Average Error: 0.1 → 0.0
Time: 2.9s
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 r263462 = x;
        double r263463 = y;
        double r263464 = r263462 * r263463;
        double r263465 = z;
        double r263466 = t;
        double r263467 = r263465 * r263466;
        double r263468 = 16.0;
        double r263469 = r263467 / r263468;
        double r263470 = r263464 + r263469;
        double r263471 = a;
        double r263472 = b;
        double r263473 = r263471 * r263472;
        double r263474 = 4.0;
        double r263475 = r263473 / r263474;
        double r263476 = r263470 - r263475;
        double r263477 = c;
        double r263478 = r263476 + r263477;
        return r263478;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r263479 = z;
        double r263480 = t;
        double r263481 = 16.0;
        double r263482 = r263480 / r263481;
        double r263483 = y;
        double r263484 = x;
        double r263485 = a;
        double r263486 = 4.0;
        double r263487 = r263485 / r263486;
        double r263488 = -r263487;
        double r263489 = b;
        double r263490 = c;
        double r263491 = fma(r263488, r263489, r263490);
        double r263492 = fma(r263483, r263484, r263491);
        double r263493 = fma(r263479, r263482, r263492);
        return r263493;
}

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 2020047 +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))