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

Average Error: 5.8 → 0.8

Time: 54.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k = -\infty:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(z \cdot \left(t \cdot x\right)\right) \cdot \left(y \cdot 18.0\right) - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot 27.0\right) \cdot j\right)\right)\\ \mathbf{elif}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k \le 2.6091300751783966 \cdot 10^{+305}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(z \cdot \left(t \cdot x\right)\right) \cdot \left(y \cdot 18.0\right) - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot 27.0\right) \cdot j\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r33807484 = x;
        double r33807485 = 18.0;
        double r33807486 = r33807484 * r33807485;
        double r33807487 = y;
        double r33807488 = r33807486 * r33807487;
        double r33807489 = z;
        double r33807490 = r33807488 * r33807489;
        double r33807491 = t;
        double r33807492 = r33807490 * r33807491;
        double r33807493 = a;
        double r33807494 = 4.0;
        double r33807495 = r33807493 * r33807494;
        double r33807496 = r33807495 * r33807491;
        double r33807497 = r33807492 - r33807496;
        double r33807498 = b;
        double r33807499 = c;
        double r33807500 = r33807498 * r33807499;
        double r33807501 = r33807497 + r33807500;
        double r33807502 = r33807484 * r33807494;
        double r33807503 = i;
        double r33807504 = r33807502 * r33807503;
        double r33807505 = r33807501 - r33807504;
        double r33807506 = j;
        double r33807507 = 27.0;
        double r33807508 = r33807506 * r33807507;
        double r33807509 = k;
        double r33807510 = r33807508 * r33807509;
        double r33807511 = r33807505 - r33807510;
        return r33807511;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r33807512 = t;
        double r33807513 = x;
        double r33807514 = 18.0;
        double r33807515 = r33807513 * r33807514;
        double r33807516 = y;
        double r33807517 = r33807515 * r33807516;
        double r33807518 = z;
        double r33807519 = r33807517 * r33807518;
        double r33807520 = r33807512 * r33807519;
        double r33807521 = a;
        double r33807522 = 4.0;
        double r33807523 = r33807521 * r33807522;
        double r33807524 = r33807523 * r33807512;
        double r33807525 = r33807520 - r33807524;
        double r33807526 = c;
        double r33807527 = b;
        double r33807528 = r33807526 * r33807527;
        double r33807529 = r33807525 + r33807528;
        double r33807530 = r33807513 * r33807522;
        double r33807531 = i;
        double r33807532 = r33807530 * r33807531;
        double r33807533 = r33807529 - r33807532;
        double r33807534 = 27.0;
        double r33807535 = j;
        double r33807536 = r33807534 * r33807535;
        double r33807537 = k;
        double r33807538 = r33807536 * r33807537;
        double r33807539 = r33807533 - r33807538;
        double r33807540 = -inf.0;
        bool r33807541 = r33807539 <= r33807540;
        double r33807542 = r33807512 * r33807513;
        double r33807543 = r33807518 * r33807542;
        double r33807544 = r33807516 * r33807514;
        double r33807545 = r33807543 * r33807544;
        double r33807546 = r33807513 * r33807531;
        double r33807547 = fma(r33807512, r33807521, r33807546);
        double r33807548 = r33807537 * r33807534;
        double r33807549 = r33807548 * r33807535;
        double r33807550 = fma(r33807522, r33807547, r33807549);
        double r33807551 = r33807545 - r33807550;
        double r33807552 = fma(r33807527, r33807526, r33807551);
        double r33807553 = 2.6091300751783966e+305;
        bool r33807554 = r33807539 <= r33807553;
        double r33807555 = r33807554 ? r33807539 : r33807552;
        double r33807556 = r33807541 ? r33807552 : r33807555;
        return r33807556;
}

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	0.8

\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18.0 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) - \left(\left(k \cdot j\right) \cdot 27.0 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.68027943805222:\\ \;\;\;\;\left(\left(18.0 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) + \left(c \cdot b - 27.0 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18.0 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) - \left(\left(k \cdot j\right) \cdot 27.0 - c \cdot b\right)\\ \end{array}\]

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)) < -inf.0 or 2.6091300751783966e+305 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k))

   1. Initial program 57.7

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

   2. Simplified 12.6

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

   4. Applied associate-*r* 12.6

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

   6. Applied associate-*r* 5.2

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

   if -inf.0 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)) < 2.6091300751783966e+305

   1. Initial program 0.3

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k = -\infty:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(z \cdot \left(t \cdot x\right)\right) \cdot \left(y \cdot 18.0\right) - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot 27.0\right) \cdot j\right)\right)\\ \mathbf{elif}\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k \le 2.6091300751783966 \cdot 10^{+305}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, c, \left(z \cdot \left(t \cdot x\right)\right) \cdot \left(y \cdot 18.0\right) - \mathsf{fma}\left(4.0, \mathsf{fma}\left(t, a, x \cdot i\right), \left(k \cdot 27.0\right) \cdot j\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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"

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18.0 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4.0)) (- (* c b) (* 27.0 (* k j)))) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))