Average Error: 0.0 → 0.0
Time: 10.2s
Precision: 64
\[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i\]
↓
\[\mathsf{fma}\left(c, i, \mathsf{fma}\left(t, z, \mathsf{fma}\left(a, b, x \cdot y\right)\right)\right)\]
\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i
↓
\mathsf{fma}\left(c, i, \mathsf{fma}\left(t, z, \mathsf{fma}\left(a, b, x \cdot y\right)\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r66711 = x;
        double r66712 = y;
        double r66713 = r66711 * r66712;
        double r66714 = z;
        double r66715 = t;
        double r66716 = r66714 * r66715;
        double r66717 = r66713 + r66716;
        double r66718 = a;
        double r66719 = b;
        double r66720 = r66718 * r66719;
        double r66721 = r66717 + r66720;
        double r66722 = c;
        double r66723 = i;
        double r66724 = r66722 * r66723;
        double r66725 = r66721 + r66724;
        return r66725;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r66726 = c;
        double r66727 = i;
        double r66728 = t;
        double r66729 = z;
        double r66730 = a;
        double r66731 = b;
        double r66732 = x;
        double r66733 = y;
        double r66734 = r66732 * r66733;
        double r66735 = fma(r66730, r66731, r66734);
        double r66736 = fma(r66728, r66729, r66735);
        double r66737 = fma(r66726, r66727, r66736);
        return r66737;
}

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

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

  3. Taylor expanded around inf 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +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)))