Average Error: 0.0 → 0.0
Time: 4.0s
Precision: 64
\[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i\]
↓
\[\mathsf{fma}\left(i, c, \mathsf{fma}\left(b, a, \mathsf{fma}\left(x, y, z \cdot t\right)\right)\right)\]
\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i
↓
\mathsf{fma}\left(i, c, \mathsf{fma}\left(b, a, \mathsf{fma}\left(x, y, z \cdot t\right)\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r72523 = x;
        double r72524 = y;
        double r72525 = r72523 * r72524;
        double r72526 = z;
        double r72527 = t;
        double r72528 = r72526 * r72527;
        double r72529 = r72525 + r72528;
        double r72530 = a;
        double r72531 = b;
        double r72532 = r72530 * r72531;
        double r72533 = r72529 + r72532;
        double r72534 = c;
        double r72535 = i;
        double r72536 = r72534 * r72535;
        double r72537 = r72533 + r72536;
        return r72537;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r72538 = i;
        double r72539 = c;
        double r72540 = b;
        double r72541 = a;
        double r72542 = x;
        double r72543 = y;
        double r72544 = z;
        double r72545 = t;
        double r72546 = r72544 * r72545;
        double r72547 = fma(r72542, r72543, r72546);
        double r72548 = fma(r72540, r72541, r72547);
        double r72549 = fma(r72538, r72539, r72548);
        return r72549;
}

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

Derivation

  1. Initial program 0.0

    \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i\]

  2. Simplified0.0

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

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(i, c, \mathsf{fma}\left(b, a, \mathsf{fma}\left(x, y, z \cdot t\right)\right)\right)\]

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Linear.V4:$cdot from linear-1.19.1.3"
  :precision binary64
  (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))