;
        return r1278513;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r1278514 = t;
        double r1278515 = -6.782931351479745e-133;
        bool r1278516 = r1278514 <= r1278515;
        double r1278517 = 4.4440741678924465e-95;
        bool r1278518 = r1278514 <= r1278517;
        double r1278519 = !r1278518;
        bool r1278520 = r1278516 || r1278519;
        double r1278521 = x;
        double r1278522 = 18.0;
        double r1278523 = y;
        double r1278524 = r1278522 * r1278523;
        double r1278525 = r1278521 * r1278524;
        double r1278526 = z;
        double r1278527 = r1278525 * r1278526;
        double r1278528 = a;
        double r1278529 = 4.0;
        double r1278530 = r1278528 * r1278529;
        double r1278531 = r1278527 - r1278530;
        double r1278532 = r1278514 * r1278531;
        double r1278533 = b;
        double r1278534 = c;
        double r1278535 = r1278533 * r1278534;
        double r1278536 = r1278521 * r1278529;
        double r1278537 = i;
        double r1278538 = r1278536 * r1278537;
        double r1278539 = j;
        double r1278540 = 27.0;
        double r1278541 = r1278539 * r1278540;
        double r1278542 = k;
        double r1278543 = r1278541 * r1278542;
        double r1278544 = r1278538 + r1278543;
        double r1278545 = r1278535 - r1278544;
        double r1278546 = r1278532 + r1278545;
        double r1278547 = 0.0;
        double r1278548 = r1278547 - r1278530;
        double r1278549 = r1278514 * r1278548;
        double r1278550 = r1278540 * r1278542;
        double r1278551 = r1278539 * r1278550;
        double r1278552 = r1278538 + r1278551;
        double r1278553 = r1278535 - r1278552;
        double r1278554 = r1278549 + r1278553;
        double r1278555 = r1278520 ? r1278546 : r1278554;
        return r1278555;
}

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	5.8
Target	1.7
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 2 regimes

2. if t < -6.782931351479745e-133 or 4.4440741678924465e-95 < t

   1. Initial program 3.5

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

   2. Simplified 3.5

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

   4. Applied associate-*l* 3.5

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

   if -6.782931351479745e-133 < t < 4.4440741678924465e-95

   1. Initial program 9.0

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

   2. Simplified 9.0

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

   4. Applied associate-*l* 9.1

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

   5. Taylor expanded around 0 6.5

      \[\leadsto t \cdot \left(\color{blue}{0} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.7829313514797449 \cdot 10^{-133} \lor \neg \left(t \le 4.444074167892446 \cdot 10^{-95}\right):\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(0 - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(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)))