Average Error: 0.1 → 0.0
Time: 3.8s
Precision: 64
\[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c\]
↓
\[\mathsf{fma}\left(z, \frac{t}{16}, \mathsf{fma}\left(y, x, \mathsf{fma}\left(-\frac{a}{4}, b, c\right)\right)\right)\]
\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c
↓
\mathsf{fma}\left(z, \frac{t}{16}, \mathsf{fma}\left(y, x, \mathsf{fma}\left(-\frac{a}{4}, b, c\right)\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r245445 = x;
        double r245446 = y;
        double r245447 = r245445 * r245446;
        double r245448 = z;
        double r245449 = t;
        double r245450 = r245448 * r245449;
        double r245451 = 16.0;
        double r245452 = r245450 / r245451;
        double r245453 = r245447 + r245452;
        double r245454 = a;
        double r245455 = b;
        double r245456 = r245454 * r245455;
        double r245457 = 4.0;
        double r245458 = r245456 / r245457;
        double r245459 = r245453 - r245458;
        double r245460 = c;
        double r245461 = r245459 + r245460;
        return r245461;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r245462 = z;
        double r245463 = t;
        double r245464 = 16.0;
        double r245465 = r245463 / r245464;
        double r245466 = y;
        double r245467 = x;
        double r245468 = a;
        double r245469 = 4.0;
        double r245470 = r245468 / r245469;
        double r245471 = -r245470;
        double r245472 = b;
        double r245473 = c;
        double r245474 = fma(r245471, r245472, r245473);
        double r245475 = fma(r245466, r245467, r245474);
        double r245476 = fma(r245462, r245465, r245475);
        return r245476;
}

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

Derivation

  1. Initial program 0.1

    \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{t}{16}, \mathsf{fma}\left(y, x, \mathsf{fma}\left(-\frac{a}{4}, b, c\right)\right)\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(z, \frac{t}{16}, \mathsf{fma}\left(y, x, \mathsf{fma}\left(-\frac{a}{4}, b, c\right)\right)\right)\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, C"
  :precision binary64
  (+ (- (+ (* x y) (/ (* z t) 16)) (/ (* a b) 4)) c))