Average Error: 0.1 → 0.0
Time: 4.9m
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 r248457 = x;
        double r248458 = y;
        double r248459 = r248457 * r248458;
        double r248460 = z;
        double r248461 = t;
        double r248462 = r248460 * r248461;
        double r248463 = 16.0;
        double r248464 = r248462 / r248463;
        double r248465 = r248459 + r248464;
        double r248466 = a;
        double r248467 = b;
        double r248468 = r248466 * r248467;
        double r248469 = 4.0;
        double r248470 = r248468 / r248469;
        double r248471 = r248465 - r248470;
        double r248472 = c;
        double r248473 = r248471 + r248472;
        return r248473;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r248474 = z;
        double r248475 = t;
        double r248476 = 16.0;
        double r248477 = r248475 / r248476;
        double r248478 = y;
        double r248479 = x;
        double r248480 = a;
        double r248481 = 4.0;
        double r248482 = r248480 / r248481;
        double r248483 = -r248482;
        double r248484 = b;
        double r248485 = c;
        double r248486 = fma(r248483, r248484, r248485);
        double r248487 = fma(r248478, r248479, r248486);
        double r248488 = fma(r248474, r248477, r248487);
        return r248488;
}

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 2020034 +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))