Average Error: 0.0 → 0.0
Time: 9.6s
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 r78671 = x;
        double r78672 = y;
        double r78673 = r78671 * r78672;
        double r78674 = z;
        double r78675 = t;
        double r78676 = r78674 * r78675;
        double r78677 = r78673 + r78676;
        double r78678 = a;
        double r78679 = b;
        double r78680 = r78678 * r78679;
        double r78681 = r78677 + r78680;
        double r78682 = c;
        double r78683 = i;
        double r78684 = r78682 * r78683;
        double r78685 = r78681 + r78684;
        return r78685;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r78686 = c;
        double r78687 = i;
        double r78688 = t;
        double r78689 = z;
        double r78690 = a;
        double r78691 = b;
        double r78692 = x;
        double r78693 = y;
        double r78694 = r78692 * r78693;
        double r78695 = fma(r78690, r78691, r78694);
        double r78696 = fma(r78688, r78689, r78695);
        double r78697 = fma(r78686, r78687, r78696);
        return r78697;
}

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)))