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

Average Error: 5.1 → 2.8

Time: 43.7s

Precision: 64

Math

\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k \le -2.582625430953297 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4.0\right) \cdot a, t, c \cdot b - \mathsf{fma}\left(27.0, k \cdot j, \left(x \cdot 4.0\right) \cdot i\right)\right)\\ \mathbf{elif}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k \le 1.9705971110438173 \cdot 10^{+295}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4.0\right) \cdot a, t, c \cdot b - \mathsf{fma}\left(27.0, k \cdot j, \left(x \cdot 4.0\right) \cdot i\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r5092530 = x;
        double r5092531 = 18.0;
        double r5092532 = r5092530 * r5092531;
        double r5092533 = y;
        double r5092534 = r5092532 * r5092533;
        double r5092535 = z;
        double r5092536 = r5092534 * r5092535;
        double r5092537 = t;
        double r5092538 = r5092536 * r5092537;
        double r5092539 = a;
        double r5092540 = 4.0;
        double r5092541 = r5092539 * r5092540;
        double r5092542 = r5092541 * r5092537;
        double r5092543 = r5092538 - r5092542;
        double r5092544 = b;
        double r5092545 = c;
        double r5092546 = r5092544 * r5092545;
        double r5092547 = r5092543 + r5092546;
        double r5092548 = r5092530 * r5092540;
        double r5092549 = i;
        double r5092550 = r5092548 * r5092549;
        double r5092551 = r5092547 - r5092550;
        double r5092552 = j;
        double r5092553 = 27.0;
        double r5092554 = r5092552 * r5092553;
        double r5092555 = k;
        double r5092556 = r5092554 * r5092555;
        double r5092557 = r5092551 - r5092556;
        return r5092557;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r5092558 = t;
        double r5092559 = x;
        double r5092560 = 18.0;
        double r5092561 = r5092559 * r5092560;
        double r5092562 = y;
        double r5092563 = r5092561 * r5092562;
        double r5092564 = z;
        double r5092565 = r5092563 * r5092564;
        double r5092566 = r5092558 * r5092565;
        double r5092567 = a;
        double r5092568 = 4.0;
        double r5092569 = r5092567 * r5092568;
        double r5092570 = r5092569 * r5092558;
        double r5092571 = r5092566 - r5092570;
        double r5092572 = c;
        double r5092573 = b;
        double r5092574 = r5092572 * r5092573;
        double r5092575 = r5092571 + r5092574;
        double r5092576 = r5092559 * r5092568;
        double r5092577 = i;
        double r5092578 = r5092576 * r5092577;
        double r5092579 = r5092575 - r5092578;
        double r5092580 = 27.0;
        double r5092581 = j;
        double r5092582 = r5092580 * r5092581;
        double r5092583 = k;
        double r5092584 = r5092582 * r5092583;
        double r5092585 = r5092579 - r5092584;
        double r5092586 = -2.582625430953297e+307;
        bool r5092587 = r5092585 <= r5092586;
        double r5092588 = -r5092568;
        double r5092589 = r5092588 * r5092567;
        double r5092590 = r5092583 * r5092581;
        double r5092591 = fma(r5092580, r5092590, r5092578);
        double r5092592 = r5092574 - r5092591;
        double r5092593 = fma(r5092589, r5092558, r5092592);
        double r5092594 = 1.9705971110438173e+295;
        bool r5092595 = r5092585 <= r5092594;
        double r5092596 = r5092595 ? r5092585 : r5092593;
        double r5092597 = r5092587 ? r5092593 : r5092596;
        return r5092597;
}

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

Derivation

1. Split input into 2 regimes

2. if (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)) < -2.582625430953297e+307 or 1.9705971110438173e+295 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k))

   1. Initial program 45.8

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

   2. Simplified 30.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot z\right) \cdot \left(x \cdot 18.0\right) - a \cdot 4.0, t, c \cdot b - \mathsf{fma}\left(27.0, k \cdot j, \left(x \cdot 4.0\right) \cdot i\right)\right)}\]

   3. Taylor expanded around 0 24.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{0} - a \cdot 4.0, t, c \cdot b - \mathsf{fma}\left(27.0, k \cdot j, \left(x \cdot 4.0\right) \cdot i\right)\right)\]

   if -2.582625430953297e+307 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)) < 1.9705971110438173e+295

   1. Initial program 0.2

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k \le -2.582625430953297 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4.0\right) \cdot a, t, c \cdot b - \mathsf{fma}\left(27.0, k \cdot j, \left(x \cdot 4.0\right) \cdot i\right)\right)\\ \mathbf{elif}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k \le 1.9705971110438173 \cdot 10^{+295}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-4.0\right) \cdot a, t, c \cdot b - \mathsf{fma}\left(27.0, k \cdot j, \left(x \cdot 4.0\right) \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))